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The Messaging Layer Security (MLS) Protocol
draft-ietf-mls-protocol-14

Document Type Active Internet-Draft (mls WG)
Authors Richard Barnes , Benjamin Beurdouche , Raphael Robert , Jon Millican , Emad Omara , Katriel Cohn-Gordon
Last updated 2022-05-03
Replaces draft-barnes-mls-protocol
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Send notices to benjamin.beurdouche@ens.fr, karthikeyan.bhargavan@inria.fr, cas.cremers@cs.ox.ac.uk, alan@wire.com, singuva@twitter.com, kwonal@mit.edu, ekr@rtfm.com, thyla.van.der@merwe.tech
draft-ietf-mls-protocol-14
Network Working Group                                          R. Barnes
Internet-Draft                                                     Cisco
Intended status: Informational                             B. Beurdouche
Expires: 4 November 2022                                 Inria & Mozilla
                                                               R. Robert
                                                                        
                                                             J. Millican
                                                                Facebook
                                                                E. Omara
                                                                  Google
                                                          K. Cohn-Gordon
                                                    University of Oxford
                                                              3 May 2022

              The Messaging Layer Security (MLS) Protocol
                       draft-ietf-mls-protocol-14

Abstract

   Messaging applications are increasingly making use of end-to-end
   security mechanisms to ensure that messages are only accessible to
   the communicating endpoints, and not to any servers involved in
   delivering messages.  Establishing keys to provide such protections
   is challenging for group chat settings, in which more than two
   clients need to agree on a key but may not be online at the same
   time.  In this document, we specify a key establishment protocol that
   provides efficient asynchronous group key establishment with forward
   secrecy and post-compromise security for groups in size ranging from
   two to thousands.

Discussion Venues

   This note is to be removed before publishing as an RFC.

   Source for this draft and an issue tracker can be found at
   https://github.com/mlswg/mls-protocol (https://github.com/mlswg/mls-
   protocol).

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF).  Note that other groups may also distribute
   working documents as Internet-Drafts.  The list of current Internet-
   Drafts is at https://datatracker.ietf.org/drafts/current/.

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   Internet-Drafts are draft documents valid for a maximum of six months
   and may be updated, replaced, or obsoleted by other documents at any
   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on 4 November 2022.

Copyright Notice

   Copyright (c) 2022 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents (https://trustee.ietf.org/
   license-info) in effect on the date of publication of this document.
   Please review these documents carefully, as they describe your rights
   and restrictions with respect to this document.  Code Components
   extracted from this document must include Revised BSD License text as
   described in Section 4.e of the Trust Legal Provisions and are
   provided without warranty as described in the Revised BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   4
     1.1.  Change Log  . . . . . . . . . . . . . . . . . . . . . . .   5
   2.  Terminology . . . . . . . . . . . . . . . . . . . . . . . . .  11
     2.1.  Presentation Langauge . . . . . . . . . . . . . . . . . .  12
       2.1.1.  Optional Value  . . . . . . . . . . . . . . . . . . .  12
       2.1.2.  Variable-size Vector Headers  . . . . . . . . . . . .  13
   3.  Operating Context . . . . . . . . . . . . . . . . . . . . . .  15
   4.  Protocol Overview . . . . . . . . . . . . . . . . . . . . . .  15
     4.1.  Cryptographic State and Evolution . . . . . . . . . . . .  16
     4.2.  Example Protocol Execution  . . . . . . . . . . . . . . .  18
     4.3.  Relationships Between Epochs  . . . . . . . . . . . . . .  22
   5.  Ratchet Tree Concepts . . . . . . . . . . . . . . . . . . . .  23
     5.1.  Ratchet Tree Terminology  . . . . . . . . . . . . . . . .  24
     5.2.  Views of a Ratchet Tree . . . . . . . . . . . . . . . . .  25
     5.3.  Ratchet Tree Nodes  . . . . . . . . . . . . . . . . . . .  27
   6.  Cryptographic Objects . . . . . . . . . . . . . . . . . . . .  29
     6.1.  Ciphersuites  . . . . . . . . . . . . . . . . . . . . . .  29
     6.2.  Hash-Based Identifiers  . . . . . . . . . . . . . . . . .  30
     6.3.  Credentials . . . . . . . . . . . . . . . . . . . . . . .  31
       6.3.1.  Uniquely Identifying Clients  . . . . . . . . . . . .  32
   7.  Message Framing . . . . . . . . . . . . . . . . . . . . . . .  33
     7.1.  Content Authentication  . . . . . . . . . . . . . . . . .  35
     7.2.  Encoding and Decoding a Plaintext . . . . . . . . . . . .  37
     7.3.  Encoding and Decoding a Ciphertext  . . . . . . . . . . .  37
       7.3.1.  Content Encryption  . . . . . . . . . . . . . . . . .  38

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       7.3.2.  Sender Data Encryption  . . . . . . . . . . . . . . .  39
   8.  Ratchet Tree Operations . . . . . . . . . . . . . . . . . . .  40
     8.1.  Parent Node Contents  . . . . . . . . . . . . . . . . . .  40
     8.2.  Leaf Node Contents  . . . . . . . . . . . . . . . . . . .  41
     8.3.  Leaf Node Validation  . . . . . . . . . . . . . . . . . .  44
     8.4.  Ratchet Tree Evolution  . . . . . . . . . . . . . . . . .  45
     8.5.  Adding and Removing Leaves  . . . . . . . . . . . . . . .  47
     8.6.  Synchronizing Views of the Tree . . . . . . . . . . . . .  49
     8.7.  Tree Hashes . . . . . . . . . . . . . . . . . . . . . . .  51
     8.8.  Parent Hash . . . . . . . . . . . . . . . . . . . . . . .  51
       8.8.1.  Using Parent Hashes . . . . . . . . . . . . . . . . .  54
       8.8.2.  Verifying Parent Hashes . . . . . . . . . . . . . . .  54
     8.9.  Update Paths  . . . . . . . . . . . . . . . . . . . . . .  55
   9.  Key Schedule  . . . . . . . . . . . . . . . . . . . . . . . .  56
     9.1.  Group Context . . . . . . . . . . . . . . . . . . . . . .  58
     9.2.  Transcript Hashes . . . . . . . . . . . . . . . . . . . .  60
     9.3.  External Initialization . . . . . . . . . . . . . . . . .  61
     9.4.  Pre-Shared Keys . . . . . . . . . . . . . . . . . . . . .  61
     9.5.  Exporters . . . . . . . . . . . . . . . . . . . . . . . .  64
     9.6.  Resumption PSK  . . . . . . . . . . . . . . . . . . . . .  65
     9.7.  Epoch Authenticators  . . . . . . . . . . . . . . . . . .  65
   10. Secret Tree . . . . . . . . . . . . . . . . . . . . . . . . .  66
     10.1.  Encryption Keys  . . . . . . . . . . . . . . . . . . . .  66
     10.2.  Deletion Schedule  . . . . . . . . . . . . . . . . . . .  67
   11. Key Packages  . . . . . . . . . . . . . . . . . . . . . . . .  69
     11.1.  KeyPackage Validation  . . . . . . . . . . . . . . . . .  71
     11.2.  KeyPackage Identifiers . . . . . . . . . . . . . . . . .  71
   12. Group Creation  . . . . . . . . . . . . . . . . . . . . . . .  71
     12.1.  Required Capabilities  . . . . . . . . . . . . . . . . .  73
     12.2.  Reinitialization . . . . . . . . . . . . . . . . . . . .  73
     12.3.  Sub-group Branching  . . . . . . . . . . . . . . . . . .  74
   13. Group Evolution . . . . . . . . . . . . . . . . . . . . . . .  75
     13.1.  Proposals  . . . . . . . . . . . . . . . . . . . . . . .  76
       13.1.1.  Add  . . . . . . . . . . . . . . . . . . . . . . . .  76
       13.1.2.  Update . . . . . . . . . . . . . . . . . . . . . . .  77
       13.1.3.  Remove . . . . . . . . . . . . . . . . . . . . . . .  77
       13.1.4.  PreSharedKey . . . . . . . . . . . . . . . . . . . .  78
       13.1.5.  ReInit . . . . . . . . . . . . . . . . . . . . . . .  78
       13.1.6.  ExternalInit . . . . . . . . . . . . . . . . . . . .  79
       13.1.7.  AppAck . . . . . . . . . . . . . . . . . . . . . . .  79
       13.1.8.  GroupContextExtensions . . . . . . . . . . . . . . .  80
       13.1.9.  External Proposals . . . . . . . . . . . . . . . . .  81
     13.2.  Commit . . . . . . . . . . . . . . . . . . . . . . . . .  82
       13.2.1.  Creating a Commit  . . . . . . . . . . . . . . . . .  85
       13.2.2.  Processing a Commit  . . . . . . . . . . . . . . . .  88
       13.2.3.  Adding Members to the Group  . . . . . . . . . . . .  90
     13.3.  Ratchet Tree Extension . . . . . . . . . . . . . . . . .  96
   14. Extensibility . . . . . . . . . . . . . . . . . . . . . . . .  99

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     14.1.  Ciphersuites . . . . . . . . . . . . . . . . . . . . . .  99
     14.2.  Proposals  . . . . . . . . . . . . . . . . . . . . . . . 100
     14.3.  Credential Extensibility . . . . . . . . . . . . . . . . 100
   15. Sequencing of State Changes . . . . . . . . . . . . . . . . . 102
     15.1.  Server-Enforced Ordering . . . . . . . . . . . . . . . . 103
     15.2.  Client-Enforced Ordering . . . . . . . . . . . . . . . . 103
   16. Application Messages  . . . . . . . . . . . . . . . . . . . . 104
     16.1.  Message Encryption and Decryption  . . . . . . . . . . . 104
     16.2.  Restrictions . . . . . . . . . . . . . . . . . . . . . . 105
     16.3.  Delayed and Reordered Application messages . . . . . . . 105
   17. Security Considerations . . . . . . . . . . . . . . . . . . . 105
     17.1.  Confidentiality of the Group Secrets . . . . . . . . . . 106
     17.2.  Authentication . . . . . . . . . . . . . . . . . . . . . 106
     17.3.  Forward Secrecy and Post-Compromise Security . . . . . . 107
     17.4.  KeyPackage Reuse . . . . . . . . . . . . . . . . . . . . 107
     17.5.  Group Fragmentation by Malicious Insiders  . . . . . . . 108
   18. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 108
     18.1.  MLS Ciphersuites . . . . . . . . . . . . . . . . . . . . 109
     18.2.  MLS Extension Types  . . . . . . . . . . . . . . . . . . 113
     18.3.  MLS Proposal Types . . . . . . . . . . . . . . . . . . . 114
     18.4.  MLS Credential Types . . . . . . . . . . . . . . . . . . 115
     18.5.  MLS Designated Expert Pool . . . . . . . . . . . . . . . 116
     18.6.  The "message/mls" MIME Type  . . . . . . . . . . . . . . 117
   19. Contributors  . . . . . . . . . . . . . . . . . . . . . . . . 117
   20. References  . . . . . . . . . . . . . . . . . . . . . . . . . 118
     20.1.  Normative References . . . . . . . . . . . . . . . . . . 118
     20.2.  Informative References . . . . . . . . . . . . . . . . . 119
   Appendix A.  Protocol Origins of Example Trees  . . . . . . . . . 121
   Appendix B.  Array-Based Trees  . . . . . . . . . . . . . . . . . 122
   Appendix C.  Link-Based Trees . . . . . . . . . . . . . . . . . . 126
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . . 128

1.  Introduction

   DISCLAIMER: This is a work-in-progress draft of MLS and has not yet
   seen significant security analysis.  It should not be used as a basis
   for building production systems.

   RFC EDITOR: PLEASE REMOVE THE FOLLOWING PARAGRAPH The source for this
   draft is maintained in GitHub.  Suggested changes should be submitted
   as pull requests at https://github.com/mlswg/mls-protocol.
   Instructions are on that page as well.  Editorial changes can be
   managed in GitHub, but any substantive change should be discussed on
   the MLS mailing list.

   A group of users who want to send each other encrypted messages needs
   a way to derive shared symmetric encryption keys.  For two parties,
   this problem has been studied thoroughly, with the Double Ratchet

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   emerging as a common solution [doubleratchet] [signal].  Channels
   implementing the Double Ratchet enjoy fine-grained forward secrecy as
   well as post-compromise security, but are nonetheless efficient
   enough for heavy use over low-bandwidth networks.

   For a group of size greater than two, a common strategy is to
   unilaterally broadcast symmetric "sender" keys over existing shared
   symmetric channels, and then for each member to send messages to the
   group encrypted with their own sender key.  Unfortunately, while this
   improves efficiency over pairwise broadcast of individual messages
   and provides forward secrecy (with the addition of a hash ratchet),
   it is difficult to achieve post-compromise security with sender keys.
   An adversary who learns a sender key can often indefinitely and
   passively eavesdrop on that member's messages.  Generating and
   distributing a new sender key provides a form of post-compromise
   security with regard to that sender.  However, it requires
   computation and communications resources that scale linearly with the
   size of the group.

   In this document, we describe a protocol based on tree structures
   that enable asynchronous group keying with forward secrecy and post-
   compromise security.  Based on earlier work on "asynchronous
   ratcheting trees" [art], the protocol presented here uses an
   asynchronous key-encapsulation mechanism for tree structures.  This
   mechanism allows the members of the group to derive and update shared
   keys with costs that scale as the log of the group size.

1.1.  Change Log

   RFC EDITOR PLEASE DELETE THIS SECTION.

   draft-13

   *  TLS syntax updates (including variable-header-length vectors) (*)

   *  Stop generating redundant PKE key pairs. (*)

   *  Move validation of identity change to the AS

   *  Add message/mls MIME type registration

   *  Split LeafNode from KeyPackage (*)

   *  Remove endpoint_id (*)

   *  Reorganize to make section layout more sane

   *  Forbid proposals by reference in external commits (*)

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   *  Domain separation for KeyPackage and Proposal references (*)

   *  Downgrade MUST to SHOULD for commit senders including all valid
      commits

   *  Stronger parent hashes for authenticated identities (*)

   *  Move wire_format to a separate tagged-union structure MLSMessage

   *  Generalize tree extend/truncate algorithms

   *  Add algorithms for link-based trees

   *  Forbid self-Update entirely (*)

   *  Consolidate resumption PSK cases (*)

   *  384 Ciphersuite Addition

   *  Remove explicit version pin on HPKE (*)

   *  Remove the requirement for Add in external commit (*)

   *  Use smaller, fixed-size hash-based identifiers (*)

   *  Be explicit that Credentials can attest to multiple identities (*)

   draft-12

   *  Use the GroupContext to derive the joiner_secret (*)

   *  Make PreSharedKeys non optional in GroupSecrets (*)

   *  Update name for this particular key (*)

   *  Truncate tree size on removal (*)

   *  Use HPKE draft-08 (*)

   *  Clarify requirements around identity in MLS groups (*)

   *  Signal the intended wire format for MLS messages (*)

   *  Inject GroupContext as HPKE info instead of AAD (*)

   *  Clarify extension handling and make extension updatable (*)

   *  Improve extensibility of Proposals (*)

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   *  Constrain proposal in External Commit (*)

   *  Remove the notion of a 'leaf index' (*)

   *  Add group_context_extensions proposal ID (*)

   *  Add RequiredCapabilities extension (*)

   *  Use cascaded KDF instead of concatenation to consolidate PSKs (*)

   *  Use key package hash to index clients in message structs (*)

   *  Don't require PublicGroupState for external init (*)

   *  Make ratchet tree section clearer.

   *  Handle non-member sender cases in MLSPlaintextTBS

   *  Clarify encoding of signatures with NIST curves

   *  Remove OPEN ISSUEs and TODOs

   *  Normalize the description of the zero vector

   draft-11

   *  Include subtree keys in parent hash (*)

   *  Pin HPKE to draft-07 (*)

   *  Move joiner secret to the end of the first key schedule epoch (*)

   *  Add an AppAck proposal

   *  Make initializations of transcript hashes consistent

   draft-10

   *  Allow new members to join via an external Commit (*)

   *  Enable proposals to be sent inline in a Commit (*)

   *  Re-enable constant-time Add (*)

   *  Change expiration extension to lifetime extension (*)

   *  Make the tree in the Welcome optional (*)

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   *  PSK injection, re-init, sub-group branching (*)

   *  Require the initial init_secret to be a random value (*)

   *  Remove explicit sender data nonce (*)

   *  Do not encrypt to joiners in UpdatePath generation (*)

   *  Move MLSPlaintext signature under the confirmation tag (*)

   *  Explicitly authenticate group membership with MLSPLaintext (*)

   *  Clarify X509Credential structure (*)

   *  Remove unneeded interim transcript hash from GroupInfo (*)

   *  IANA considerations

   *  Derive an authentication secret

   *  Use Extract/Expand from HPKE KDF

   *  Clarify that application messages MUST be encrypted

   draft-09

   *  Remove blanking of nodes on Add (*)

   *  Change epoch numbers to uint64 (*)

   *  Add PSK inputs (*)

   *  Add key schedule exporter (*)

   *  Sign the updated direct path on Commit, using "parent hashes" and
      one signature per leaf (*)

   *  Use structured types for external senders (*)

   *  Redesign Welcome to include confirmation and use derived keys (*)

   *  Remove ignored proposals (*)

   *  Always include an Update with a Commit (*)

   *  Add per-message entropy to guard against nonce reuse (*)

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   *  Use the same hash ratchet construct for both application and
      handshake keys (*)

   *  Add more ciphersuites

   *  Use HKDF to derive key pairs (*)

   *  Mandate expiration of ClientInitKeys (*)

   *  Add extensions to GroupContext and flesh out the extensibility
      story (*)

   *  Rename ClientInitKey to KeyPackage

   draft-08

   *  Change ClientInitKeys so that they only refer to one ciphersuite
      (*)

   *  Decompose group operations into Proposals and Commits (*)

   *  Enable Add and Remove proposals from outside the group (*)

   *  Replace Init messages with multi-recipient Welcome message (*)

   *  Add extensions to ClientInitKeys for expiration and downgrade
      resistance (*)

   *  Allow multiple Proposals and a single Commit in one MLSPlaintext
      (*)

   draft-07

   *  Initial version of the Tree based Application Key Schedule (*)

   *  Initial definition of the Init message for group creation (*)

   *  Fix issue with the transcript used for newcomers (*)

   *  Clarifications on message framing and HPKE contexts (*)

   draft-06

   *  Reorder blanking and update in the Remove operation (*)

   *  Rename the GroupState structure to GroupContext (*)

   *  Rename UserInitKey to ClientInitKey

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   *  Resolve the circular dependency that draft-05 introduced in the
      confirmation MAC calculation (*)

   *  Cover the entire MLSPlaintext in the transcript hash (*)

   draft-05

   *  Common framing for handshake and application messages (*)

   *  Handshake message encryption (*)

   *  Convert from literal state to a commitment via the "tree hash" (*)

   *  Add credentials to the tree and remove the "roster" concept (*)

   *  Remove the secret field from tree node values

   draft-04

   *  Updating the language to be similar to the Architecture document

   *  ECIES is now renamed in favor of HPKE (*)

   *  Using a KDF instead of a Hash in TreeKEM (*)

   draft-03

   *  Added ciphersuites and signature schemes (*)

   *  Re-ordered fields in UserInitKey to make parsing easier (*)

   *  Fixed inconsistencies between Welcome and GroupState (*)

   *  Added encryption of the Welcome message (*)

   draft-02

   *  Removed ART (*)

   *  Allowed partial trees to avoid double-joins (*)

   *  Added explicit key confirmation (*)

   draft-01

   *  Initial description of the Message Protection mechanism. (*)

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   *  Initial specification proposal for the Application Key Schedule
      using the per-participant chaining of the Application Secret
      design. (*)

   *  Initial specification proposal for an encryption mechanism to
      protect Application Messages using an AEAD scheme. (*)

   *  Initial specification proposal for an authentication mechanism of
      Application Messages using signatures. (*)

   *  Initial specification proposal for a padding mechanism to
      improving protection of Application Messages against traffic
      analysis. (*)

   *  Inversion of the Group Init Add and Application Secret derivations
      in the Handshake Key Schedule to be ease chaining in case we
      switch design. (*)

   *  Removal of the UserAdd construct and split of GroupAdd into Add
      and Welcome messages (*)

   *  Initial proposal for authenticating handshake messages by signing
      over group state and including group state in the key schedule (*)

   *  Added an appendix with example code for tree math

   *  Changed the ECIES mechanism used by TreeKEM so that it uses nonces
      generated from the shared secret

   draft-00

   *  Initial adoption of draft-barnes-mls-protocol-01 as a WG item.

2.  Terminology

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
   "OPTIONAL" in this document are to be interpreted as described in BCP
   14 [RFC2119] [RFC8174] when, and only when, they appear in all
   capitals, as shown here.

   Client:  An agent that uses this protocol to establish shared
      cryptographic state with other clients.  A client is defined by
      the cryptographic keys it holds.

   Group:  A group represents a logical collection of clents that share

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      a common secret value at any given time.  Its state is represented
      as a linear sequence of epochs in which each epoch depends on its
      predecessor.

   Epoch:  A state of a group in which a specific set of authenticated
      clients hold shared cryptographic state.

   Member:  A client that is included in the shared state of a group,
      hence has access to the group's secrets.

   Key Package:  A signed object describing a client's identity and
      capabilities, and including a hybrid public-key encryption (HPKE
      [RFC9180]) public key that can be used to encrypt to that client,
      and which other clients can use to introduce the client to a new
      group.

   Signature Key:  A signing key pair used to authenticate the sender of
      a message.

   Handshake Message:  An MLSPlaintext or MLSCiphertext message carrying
      an MLS Proposal or Commit object, as opposed to application data.

   Application Message:  An MLSCiphertext message carrying application
      data.

   Terminology specific to tree computations is described in
   Section 5.1.

   In general, symmetric values are referred to as "keys" or "secrets"
   interchangeably.  Either term denotes a value that MUST be kept
   confidential to a Client.  When labelling individual values, we
   typically use "secret" to refer to a value that is used derive
   further secret values, and "key" to refer to a value that is used
   with an algorithm such as HMAC or an AEAD algorithm.

2.1.  Presentation Langauge

   We use the TLS presentation language [RFC8446] to describe the
   structure of protocol messages.  In addition to the base syntax, we
   add two additional features, the ability for fields to be optional
   and the ability for vectors to have variable-size length headers.

2.1.1.  Optional Value

   An optional value is encoded with a presence-signaling octet,
   followed by the value itself if present.  When decoding, a presence
   octet with a value other than 0 or 1 MUST be rejected as malformed.

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   struct {
       uint8 present;
       select (present) {
           case 0: struct{};
           case 1: T value;
       }
   } optional<T>;

2.1.2.  Variable-size Vector Headers

   In the TLS presentation language, vectors are encoded as a sequence
   of encoded elements prefixed with a length.  The length field has a
   fixed size set by specifying the minimum and maximum lengths of the
   encoded sequence of elements.

   In MLS, there are several vectors whose sizes vary over significant
   ranges.  So instead of using a fixed-size length field, we use a
   variable-size length using a variable-length integer encoding based
   on the one in Section 16 of [RFC9000].  (They differ only in that the
   one here requires a minimum-size encoding.)  Instead of presenting
   min and max values, the vector description simply includes a V.  For
   example:

   struct {
       uint32 fixed<0..255>;
       opaque variable<V>;
   } StructWithVectors;

   Such a vector can represent values with length from 0 bytes to 2^30
   bytes.  The variable-length integer encoding reserves the two most
   significant bits of the first byte to encode the base 2 logarithm of
   the integer encoding length in bytes.  The integer value is encoded
   on the remaining bits, in network byte order.  The encoded value MUST
   use the smallest number of bits required to represent the value.
   When decoding, values using more bits than necessary MUST be treated
   as malformed.

   This means that integers are encoded on 1, 2, or 4 bytes and can
   encode 6-, 14-, or 30-bit values respectively.

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          +========+=========+=============+=======+============+
          | Prefix | Length  | Usable Bits | Min   | Max        |
          +========+=========+=============+=======+============+
          | 00     | 1       | 6           | 0     | 63         |
          +--------+---------+-------------+-------+------------+
          | 01     | 2       | 14          | 64    | 16383      |
          +--------+---------+-------------+-------+------------+
          | 10     | 4       | 30          | 16384 | 1073741823 |
          +--------+---------+-------------+-------+------------+
          | 11     | invalid | -           | -     | -          |
          +--------+---------+-------------+-------+------------+

                   Table 1: Summary of Integer Encodings

   Vectors that start with "11" prefix are invalid and MUST be rejected.

   For example, the four byte sequence 0x9d7f3e7d decodes to 494878333;
   the two byte sequence 0x7bbd decodes to 15293; and the single byte
   0x25 decodes to 37.

   The following figure adapts the pseudocode provided in [RFC9000] to
   add a check for minimum-length encoding:

   ReadVarint(data):
     // The length of variable-length integers is encoded in the
     // first two bits of the first byte.
     v = data.next_byte()
     prefix = v >> 6
     length = 1 << prefix

     // Once the length is known, remove these bits and read any
     // remaining bytes.
     v = v & 0x3f
     repeat length-1 times:
       v = (v << 8) + data.next_byte()
     return v

     // Check that the encoder used the minimum bits required
     if length > 1 && v < (1 << (length - 1)):
       raise an exception

   The use of variable-size integers for vector lengths allows vectors
   to grow very large, up to 2^30 bytes.  Implementations should take
   care not to allow vectors to overflow available storage.  To
   facilitate debugging of potential interoperatbility problems,
   implementations should provide a clear error when such an overflow
   condition occurs.

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3.  Operating Context

   MLS is designed to operate in the context described in
   [I-D.ietf-mls-architecture].  In particular, we assume that the
   following services are provided:

   *  A Delivery Service that routes MLS messages among the participants
      in the protocol.  The following types of delivery are typically
      required:

      -  Pre-publication of KeyPackage objects for clients

      -  Broadcast delivery of Proposal and Commit messages to members
         of a group

      -  Unicast delivery of Welcome messages to new members of a group

   *  An Authentication Service that enables group members to
      authenticate the credentials presented by other group members.

4.  Protocol Overview

   The core functionality of MLS is continuous group authenticated key
   exchange (AKE).  As with other authenticated key exchange protocols
   (such as TLS), the participants in the protocol agree on a common
   secret value, and each participant can verify the identity of the
   other participants.  MLS provides group AKE in the sense that there
   can be more than two participants in the protocol, and continuous
   group AKE in the sense that the set of participants in the protocol
   can change over time.

   The core organizing principles of MLS are _groups_ and _epochs_.  A
   group represents a logical collection of clients that share a common
   secret value at any given time.  The history of a group is divided
   into a linear sequence of epochs.  In each epoch, a set of
   authenticated _members_ agree on an _epoch secret_ that is known only
   to the members of the group in that epoch.  The set of members
   involved in the group can change from one epoch to the next, and MLS
   ensures that only the members in the current epoch have access to the
   epoch secret.  From the epoch secret, members derive further shared
   secrets for message encryption, group membership authentication, etc.

   The creator of an MLS group creates the group's first epoch
   unilaterally, with no protocol interactions.  Thereafter, the members
   of the group advance their shared cryptographic state from one epoch
   to another by exchanging MLS messages:

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   *  A _KeyPackage_ object describes a client's capabilities and
      provides keys that can be used to add the client to a group.

   *  A _Proposal_ message proposes a change to be made in the next
      epoch, such as adding or removing a member

   *  A _Commit_ message initiates a new epoch by instructing members of
      the group to implement a collection of proposals

   *  A _Welcome_ message provides a new member to the group with the
      information to initialize their state for the epoch in which they
      were added or in which they want to add themselves to the group

   KeyPackage and Welcome messages are used to initiate a group or
   introduce new members, so they are exchanged between group members
   and clients not yet in the group.

   Proposal and Commit messages are sent from one member of a group to
   the others.  MLS provides a common framing layer for sending messages
   within a group: An _MLSPlaintext_ message provides sender
   authentication for unencrypted Proposal and Commit messages.  An
   _MLSCiphertext_ message provides encryption and authentication for
   both Proposal/Commit messages as well as any application data.

4.1.  Cryptographic State and Evolution

   The cryptographic state at the core of MLS is divided into three
   areas of responsibility:

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                          .-    ...    -.
                         |               |
                         |       |       |
                         |       |       | Key Schedule
                         |       V       |
                         |  epoch_secret |
.                        |       |       |                             .
|\ Ratchet               |       |       |                     Secret /|
| \ Tree                 |       |       |                      Tree / |
|  \                     |       |       |                          /  |
|   \                    |       V       |                         /   |
|    +--> commit_secret --> epoch_secret --> encryption_secret -->+    |
|   /                    |       |       |                         \   |
|  /                     |       |       |                          \  |
| /                      |       |       |                           \ |
|/                       |       |       |                            \|
'                        |       V       |                             '
                         |  epoch_secret |
                         |       |       |
                         |       |       |
                         |       V       |
                         |               |
                          '-    ...    -'

              Figure 1: Overview of MLS group evolution

   *  A _ratchet tree_ that represents the membership of the group,
      providing group members a way to authenticate each other and
      efficiently encrypt messages to subsets of the group.  Each epoch
      has a distinct ratchet tree.  It seeds the _key schedule_.

   *  A _key schedule_ that describes the chain of key derivations used
      to progress from epoch to epoch (mainly using the _init_secret_
      and _epoch_secret_); and to derive a variety of other secrets (see
      Table 3) used during the current epoch.  One of these (the
      _encryption_secret_) is the root of _secret_tree_.

   *  A _secret tree_ derived from the key schedule that represents
      shared secrets used by the members of the group to provide
      confidentiality and forward secrecy for MLS messages.  Each epoch
      has a distinct secret tree.

   Each member of the group maintains a view of these facets of the
   group's state.  MLS messages are used to initialize these views and
   keep them in sync as the group transitions between epochs.

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   Each new epoch is initiated with a Commit message.  The Commit
   instructs existing members of the group to update their views of the
   ratchet tree by applying a set of Proposals, and uses the updated
   ratchet tree to distribute fresh entropy to the group.  This fresh
   entropy is provided only to members in the new epoch, not to members
   who have been removed, so it maintains the confidentiality of the
   epoch secret (in other words, it provides post-compromise security
   with respect to those members).

   For each Commit that adds member(s) to the group, there is a single
   corresponding Welcome message.  The Welcome message provides all the
   new members with the information they need to initialize their views
   of the key schedule and ratchet tree, so that these views are
   equivalent to the views held by other members of the group in this
   epoch.

   In addition to defining how one epoch secret leads to the next, the
   key schedule also defines a collection of secrets that are derived
   from the epoch secret.  For example:

   *  An _encryption secret_ that is used to initialize the secret tree
      for the epoch.

   *  A _confirmation key_ that is used to confirm that all members
      agree on the shared state of the group.

   *  A _resumption secret_ that members can use to prove their
      membership in the group, e.g., in the case of branching a
      subgroup.

   Finally, an _init secret_ is derived that is used to initialize the
   next epoch.

4.2.  Example Protocol Execution

   There are three major operations in the lifecycle of a group:

   *  Adding a member, initiated by a current member;

   *  Updating the leaf secret of a member;

   *  Removing a member.

   Each of these operations is "proposed" by sending a message of the
   corresponding type (Add / Update / Remove).  The state of the group
   is not changed, however, until a Commit message is sent to provide
   the group with fresh entropy.  In this section, we show each proposal
   being committed immediately, but in more advanced deployment cases an

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   application might gather several proposals before committing them all
   at once.  In the illustrations below, we show the Proposal and Commit
   messages directly, while in reality they would be sent encapsulated
   in MLSPlaintext or MLSCiphertext objects.

   Before the initialization of a group, clients publish KeyPackages to
   a directory provided by the Service Provider.

                                                                 Group
  A                B                C            Directory       Channel
  |                |                |                |              |
  | KeyPackageA    |                |                |              |
  +------------------------------------------------->|              |
  |                |                |                |              |
  |                | KeyPackageB    |                |              |
  |                +-------------------------------->|              |
  |                |                |                |              |
  |                |                | KeyPackageC    |              |
  |                |                +--------------->|              |
  |                |                |                |              |

    Figure 2: Clients A, B, and C publish KeyPackages to the directory

   When a client A wants to establish a group with B and C, it first
   initializes a group state containing only itself and downloads
   KeyPackages for B and C.  For each member, A generates an Add and
   Commit message adding that member, and broadcasts them to the group.
   It also generates a Welcome message and sends this directly to the
   new member (there's no need to send it to the group).  Only after A
   has received its Commit message back from the server does it update
   its state to reflect the new member's addition.

   Upon receiving the Welcome message, the new member will be able to
   read and send new messages to the group.  However, messages sent
   before they were added to the group will not be accessible.

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                                                                  Group
   A              B              C          Directory            Channel
   |              |              |              |                   |
   |         KeyPackageB, KeyPackageC           |                   |
   |<-------------------------------------------+                   |
   |              |              |              |                   |
   |              |              |              | Add(A->AB)        |
   |              |              |              | Commit(Add)       |
   +--------------------------------------------------------------->|
   |              |              |              |                   |
   |  Welcome(B)  |              |              |                   |
   +------------->|              |              |                   |
   |              |              |              |                   |
   |              |              |              | Add(A->AB)        |
   |              |              |              | Commit(Add)       |
   |<---------------------------------------------------------------+
   |              |              |              |                   |
   |              |              |              |                   |
   |              |              |              | Add(AB->ABC)      |
   |              |              |              | Commit(Add)       |
   +--------------------------------------------------------------->|
   |              |              |              |                   |
   |              |  Welcome(C)  |              |                   |
   +---------------------------->|              |                   |
   |              |              |              |                   |
   |              |              |              | Add(AB->ABC)      |
   |              |              |              | Commit(Add)       |
   |<---------------------------------------------------------------+
   |              |<------------------------------------------------+
   |              |              |              |                   |

          Figure 3: Client A creates a group with clients B and C

   Subsequent additions of group members proceed in the same way.  Any
   member of the group can download a KeyPackage for a new client and
   broadcast Add and Commit messages that the current group will use to
   update their state, and a Welcome message that the new client can use
   to initialize its state and join the group.

   To enforce the forward secrecy and post-compromise security of
   messages, each member periodically updates the keys that represent
   them to the group.  A member does this by sending a Commit (possibly
   with no proposals), or by sending an Update message that is committed
   by another member.  Once the other members of the group have
   processed these messages, the group's secrets will be unknown to an
   attacker that had compromised the sender's prior leaf secret.

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   Update messages should be sent at regular intervals of time as long
   as the group is active, and members that don't update should
   eventually be removed from the group.  It's left to the application
   to determine an appropriate amount of time between Updates.

                                                             Group
   A              B     ...      Z          Directory        Channel
   |              |              |              |              |
   |              | Update(B)    |              |              |
   |              +------------------------------------------->|
   | Commit(Upd)  |              |              |              |
   +---------------------------------------------------------->|
   |              |              |              |              |
   |              |              |              | Update(B)    |
   |              |              |              | Commit(Upd)  |
   |<----------------------------------------------------------+
   |              |<-------------------------------------------+
   |              |              |<----------------------------+
   |              |              |              |              |

        Figure 4: Client B proposes to update its key, and client A
       commits the proposal.  As a result, the keys for both B and A
      updated, so the group has post-compromise security with respect
                              to both of them.

   Members are removed from the group in a similar way.  Any member of
   the group can send a Remove proposal followed by a Commit message.
   The Commit message provides new entropy to all members of the group
   except the removed member.  This new entropy is added to the epoch
   secret for the new epoch, so that it is not known to the removed
   member.  Note that this does not necessarily imply that any member is
   actually allowed to evict other members; groups can enforce access
   control policies on top of these basic mechanism.

                                                             Group
   A              B     ...      Z          Directory       Channel
   |              |              |              |              |
   |              |              | Remove(B)    |              |
   |              |              | Commit(Rem)  |              |
   |              |              +---------------------------->|
   |              |              |              |              |
   |              |              |              | Remove(B)    |
   |              |              |              | Commit(Rem)  |
   |<----------------------------------------------------------+
   |              |              |<----------------------------+
   |              |              |              |              |

             Figure 5: Client Z removes client B from the group

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4.3.  Relationships Between Epochs

   A group has a single linear sequence of epochs.  Groups and epochs
   are generally independent of one-another.  However, it can sometimes
   be useful to link epochs cryptographically, either within a group or
   across groups.  MLS derives a resumption pre-shared key (PSK) from
   each epoch to allow entropy extracted from one epoch to be injected
   into a future epoch.  This link guarantees that members entering the
   new epoch agree on a key if and only if they were members of the
   group during the epoch from which the resumption key was extracted.

   MLS supports two ways to tie a new group to an existing group.  Re-
   initialization closes one group and creates a new group comprising
   the same members with different parameters.  Branching starts a new
   group with a subset of the original group's participants (with no
   effect on the original group).  In both cases, the new group is
   linked to the old group via a resumption PSK.

   epoch_A_[n-1]
        |
        |
        |<-- ReInit
        |
        V
   epoch_A_[n]           epoch_B_[0]
        .                     |
        .  PSK(usage=reinit)  |
        .....................>|
                              |
                              V
                         epoch_B_[1]

                      Figure 6: Reinitializing a group

   epoch_A_[n]           epoch_B_[0]
        |                     |
        |  PSK(usage=branch)  |
        |....................>|
        |                     |
        V                     V
   epoch_A_[n+1]         epoch_B_[1]

                        Figure 7: Branching a group

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   Applications may also choose to use resumption PSKs to link epochs in
   other ways.  For example, the following figure shows a case where a
   resumption PSK from epoch n is injected into epoch n+k.  This
   demonstrates that the members of the group at epoch n+k were also
   members at epoch n, irrespective of any changes to these members'
   keys due to Updates or Commits.

   epoch_A_[n]
        |
        |  PSK(usage=application)
        |.....................
        |                    .
        |                    .
       ...                  ...
        |                    .
        |                    .
        V                    .
   epoch_A_[n+k-1]           .
        |                    .
        |                    .
        |<....................
        |
        V
   epoch_A_[n+k]

            Figure 8: Reinjecting entropy from an earlier epoch

5.  Ratchet Tree Concepts

   The protocol uses "ratchet trees" for deriving shared secrets among a
   group of clients.  A ratchet tree is an arrangement of secrets and
   key pairs among the members of a group in a way that allows for
   secrets to be efficiently updated to reflect changes in the group.

   Ratchet trees allow a group to efficiently remove any member by
   encrypting new entropy to a subset of the group.  A ratchet tree
   assigns shared keys to subgroups of the overall group, so that, for
   example, encrypting to all but one member of the group requires only
   log(N) encryptions, instead of the N-1 encryptions that would be
   needed to encrypt to each participant individually (where N is the
   number of members in the group).

   This remove operation allows MLS to efficiently achieve post-
   compromise security.  In an Update proposal or a full Commit message,
   an old (possibly compromised) representation of a member is
   efficiently removed from the group and replaced with a freshly
   generated instance.

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5.1.  Ratchet Tree Terminology

   Trees consist of _nodes_. A node is a _leaf_ if it has no children,
   and a _parent_ otherwise; note that all parents in our trees have
   precisely two children, a _left_ child and a _right_ child.  A node
   is the _root_ of a tree if it has no parents, and _intermediate_ if
   it has both children and parents.  The _descendants_ of a node are
   that node's children, and the descendants of its children, and we say
   a tree _contains_ a node if that node is a descendant of the root of
   the tree, or if the node itself is the root of the tree.  Nodes are
   _siblings_ if they share the same parent.

   A _subtree_ of a tree is the tree given by any node (the _head_ of
   the subtree) and its descendants.  The _size_ of a tree or subtree is
   the number of leaf nodes it contains.  For a given parent node, its
   _left subtree_ is the subtree with its left child as head
   (respectively _right subtree_).

   All trees used in this protocol are left-balanced binary trees.  A
   binary tree is _full_ (and _balanced_) if its size is a power of two
   and for any parent node in the tree, its left and right subtrees have
   the same size.

   A binary tree is _left-balanced_ if for every parent, either the
   parent is balanced, or the left subtree of that parent is the largest
   full subtree that could be constructed from the leaves present in the
   parent's own subtree.  Given a list of n items, there is a unique
   left-balanced binary tree structure with these elements as leaves.

   (Note that left-balanced binary trees are the same structure that is
   used for the Merkle trees in the Certificate Transparency protocol
   [I-D.ietf-trans-rfc6962-bis].)

   The _direct path_ of a root is the empty list, and of any other node
   is the concatenation of that node's parent along with the parent's
   direct path.  The _copath_ of a node is the node's sibling
   concatenated with the list of siblings of all the nodes in its direct
   path, excluding the root.

   For example, in the below tree:

   *  The direct path of C is (W, V, X)

   *  The copath of C is (D, U, Z)

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                 X = root
                 |
           .-----+-----.
          /             \
         V               Z
         |               |
       .-+-.           .-+
      /     \         /   \
     U       W       Y     +
    / \     / \     / \    |
   A   B   C   D   E   F   G

   0   1   2   3   4   5   6

                Figure 9: A complete tree with seven members

   A tree with n leaves has 2*n - 1 nodes.  For example, the above tree
   has 7 leaves (A, B, C, D, E, F, G) and 13 nodes.

   Each leaf is given an _index_ (or _leaf index_), starting at 0 from
   the left to n-1 at the right.

   Finally, a node in the tree may also be _blank_, indicating that no
   value is present at that node (i.e. no keying material).  This is
   often the case when a leaf was recently removed from the tree.

   There are multiple ways that an implementation might represent a
   ratchet tree in memory.  For example, left-balanced binary trees can
   be represented as an array of nodes, with node relationships computed
   based on nodes' indices in the array.  Or a more traditional
   representation of linked node objects may be used.  Appendix B and
   Appendix C provide some details on how to implement the tree
   operations required for MLS in these representations.  MLS places no
   requirements on implementations' internal representations of ratchet
   trees.  An implementation MAY use any tree representation and
   associated algorithms, as long as they produce correct protocol
   messages.

5.2.  Views of a Ratchet Tree

   We generally assume that each participant maintains a complete and
   up-to-date view of the public state of the group's ratchet tree,
   including the public keys for all nodes and the credentials
   associated with the leaf nodes.

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   No participant in an MLS group knows the private key associated with
   every node in the tree.  Instead, each member is assigned to a leaf
   of the tree, which determines the subset of private keys it knows.
   The credential stored at that leaf is one provided by the member.

   In particular, MLS maintains the members' views of the tree in such a
   way as to maintain the _tree invariant_:

   The private key for a node in the tree is known to a member of
   the group only if the node's subtree contains that member's leaf.

   In other words, if a node is not blank, then it holds a public key.
   The corresponding private key is known only to members occupying
   leaves below that node.

   The reverse implication is not true: A member may not know the
   private keys of all the intermediate nodes they're below.  Such a
   member has an _unmerged_ leaf.  Encrypting to an intermediate node
   requires encrypting to the node's public key, as well as the public
   keys of all the unmerged leaves below it.  A leaf is unmerged when it
   is first added, because the process of adding the leaf does not give
   it access to all of the nodes above it in the tree.  Leaves are
   "merged" as they receive the private keys for nodes, as described in
   Section 8.4.

   For example, consider a four-member group (A, B, C, D) where the node
   above the right two members is blank.  (This is what it would look
   like if A created a group with B, C, and D.)  Then the public state
   of the tree and the views of the private keys of the tree held by
   each participant would be as follows, where _ represents a blank
   node, ? represents an unknown private key, and pk(X) represents the
   public key corresponding to the private key X:

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            Public Tree
   ============================
               pk(ABCD)
             /          \
       pk(AB)            _
        / \             / \
   pk(A)   pk(B)   pk(C)   pk(D)

    Private @ A       Private @ B       Private @ C       Private @ D
   =============     =============     =============     =============
        ABCD              ABCD              ABCD              ABCD
       /   \             /   \             /   \             /   \
     AB      _         AB      _         ?       _         ?       _
    / \     / \       / \     / \       / \     / \       / \     / \
   A   ?   ?   ?     ?   B   ?   ?     ?   ?   C   ?     ?   ?   ?   D

   Note how the tree invariant applies: Each member knows only their own
   leaf, and the private key AB is known only to A and B.

5.3.  Ratchet Tree Nodes

   A particular instance of a ratchet tree includes the same parameters
   that define an instance of HPKE, namely:

   *  A Key Encapsulation Mechanism (KEM), including a DeriveKeyPair
      function that creates a key pair for the KEM from a symmetric
      secret

   *  A Key Derivation Function (KDF), including Extract and Expand
      functions

   *  An AEAD encryption scheme

   Each non-blank node in a ratchet tree contains up to five values:

   *  A public key

   *  A private key (only within the member's direct path, see below)

   *  A credential (only for leaf nodes)

   *  An ordered list of "unmerged" leaves (see Section 5.2)

   *  A hash of certain information about the node's parent, as of the
      last time the node was changed (see Section 8.8).

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   The _resolution_ of a node is an ordered list of non-blank nodes that
   collectively cover all non-blank descendants of the node.  The
   resolution of a non-blank node with no unmerged leaves is just the
   node itself.  More generally, the resolution of a node is effectively
   a depth-first, left-first enumeration of the nearest non-blank nodes
   below the node:

   *  The resolution of a non-blank node comprises the node itself,
      followed by its list of unmerged leaves, if any

   *  The resolution of a blank leaf node is the empty list

   *  The resolution of a blank intermediate node is the result of
      concatenating the resolution of its left child with the resolution
      of its right child, in that order

   For example, consider the following subtree, where the _ character
   represents a blank node and unmerged leaves are indicated in square
   brackets:

          ...
          /
         _
         |
       .-+-.
      /     \
     _       Z[C]
    / \     / \
   A   _   C   D

   0   1   2   3

             Figure 10: A tree with blanks and unmerged leaves

   In this tree, we can see all of the above rules in play:

   *  The resolution of node Z is the list [Z, C]

   *  The resolution of leaf 1 is the empty list []

   *  The resolution of top node is the list [A, Z, C]

   Every node, regardless of whether the node is blank or populated, has
   a corresponding _hash_ that summarizes the contents of the subtree
   below that node.  The rules for computing these hashes are described
   in Section 8.7.

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6.  Cryptographic Objects

6.1.  Ciphersuites

   Each MLS session uses a single ciphersuite that specifies the
   following primitives to be used in group key computations:

   *  HPKE parameters:

      -  A Key Encapsulation Mechanism (KEM)

      -  A Key Derivation Function (KDF)

      -  An AEAD encryption algorithm

   *  A hash algorithm

   *  A MAC algorithm

   *  A signature algorithm

   MLS uses HPKE for public-key encryption [RFC9180].  The DeriveKeyPair
   function associated to the KEM for the ciphersuite maps octet strings
   to HPKE key pairs.  As in HPKE, MLS assumes that an AEAD algorithm
   produces a single ciphertext output from AEAD encryption (aligning
   with [RFC5116]), as opposed to a separate ciphertext and tag.

   Ciphersuites are represented with the CipherSuite type.  HPKE public
   keys are opaque values in a format defined by the underlying protocol
   (see the Cryptographic Dependencies section of the HPKE specification
   for more information).

   opaque HPKEPublicKey<V>;

   The signature algorithm specified in the ciphersuite is the mandatory
   algorithm to be used for signatures in MLSMessageAuth and the tree
   signatures.  It MUST be the same as the signature algorithm specified
   in the credentials in the leaves of the tree (including the leaf node
   information in KeyPackages used to add new members).

   Like HPKE public keys, signature public keys are represented as
   opaque values in a format defined by the cipher suite's signature
   scheme.

   opaque SignaturePublicKey<V>;

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   For ciphersuites using Ed25519 or Ed448 signature schemes, the public
   key is in the format specified in [RFC8032].  For ciphersuites using
   ECDSA with the NIST curves (P-256, P-384, or P-521), the public key
   is the output of the uncompressed Elliptic-Curve-Point-to-Octet-
   String conversion according to [SECG].

   To disambiguate different signatures used in MLS, each signed value
   is prefixed by a label as shown below:

   SignWithLabel(SignatureKey, Label, Content) =
       Signature.Sign(SignatureKey, SignContent)

   VerifyWithLabel(VerificationKey, Label, Content) =
       Signature.Verify(VerificationKey, SignContent)

   Where SignContent is specified as:

   struct {
       opaque label<V> = "MLS 1.0 " + Label;
       opaque content<V> = Content;
   } SignContent;

   Here, the functions Signature.Sign and Signature.Verify are defined
   by the signature algorithm.

   The ciphersuites are defined in section Section 18.1.

6.2.  Hash-Based Identifiers

   Some MLS messages refer to other MLS objects by hash.  For example,
   Welcome messages refer to KeyPackages for the members being welcomed,
   and Commits refer to Proposals they cover.  These identifiers are
   computed as follows:

   opaque HashReference[16];

   HashReference KeyPackageRef;
   HashReference LeafNodeRef;
   HashReference ProposalRef;

   MakeKeyPackageRef(value) = KDF.expand(
     KDF.extract("", value), "MLS 1.0 KeyPackage Reference", 16)

   MakeLeafNodeRef(value) = KDF.expand(
     KDF.extract("", value), "MLS 1.0 Leaf Node Reference", 16)

   MakeProposalRef(value) = KDF.expand(
     KDF.extract("", value), "MLS 1.0 Proposal Reference", 16)

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   For a KeyPackageRef, the value input is the encoded KeyPackage, and
   the ciphersuite specified in the KeyPackage determines the KDF used.
   For a LeafNodeRef, the value input is the LeafNode object for the
   leaf node in question.  For a ProposalRef, the value input is the
   MLSMessageContentAuth carrying the proposal.  In the latter two
   cases, the KDF is determined by the group's ciphersuite.

6.3.  Credentials

   Each member of a group presents a credential that associates an
   identity with the member's key material.  This information is
   verified according to the Authentication Service in use for a group.

   A Credential can provide multiple identifiers for the client.  It is
   up to the application to decide which identifier or identifiers to
   use at the application level.  For example, a certificate in an
   X509Credential may attest to several domain names or email addresses
   in its subjectAltName extension.  An application may decide to
   present all of these to a user, or if it knows a "desired" domain
   name or email address, it can check that the desired identifier is
   among those attested.  Using the terminology from [RFC6125], a
   Credential provides "presented identifiers", and it is up to the
   application to supply a "reference identifier" for the authenticated
   client, if any.

   Credentials MAY also include information that allows a relying party
   to verify the identity / signing key binding, such as a signature
   from a trusted authority.

   // See IANA registry for registered values
   uint16 CredentialType;

   struct {
       opaque cert_data<V>;
   } Certificate;

   struct {
       CredentialType credential_type;
       select (credential_type) {
           case basic:
               opaque identity<V>;

           case x509:
               Certificate chain<V>;
       };
   } Credential;

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   A BasicCredential is a bare assertion of an identity, without any
   additional information.  The format of the encoded identity is
   defined by the application.

   For an X.509 credential, each entry in the chain represents a single
   DER-encoded X.509 certificate.  The chain is ordered such that the
   first entry (chain[0]) is the end-entity certificate and each
   subsequent certificate in the chain MUST be the issuer of the
   previous certificate.  The public key encoded in the
   subjectPublicKeyInfo of the end-entity certificate MUST be identical
   to the signature_key in the LeafNode containing this credential.

   The signatures used in this document are encoded as specified in
   [RFC8446].  In particular, ECDSA signatures are DER-encoded and EdDSA
   signatures are defined as the concatenation of r and s as specified
   in [RFC8032].

   Each new credential that has not already been validated by the
   application MUST be validated against the Authentication Service.
   Applications SHOULD require that a client present the same set of
   identifiers throughout its presence in the group, even if its
   Credential is changed in a Commit or Update.  If an application
   allows clients to change identifiers over time, then each time the
   client presents a new credential, the application MUST verify that
   the set of identifiers in the credential is acceptable to the
   application for this client.

6.3.1.  Uniquely Identifying Clients

   MLS implementations will presumably provide applications with a way
   to request protocol operations with regard to other clients (e.g.,
   removing clients).  Such functions will need to refer to the other
   clients using some identifier.  MLS clients have a few types of
   identifiers, with different operational properties.

   The Credentials presented by the clients in a group authenticate
   application-level identifiers for the clients.  These identifiers may
   not uniquely identify clients.  For example, if a user has multiple
   devices that are all present in an MLS group, then those devices'
   clients could all present the user's application-layer identifiers.

   Internally to the protocol, group members are uniquely identified by
   their leaves, expressed as LeafNodeRef objects.  These identifiers
   are unstable: They change whenever the member sends a Commit, or
   whenever an Update proposal from the member is committed.

   MLS provides two unique client identifiers that are stable across
   epochs:

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   *  The index of a client among the leaves of the tree

   *  The epoch_id field in the key package

   The application may also provide application-specific unique
   identifiers in the extensions field of KeyPackage or LeafNode
   objects.

7.  Message Framing

   Handshake and application messages use a common framing structure.
   This framing provides encryption to ensure confidentiality within the
   group, as well as signing to authenticate the sender within the
   group.

   The main structure is MLSMessageContent, which contains the content
   of the message.  This structure is authenticated using MLSMessageAuth
   (see Section 7.1).  The two structures are combined in
   MLSMessageContentAuth, which can then be encoded/decoded from/to
   MLSPlaintext or MLSCiphertext, which are then included in the
   MLSMessage structure.

   MLSCiphertext represents a signed and encrypted message, with
   protections for both the content of the message and related metadata.
   MLSPlaintext represents a message that is only signed, and not
   encrypted.  Applications MUST use MLSCiphertext to encrypt
   application messages and SHOULD use MLSCiphertext to encode handshake
   messages, but MAY transmit handshake messages encoded as MLSPlaintext
   objects in cases where it is necessary for the Delivery Service to
   examine such messages.

   enum {
       reserved(0),
       mls10(1),
       (255)
   } ProtocolVersion;

   enum {
       reserved(0),
       application(1),
       proposal(2),
       commit(3),
       (255)
   } ContentType;

   enum {
       reserved(0),
       member(1),

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       external(2),
       new_member(3),
       (255)
   } SenderType;

   struct {
       SenderType sender_type;
       switch (sender_type) {
           case member:
               LeafNodeRef member_ref;
           case external:
               uint32 sender_index;
           case new_member:
               struct{};
       }
   } Sender;

   enum {
     reserved(0),
     mls_plaintext(1),
     mls_ciphertext(2),
     mls_welcome(3),
     mls_group_info(4),
     mls_key_package(5),
     (255)
   } WireFormat;

   struct {
       opaque group_id<V>;
       uint64 epoch;
       Sender sender;
       opaque authenticated_data<V>;

       ContentType content_type;
       select (MLSMessageContent.content_type) {
           case application:
             opaque application_data<V>;
           case proposal:
             Proposal proposal;
           case commit:
             Commit commit;
       }
   } MLSMessageContent;

   struct {
       ProtocolVersion version = mls10;
       WireFormat wire_format;
       select (MLSMessage.wire_format) {

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           case mls_plaintext:
               MLSPlaintext plaintext;
           case mls_ciphertext:
               MLSCiphertext ciphertext;
           case mls_welcome:
               Welcome welcome;
           case mls_group_info:
               GroupInfo group_info;
           case mls_key_package:
               KeyPackage key_package;
       }
   } MLSMessage;

   External sender types are sent as MLSPlaintext, see Section 13.1.9
   for their use.

   The following structure is used to fully describe the data
   transmitted in plaintexts or ciphertexts.

   struct {
       WireFormat wire_format;
       MLSMessageContent content;
       MLSMessageAuth auth;
   } MLSMessageContentAuth;

7.1.  Content Authentication

   MLSMessageContent is authenticated using the MLSMessageAuth
   structure.

   struct {
       opaque mac_value<V>;
   } MAC;

   struct {
       // SignWithLabel(., "MLSMessageContentTBS", MLSMessageContentTBS)
       opaque signature<V>;
       select (MLSMessageContent.content_type) {
           case commit:
               // MAC(confirmation_key,
               //     GroupContext.confirmed_transcript_hash)
               MAC confirmation_tag;
           case application:
           case proposal:
               struct{};
       }
   } MLSMessageAuth;

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   The signature is computed using SignWithLabel with label
   "MLSMessageContentTBS" and with a content that covers the message
   content and the wire format that will be used for this message.  If
   the sender's sender_type is Member, the content also covers the
   GroupContext for the current epoch, so that signatures are specific
   to a given group and epoch.

   The sender MUST use the private key corresponding to the following
   signature key depending on the sender's sender_type:

   *  member: The signature key contained in the Credential at the leaf
      with the sender's LeafNodeRef

   *  external: The signature key contained in the Credential at the
      index indicated by the sender_index in the external_senders group
      context extension (see Section Section 13.1.9.1).  In this case,
      the content_type of the message MUST NOT be commit, since only
      members of the group or new joiners can send Commit messages.

   *  new_member: The signature key depends on the content_type:

      -  proposal: The signature key in the credential contained in
         KeyPackage in the Add proposal (see Section Section 13.1.9).

      -  commit: The signature key in the credential contained in the
         KeyPackage in the Commit's path (see Section Section 9.3).

   struct {
       ProtocolVersion version = mls10;
       WireFormat wire_format;
       MLSMessageContent content;
       select (MLSMessageContentTBS.content.sender.sender_type) {
           case member:
           case new_member:
               GroupContext context;

           case external:
               struct{};
       }
   } MLSMessageContentTBS;

   Recipients of an MLSMessage MUST verify the signature with the key
   depending on the sender_type of the sender as described above.

   The confirmation tag value confirms that the members of the group
   have arrived at the same state of the group.

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   A MLSMessageAuth is said to be valid when both the signature and
   confirmation_tag fields are valid.

7.2.  Encoding and Decoding a Plaintext

   Plaintexts are encoded using the MLSPlaintext structure.

   struct {
       MLSMessageContent content;
       MLSMessageAuth auth;
       select(MLSPlaintext.content.sender.sender_type) {
           case member:
               MAC membership_tag;
           case external:
           case new_member:
               struct{};
       }
   } MLSPlaintext;

   The membership_tag field in the MLSPlaintext object authenticates the
   sender's membership in the group.  For messages sent by members, it
   MUST be set to the following value:

   struct {
     MLSMessageContentTBS content_tbs;
     MLSMessageAuth auth;
   } MLSMessageContentTBM;

   membership_tag = MAC(membership_key, MLSMessageContentTBM)

   When decoding a MLSPlaintext into a MLSMessageContentAuth, the
   application MUST check membership_tag, and MUST check that the
   MLSMessageAuth is valid.

7.3.  Encoding and Decoding a Ciphertext

   Ciphertexts are encoded using the MLSCiphertext structure.

   struct {
       opaque group_id<V>;
       uint64 epoch;
       ContentType content_type;
       opaque authenticated_data<V>;
       opaque encrypted_sender_data<V>;
       opaque ciphertext<V>;
   } MLSCiphertext;

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   encrypted_sender_data and ciphertext are encrypted using the AEAD
   function specified by the ciphersuite in use, using as input the
   structures MLSSenderData and MLSCiphertextContent.

7.3.1.  Content Encryption

   The ciphertext content is encoded using the MLSCiphertextContent
   structure.

   struct {
       select (MLSCiphertext.content_type) {
           case application:
             opaque application_data<V>;

           case proposal:
             Proposal proposal;

           case commit:
             Commit commit;
       }

       MLSMessageAuth auth;
       opaque padding<V>;
   } MLSCiphertextContent;

   In the MLS key schedule, the sender creates two distinct key ratchets
   for handshake and application messages for each member of the group.
   When encrypting a message, the sender looks at the ratchets it
   derived for its own member and chooses an unused generation from
   either the handshake or application ratchet depending on the content
   type of the message.  This generation of the ratchet is used to
   derive a provisional nonce and key.

   Before use in the encryption operation, the nonce is XORed with a
   fresh random value to guard against reuse.  Because the key schedule
   generates nonces deterministically, a client must keep persistent
   state as to where in the key schedule it is; if this persistent state
   is lost or corrupted, a client might reuse a generation that has
   already been used, causing reuse of a key/nonce pair.

   To avoid this situation, the sender of a message MUST generate a
   fresh random 4-byte "reuse guard" value and XOR it with the first
   four bytes of the nonce from the key schedule before using the nonce
   for encryption.  The sender MUST include the reuse guard in the
   reuse_guard field of the sender data object, so that the recipient of
   the message can use it to compute the nonce to be used for
   decryption.

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   +-+-+-+-+---------...---+
   |   Key Schedule Nonce  |
   +-+-+-+-+---------...---+
              XOR
   +-+-+-+-+---------...---+
   | Guard |       0       |
   +-+-+-+-+---------...---+
              ===
   +-+-+-+-+---------...---+
   | Encrypt/Decrypt Nonce |
   +-+-+-+-+---------...---+

   The Additional Authenticated Data (AAD) input to the encryption
   contains an object of the following form, with the values used to
   identify the key and nonce:

   struct {
       opaque group_id<V>;
       uint64 epoch;
       ContentType content_type;
       opaque authenticated_data<V>;
   } MLSCiphertextContentAAD;

   When decoding a MLSCiphertextContent, the application MUST check that
   the MLSMessageAuth is valid.

7.3.2.  Sender Data Encryption

   The "sender data" used to look up the key for the content encryption
   is encrypted with the ciphersuite's AEAD with a key and nonce derived
   from both the sender_data_secret and a sample of the encrypted
   content.  Before being encrypted, the sender data is encoded as an
   object of the following form:

   struct {
       LeafNodeRef sender;
       uint32 generation;
       opaque reuse_guard[4];
   } MLSSenderData;

   MLSSenderData.sender is assumed to be a member sender type.  When
   constructing an MLSSenderData from a Sender object, the sender MUST
   verify Sender.sender_type is member and use Sender.sender for
   MLSSenderData.sender.

   The reuse_guard field contains a fresh random value used to avoid
   nonce reuse in the case of state loss or corruption, as described in
   Section 7.3.1.

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   The key and nonce provided to the AEAD are computed as the KDF of the
   first KDF.Nh bytes of the ciphertext generated in the previous
   section.  If the length of the ciphertext is less than KDF.Nh, the
   whole ciphertext is used without padding.  In pseudocode, the key and
   nonce are derived as:

   ciphertext_sample = ciphertext[0..KDF.Nh-1]

   sender_data_key = ExpandWithLabel(sender_data_secret, "key",
                         ciphertext_sample, AEAD.Nk)
   sender_data_nonce = ExpandWithLabel(sender_data_secret, "nonce",
                         ciphertext_sample, AEAD.Nn)

   The Additional Authenticated Data (AAD) for the SenderData ciphertext
   is the first three fields of MLSCiphertext:

   struct {
       opaque group_id<V>;
       uint64 epoch;
       ContentType content_type;
   } MLSSenderDataAAD;

   When parsing a SenderData struct as part of message decryption, the
   recipient MUST verify that the LeafNodeRef indicated in the sender
   field identifies a member of the group.

8.  Ratchet Tree Operations

   The ratchet tree for an epoch describes the membership of a group in
   that epoch, providing public-key encryption (HPKE) keys that can be
   used to encrypt to subsets of the group as well as information to
   authenticate the members.  In order to reflect changes to the
   membership of the group from one epoch to the next, corresponding
   changes are made to the ratchet tree.  In this section, we describe
   the content of the tree and the required operations.

8.1.  Parent Node Contents

   As discussed in Section 5.3, the nodes of a ratchet tree contain
   several types of data describing individual members (for leaf nodes)
   or subgroups of the group (for parent nodes).  Parent nodes are
   simpler:

   struct {
       HPKEPublicKey encryption_key;
       opaque parent_hash<V>;
       uint32 unmerged_leaves<V>;
   } ParentNode;

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   The encryption_key field contains an HPKE public key whose private
   key is held only by the members at the leaves among its descendants.
   The parent_hash field contains a hash of this node's parent node, as
   described in Section 8.8.  The unmerged_leaves field lists the leaves
   under this parent node that are unmerged, according to their indices
   among all the leaves in the tree.

8.2.  Leaf Node Contents

   A leaf node in the tree describes all the details of an individual
   client's appearance in the group, signed by that client.  It is also
   used in client KeyPackage objects to store the information that will
   be needed to add a client to a group.

   enum {
       reserved(0),
       key_package(1),
       update(2),
       commit(3),
       (255)
   } LeafNodeSource;

   struct {
       ProtocolVersion versions<V>;
       CipherSuite ciphersuites<V>;
       ExtensionType extensions<V>;
       ProposalType proposals<V>;
       CredentialType credentials<V>;
   } Capabilities;

   struct {
       uint64 not_before;
       uint64 not_after;
   } Lifetime;

   // See IANA registry for registered values
   uint16 ExtensionType;

   struct {
       ExtensionType extension_type;
       opaque extension_data<V>;
   } Extension;

   struct {
       HPKEPublicKey encryption_key;
       SignaturePublicKey signature_key;
       Credential credential;
       Capabilities capabilities;

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       LeafNodeSource leaf_node_source;
       select (leaf_node_source) {
           case key_package:
               Lifetime lifetime;

           case update:
               struct {}

           case commit:
               opaque parent_hash<V>;
       }

       Extension extensions<V>;
       // SignWithLabel(., "LeafNodeTBS", LeafNodeTBS)
       opaque signature<V>;
   } LeafNode;

   struct {
       HPKEPublicKey encryption_key;
       SignaturePublicKey signature_key;
       Credential credential;
       Capabilities capabilities;

       LeafNodeSource leaf_node_source;
       select (leaf_node_source) {
           case key_package:
               Lifetime lifetime;

           case update:
               struct{};

           case commit:
               opaque parent_hash<V>;
       }

       Extension extensions<V>;

       select (leaf_node_source) {
           case key_package:
               struct{};

           case update:
               opaque group_id<V>;

           case commit:
               opaque group_id<V>;
       }
   } LeafNodeTBS;

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   The encryption_key field conains an HPKE public key whose private key
   is held only by the member occupying this leaf (or in the case of a
   LeafNode in a KeyPackage object, the issuer of the KeyPackage).  The
   credential contains authentication information for this member, as
   described in Section 6.3.

   The capabilities field indicates what protocol versions,
   ciphersuites, extensions, credential types, and non-default proposal
   types are supported by a client.  Proposal and extension types
   defined in this document are considered "default" and thus need not
   be listed, while any credential types the application wishes to use
   MUST be listed.  Extensions that appear in the extensions field of a
   LeafNode MUST be included in the extensions field of the capabilities
   field, and the credential type used in the LeafNode MUST be included
   in the credentials field of the capabilities field.

   The leaf_node_source field indicates how this LeafNode came to be
   added to the tree.  This signal tells other members of the group
   whether the leaf node is required to have a lifetime or parent_hash,
   and whether the group_id is added as context to the signature.
   Whether these fields can be computed by the client represented by the
   LeafNode depends on when the LeafNode was created.  For example, a
   KeyPackage is created before the client knows which group it will be
   used with, so its signature can't bind to a group_id.

   In the case where the leaf was added to the tree based on a pre-
   published KeyPackage, the lifetime field represents the times between
   which clients will consider a LeafNode valid.  These times are
   represented as absolute times, measured in seconds since the Unix
   epoch (1970-01-01T00:00:00Z).  Applications MUST define a maximum
   total lifetime that is acceptable for a LeafNode, and reject any
   LeafNode where the total lifetime is longer than this duration.

   In the case where the leaf node was inserted into the tree via a
   Commit message, the parent_hash field contains the parent hash for
   this leaf node (see Section 8.8).

   The LeafNodeTBS structure covers the fields above the signature in
   the LeafNode.  In addition, when the leaf node was created in the
   context of a group (the update and commit cases), the group ID of the
   group is added as context to the signature.

   LeafNode objects stored in the group's ratchet tree are updated
   according to the evolution of the tree.  Each modification of
   LeafNode content MUST be reflected by a change in its signature.
   This allows other members to verify the validity of the LeafNode at
   any time, particularly in the case of a newcomer joining the group.

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8.3.  Leaf Node Validation

   The validity of a LeafNode needs to be verified at a few stages:

   *  When a LeafNode is downloaded in a KeyPackage, before it is used
      to add the client to the group

   *  When a LeafNode is received by a group member in an Add, Update,
      or Commit message

   *  When a client joining a group receives LeafNode objects for the
      other members of the group in the group's ratchet tree

   The client verifies the validity of a LeafNode using the following
   steps:

   *  Verify that the credential in the LeafNode is valid according to
      the authentication service and the client's local policy.  These
      actions MUST be the same regardless of at what point in the
      protocol the LeafNode is being verified with the following
      exception: If the LeafNode is an update to another LeafNode, the
      authentication service MUST additionally validate that the set of
      identities attested by the credential in the new LeafNode is
      acceptable relative to the identities attested by the old
      credential.

   *  Verify that the signature on the LeafNode is valid using the
      public key in the LeafNode's credential

   *  Verify that the LeafNode is compatible with the group's
      parameters.  If the GroupContext has a required_capabilities
      extension, then the required extensions, proposals, and credential
      types MUST be listed in the LeafNode's capabilities field.

   *  Verify that the credential type is supported by all members of the
      group, as specified by the capabilities field of each member's
      LeafNode, and that the capabilities field of this LeafNode
      indicates support for all the credential types currently in use by
      other members.

   *  Verify the lifetime field:

      -  When validating a LeafNode in a KeyPackage before sending an
         Add proposal, the current time MUST be within the lifetime
         range.  A KeyPackage containing a LeafNode that is expired or
         not yet valid MUST NOT be sent in an Add proposal.

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      -  When receiving an Add or validating a tree, checking the
         lifetime is RECOMMENDED, if it is feasible in a given
         application context.  Because of the asynchronous nature of
         MLS, the lifetime may have been valid when the leaf node was
         proposed for addition, even if it is expired at these later
         points in the protocol.

   *  Verify that the leaf_node_source field has the appropriate value
      for the context in which the LeafNode is being validated (as
      defined in Section 8.2).

   *  Verify that the following fields in the LeafNode are unique among
      the members of the group (including any other members added in the
      same Commit):

      -  encryption_key

      -  signature_key

   *  Verify that the extensions in the leaf node are supported.  The ID
      for each extension in the extensions field MUST be listed in the
      field capabilities.extensions of the LeafNode.

8.4.  Ratchet Tree Evolution

   In order to provide forward secrecy and post-compromise security,
   whenever a member initiates an epoch change (i.e., commits; see
   Section 13.2), they refresh the key pairs of their leaf and of all
   nodes on their leaf's direct path (all nodes for which they know the
   secret key).

   The member initiating the epoch change generates the fresh key pairs
   using the following procedure.  The procedure is designed in a way
   that allows group members to efficiently communicate the fresh secret
   keys to other group members, as described in Section 8.9.

   Recall the definition of resolution from Section 5.3.  To begin with,
   the generator of the UpdatePath updates its leaf and its leaf's
   _filtered direct path_ with new key pairs.  The filtered direct path
   of a node is obtained from the node's direct path by removing all
   nodes whose child on the nodes's copath has an empty resolution (any
   unmerged leaves of the copath child count towards its resolution).
   Such a removed node does not need a key pair, since after blanking
   it, its resolution consists of a single node on the filtered direct
   path.  Using the key pair of the node in the resolution is equivalent
   to using the key pair of the removed node.

   *  Blank all the nodes on the direct path from the leaf to the root.

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   *  Generate a fresh HPKE key pair for the leaf.

   *  Generate a sequence of path secrets, one for each node on the
      leaf's filtered direct path, as follows.  In this setting,
      path_secret[0] refers to the first parent node in the filtered
      direct path, path_secret[1] to the second parent node, and so on.

   path_secret[0] is sampled at random
   path_secret[n] = DeriveSecret(path_secret[n-1], "path")

   *  Compute the sequence of HPKE key pairs (node_priv,node_pub), one
      for each node on the leaf's direct path, as follows.

   node_secret[n] = DeriveSecret(path_secret[n], "node")
   node_priv[n], node_pub[n] = KEM.DeriveKeyPair(node_secret[n])

   The node secret is derived as a temporary intermediate secret so that
   each secret is only used with one algorithm: The path secret is used
   as an input to DeriveSecret and the node secret is used as an input
   to DeriveKeyPair.

   For example, suppose there is a group with four members, with C an
   unmerged leaf at Z:

         Y
         |
       .-+-.
      /     \
     X       Z[C]
    / \     / \
   A   B   C   D

   0   1   2   3

               Figure 11: A full tree with one unmerged leaf

   If member B subsequently generates an UpdatePath based on a secret
   "leaf_secret", then it would generate the following sequence of path
   secrets:

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   path_secret[1] ---> node_secret[1] -------> node_priv[1], node_pub[1]

        ^
        |
        |
   path_secret[0] ---> node_secret[0] -------> node_priv[0], node_pub[0]

        ^
        |
        |
   leaf_secret ------> leaf_node_secret --+--> leaf_priv, leaf_pub
                                              |                   |
                                               '-------. .-------'
                                                        |
                                                    leaf_node

   After applying the UpdatePath, the tree will have the following
   structure, where lp and np[i] represent the leaf_priv and node_priv
   values generated as described above:

   node_priv[1] --------> Y'
                          |
                        .-+-.
                       /     \
   node_priv[0] ----> X'      Z[C]
                     / \     / \
                    A   B   C   D
                        ^
   leaf_priv -----------+
                    0   1   2   3

8.5.  Adding and Removing Leaves

   In addition to the path-based updates to the tree described above, it
   is also necessary to add and remove leaves of the tree in order to
   reflect changes to the membership of the group (see Section 13.1.1
   and Section 13.1.3).  Leaves are always added and removed at the
   right edge of the tree: Either a new rightmost leaf is added, or the
   rightmost leaf is removed.  Nodes' parent/child node relationships
   are then updated to maintain the tree's left-balanced structure.
   These operations are also known as _extending_ and _truncating_ the
   tree.

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   To add a new leaf: Add leaf L as the new rightmost leaf of the tree.
   Add a blank parent node P whose right child is L.  P is attached to
   the tree as the right child of the only appropriate node to make the
   updated tree left-balanced (or set it as a new root).  The former
   right child of P's parent becomes P's left child (or the old root
   becomes P's left child if P is the new root).

                      _ <-- new parent              _
                    __|_                          __|__
                   /    \                        /     \
     X    ===>    X     |               ===>    X       _ <-- new parent
    / \          / \    |                      / \     / \
   A   B        A   B   C <-- new leaf        A   B   C   D <-- new leaf

   To remove the rightmost leaf: Remove the rightmost leaf node L and
   its parent node P.  If P was the root of the tree, P's left child is
   now the root of the tree.  Otherwise, set the right child of P's
   parent to be P's left child.

         Y                                    Y
       __|__                                __|_
      /     \                              /    \
     X       Z <-- remove parent  ===>    X     | <-- reassign child
    / \     / \                          / \    |
   A   B   C   D <-- remove leaf        A   B   C

         Y <-- remove parent
       __|_
      /    \
     X     |                  ===>    X <-- reassign root
    / \    |                         / \
   A   B   C <-- remove leaf        A   B

   Note that in the rest of the protocol, the rightmost leaf will only
   be removed when it is blank.

   Concrete algorithms for these operations on array-based and link-
   based trees are provided in Appendix B and Appendix C.  The concrete
   algorithms are non-normative.  An implementation MAY use any
   algorithm that produces the correct tree in its internal
   representation.

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8.6.  Synchronizing Views of the Tree

   After generating fresh key material and applying it to ratchet
   forward their local tree state as described in the Section 8.4, the
   generator must broadcast this update to other members of the group in
   a Commit message, who apply it to keep their local views of the tree
   in sync with the sender's.  More specifically, when a member commits
   a change to the tree (e.g., to add or remove a member), it transmits
   an UpdatePath containing a set of public keys and encrypted path
   secrets for intermediate nodes in the filtered direct path of its
   leaf.  The other members of the group use these values to update
   their view of the tree, aligning their copy of the tree to the
   sender's.

   An UpdatePath contains the following information for each node in the
   filtered direct path of the sender's leaf, including the root:

   *  The public key for the node

   *  Zero or more encrypted copies of the path secret corresponding to
      the node

   The path secret value for a given node is encrypted for the subtree
   corresponding to the parent's non-updated child, that is, the child
   on the copath of the sender's leaf node.  There is one encryption of
   the path secret to each public key in the resolution of the non-
   updated child.

   The recipient of an UpdatePath processes it with the following steps:

   1.  Compute the updated path secrets.

       *  Identify a node in the filtered direct path for which the
          recipient is in the subtree of the non-updated child.

       *  Identify a node in the resolution of the copath node for which
          the recipient has a private key.

       *  Decrypt the path secret for the parent of the copath node
          using the private key from the resolution node.

       *  Derive path secrets for ancestors of that node using the
          algorithm described above.

       *  The recipient SHOULD verify that the received public keys
          agree with the public keys derived from the new path_secret
          values.

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   2.  Merge the updated path secrets into the tree.

       *  Blank all nodes on the direct path of the sender's leaf.

       *  For all nodes on the filtered direct path of the sender's
          leaf,

          -  Set the public key to the received public key.

          -  Set the list of unmerged leaves to the empty list.

          -  Store the updated hash of the next node on the filtered
             direct path (represented as a ParentNode struct), going
             from root to leaf, so that each hash incorporates all the
             non-blank nodes above it.  The root node always has a zero-
             length hash for this value.

       *  For nodes where a path secret was recovered in step 1
          ("Compute the updated path secrets"), compute and store the
          node's updated private key.

   For example, in order to communicate the example update described in
   the previous section, the sender would transmit the following values:

   +=============+====================================================+
   | Public Key  | Ciphertext(s)                                      |
   +=============+====================================================+
   | node_pub[1] | E(pk(Z), path_secret[1]), E(pk(C), path_secret[1]) |
   +-------------+----------------------------------------------------+
   | node_pub[0] | E(pk(A), path_secret[0])                           |
   +-------------+----------------------------------------------------+

                                 Table 2

   In this table, the value node_pub[i] represents the public key
   derived from node_secret[i], pk(X) represents the current public key
   of node X, and E(K, S) represents the public-key encryption of the
   path secret S to the public key K (using HPKE).

   After processing the update, each recipient MUST delete outdated key
   material, specifically:

   *  The path secrets used to derive each updated node key pair.

   *  Each outdated node key pair that was replaced by the update.

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8.7.  Tree Hashes

   To allow group members to verify that they agree on the public
   cryptographic state of the group, this section defines a scheme for
   generating a hash value (called the "tree hash") that represents the
   contents of the group's ratchet tree and the members' leaf nodes.
   The tree hash of a tree is the tree hash of its root node, which we
   define recursively, starting with the leaves.

   The tree hash of a leaf node is the hash of leaf's LeafNodeHashInput
   object which might include a LeafNode object depending on whether or
   not it is blank.

   struct {
       uint32 leaf_index;
       optional<LeafNode> leaf_node;
   } LeafNodeHashInput;

   Now the tree hash of any non-leaf node is recursively defined to be
   the hash of its ParentNodeHashInput.  This includes an optional
   ParentNode object depending on whether the node is blank or not.

   struct {
       optional<ParentNode> parent_node;
       opaque left_hash<V>;
       opaque right_hash<V>;
   } ParentNodeHashInput;

   The left_hash and right_hash fields hold the tree hashes of the
   node's left and right children, respectively.

8.8.  Parent Hash

   The parent_hash field in ratchet tree nodes carries information to
   authenticate the information in the ratchet tree.  Parent hashes
   chain together so that the signature on a leaf node, by covering the
   leaf node's parent hash, indirectly includes information about the
   structure of the tree at the time the leaf node was last updated.

   Consider a ratchet tree with a non-blank parent node P and children V
   and S.

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           ...
           /
          P
        __|__
       /     \
      V       S
     / \     / \
   ... ... ... ...

   The parent hash of P changes whenever an UpdatePath object is applied
   to the ratchet tree along a path from a leaf U traversing node V (and
   hence also P).  The new "Parent hash of P (with copath child S)" is
   obtained by hashing P's ParentHashInput struct.

   struct {
       HPKEPublicKey encryption_key;
       opaque parent_hash<V>;
       opaque original_sibling_tree_hash<V>;
   } ParentHashInput;

   The field encryption_key contains the HPKE public key of P.  If P is
   the root, then the parent_hash field is set to a zero-length octet
   string.  Otherwise, parent_hash is the Parent Hash of the next node
   after P on the filtered direct path of U.  This way, P's Parent Hash
   fixes the new HPKE public key of each node V on the path from P to
   the root.  Note that the path from P to the root may contain some
   blank nodes that are not fixed by P's Parent Hash.  However, for each
   node that has an HPKE key, this key is fixed by P's Parent Hash.

   Finally, original_sibling_tree_hash is the tree hash of S in the
   ratchet tree modified as follows:

   *  Extend the subtree of S by adding blank leaves until it is full,
      i.e., until its number of leaves is a power of 2 (see
      Section 8.5).

   *  For each leaf L in P.unmerged_leaves, blank L and remove it from
      the unmerged_leaves sets of all parent nodes.

   Observe that original_sibling_tree_hash does not change between
   updates of P.  This property is crucial for the correctness of the
   protocol.

   For example, in the following tree:

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                 W [F]
           ______|_____
          /             \
         U               Y [F]
       __|__           __|__
      /     \         /     \
     T       _       _      |
    / \     / \     / \     |
   A   B   C   D   E   F    G

          Figure 12: A tree illustrating parent hash computations.

   With P = W and S = Y, original_sibling_tree_hash is the tree hash of
   the following tree:

         Y
       __|__
      /     \
     _       _
    / \     / \
   E   _   G   _

   Because W.unmerged_leaves includes F, F is blanked and removed from
   Y.unmerged_leaves.

   Note that no recomputation is needed if the tree hash of S is
   unchanged since the last time P was updated.  This is the case for
   computing or processing a Commit whose UpdatePath traverses P, since
   the Commit itself resets P.  (In other words, it is only necessary to
   recompute the original sibling tree hash when validating group's tree
   on joining.)  More generally, if none of the entries in
   P.unmerged_leaves is in the subtree under S (and thus no nodes were
   truncated), then the original tree hash at S is the tree hash of S in
   the current tree.

   If it is necessary to recompute the original tree hash of a node, the
   efficiency of recomputation can be improved by caching intermediate
   tree hashes, to avoid recomputing over the subtree when the subtree
   is included in multiple parent hashes.  A subtree hash can be reused
   as long as the intersection of the parent's unmerged leaves with the
   subtree is the same as in the earlier computation.

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8.8.1.  Using Parent Hashes

   The Parent Hash of P appears in three types of structs.  If V is
   itself a parent node then P's Parent Hash is stored in the
   parent_hash field of the structs ParentHashInput and ParentNode of
   the node before P on the filtered direct path of U.  (The ParentNode
   struct is used to encapsulate all public information about that node
   that must be conveyed to a new member joining the group as well as to
   define its Tree Hash.)

   If, on the other hand, V is the leaf U and its LeafNode has
   leaf_node_source set to commit, then the Parent Hash of P (with V's
   sibling as copath child) is stored in the parent_hash field.  This is
   true in particular of the LeafNode object sent in the leaf_node field
   of an UpdatePath.  The signature of such a LeafNode thus also attests
   to which keys the group member introduced into the ratchet tree and
   to whom the corresponding secret keys were sent.  This helps prevent
   malicious insiders from constructing artificial ratchet trees with a
   node V whose HPKE secret key is known to the insider yet where the
   insider isn't assigned a leaf in the subtree rooted at V.  Indeed,
   such a ratchet tree would violate the tree invariant.

8.8.2.  Verifying Parent Hashes

   Parent hashes are verified at two points in the protocol: When
   joining a group and when processing a Commit.

   The parent hash in a node U is valid with respect to a parent node P
   if the following criteria hold:

   *  U is a descendant of P in the tree

   *  The nodes between U and P in the tree are all blank

   *  The parent_hash field of U is equal to the parent hash of P with
      copath child S, where S is the child of P that is not on the path
      from U to P.

   A parent node P is "parent-hash valid" if it can be chained back to a
   leaf node in this way.  That is, if there is leaf node L and a
   sequence of parent nodes P_1, ..., P_N such that P_N = P and each
   step in the chain is authenticated by a parent hash: L's parent hash
   is valid with respect to P_1, P_1's parent hash is valid with respect
   to P_2, and so on.

   When joining a group, the new member MUST authenticate that each non-
   blank parent node P is parent-hash valid.  This can be done "bottom
   up" by building chains up from leaves and verifying that all non-

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   blank parent nodes are covered by exactly one such chain, or "top
   down" by verifying that there is exactly one descendant of each non-
   blank parent node for which the parent node is parent-hash valid.

   When processing a Commit message that includes an UpdatePath, clients
   MUST recompute the expected value of parent_hash for the committer's
   new leaf and verify that it matches the parent_hash value in the
   supplied leaf_node.  After being merged into the tree, the nodes in
   the UpdatePath form a parent-hash chain from the committer's leaf to
   the root.

8.9.  Update Paths

   As described in Section 13.2, each MLS Commit message may optionally
   transmit a LeafNode and parent node values along its direct path.
   The path contains a public key and encrypted secret value for all
   intermediate nodes in the filtered direct path from the leaf to the
   root.  The path is ordered from the closest node to the leaf to the
   root; each node MUST be the parent of its predecessor.

   struct {
       opaque kem_output<V>;
       opaque ciphertext<V>;
   } HPKECiphertext;

   struct {
       HPKEPublicKey encryption_key;
       HPKECiphertext encrypted_path_secret<V>;
   } UpdatePathNode;

   struct {
       LeafNode leaf_node;
       UpdatePathNode nodes<V>;
   } UpdatePath;

   For each UpdatePathNode, the resolution of the corresponding copath
   node MUST be filtered by removing all new leaf nodes added as part of
   this MLS Commit message.  The number of ciphertexts in the
   encrypted_path_secret vector MUST be equal to the length of the
   filtered resolution, with each ciphertext being the encryption to the
   respective resolution node.

   The HPKECiphertext values are computed as

   kem_output, context = SetupBaseS(node_public_key, group_context)
   ciphertext = context.Seal("", path_secret)

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   where node_public_key is the public key of the node that the path
   secret is being encrypted for, group_context is the current
   GroupContext object for the group, and the functions SetupBaseS and
   Seal are defined according to [RFC9180].

   Decryption is performed in the corresponding way, using the private
   key of the resolution node.

9.  Key Schedule

   Group keys are derived using the Extract and Expand functions from
   the KDF for the group's ciphersuite, as well as the functions defined
   below:

   ExpandWithLabel(Secret, Label, Context, Length) =
       KDF.Expand(Secret, KDFLabel, Length)

   DeriveSecret(Secret, Label) =
       ExpandWithLabel(Secret, Label, "", KDF.Nh)

   Where KDFLabel is specified as:

   struct {
       uint16 length = Length;
       opaque label<V> = "MLS 1.0 " + Label;
       opaque context<V> = Context;
   } KDFLabel;

   The value KDF.Nh is the size of an output from KDF.Extract, in bytes.
   In the below diagram:

   *  KDF.Extract takes its salt argument from the top and its Input Key
      Material (IKM) argument from the left

   *  DeriveSecret takes its Secret argument from the incoming arrow

   *  0 represents an all-zero byte string of length KDF.Nh.

   When processing a handshake message, a client combines the following
   information to derive new epoch secrets:

   *  The init secret from the previous epoch

   *  The commit secret for the current epoch

   *  The GroupContext object for current epoch

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   Given these inputs, the derivation of secrets for an epoch proceeds
   as shown in the following diagram:

                    init_secret_[n-1]
                          |
                          |
                          V
    commit_secret --> KDF.Extract
                          |
                          |
                          V
                  ExpandWithLabel(., "joiner", GroupContext_[n], KDF.Nh)
                          |
                          |
                          V
                     joiner_secret
                          |
                          |
                          V
psk_secret (or 0) --> KDF.Extract
                          |
                          |
                          +--> DeriveSecret(., "welcome")
                          |    = welcome_secret
                          |
                          V
                  ExpandWithLabel(., "epoch", GroupContext_[n], KDF.Nh)
                          |
                          |
                          V
                     epoch_secret
                          |
                          |
                          +--> DeriveSecret(., <label>)
                          |    = <secret>
                          |
                          V
                    DeriveSecret(., "init")
                          |
                          |
                          V
                    init_secret_[n]

   A number of values are derived from the epoch secret for different
   purposes:

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    +==================+=====================+=======================+
    | Label            | Secret              | Purpose               |
    +==================+=====================+=======================+
    | "sender data"    | sender_data_secret  | Deriving keys to      |
    |                  |                     | encrypt sender data   |
    +------------------+---------------------+-----------------------+
    | "encryption"     | encryption_secret   | Deriving message      |
    |                  |                     | encryption keys (via  |
    |                  |                     | the secret tree)      |
    +------------------+---------------------+-----------------------+
    | "exporter"       | exporter_secret     | Deriving exported     |
    |                  |                     | secrets               |
    +------------------+---------------------+-----------------------+
    | "external"       | external_secret     | Deriving the external |
    |                  |                     | init key              |
    +------------------+---------------------+-----------------------+
    | "confirm"        | confirmation_key    | Computing the         |
    |                  |                     | confirmation MAC for  |
    |                  |                     | an epoch              |
    +------------------+---------------------+-----------------------+
    | "membership"     | membership_key      | Computing the         |
    |                  |                     | membership MAC for an |
    |                  |                     | MLSPlaintext          |
    +------------------+---------------------+-----------------------+
    | "resumption"     | resumption_psk      | Proving membership in |
    |                  |                     | this epoch (via a PSK |
    |                  |                     | injected later)       |
    +------------------+---------------------+-----------------------+
    | "authentication" | epoch_authenticator | Confirming that two   |
    |                  |                     | clients have the same |
    |                  |                     | view of the group     |
    +------------------+---------------------+-----------------------+

                      Table 3: Epoch-derived secrets

   The external_secret is used to derive an HPKE key pair whose private
   key is held by the entire group:

   external_priv, external_pub = KEM.DeriveKeyPair(external_secret)

   The public key external_pub can be published as part of the GroupInfo
   struct in order to allow non-members to join the group using an
   external commit.

9.1.  Group Context

   Each member of the group maintains a GroupContext object that
   summarizes the state of the group:

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   struct {
       opaque group_id<V>;
       uint64 epoch;
       opaque tree_hash<V>;
       opaque confirmed_transcript_hash<V>;
       Extension extensions<V>;
   } GroupContext;

   The fields in this state have the following semantics:

   *  The group_id field is an application-defined identifier for the
      group.

   *  The epoch field represents the current version of the group.

   *  The tree_hash field contains a commitment to the contents of the
      group's ratchet tree and the credentials for the members of the
      group, as described in Section 8.7.

   *  The confirmed_transcript_hash field contains a running hash over
      the messages that led to this state.

   *  The extensions field contains the details of any protocol
      extensions that apply to the group.

   When a new member is added to the group, an existing member of the
   group provides the new member with a Welcome message.  The Welcome
   message provides the information the new member needs to initialize
   its GroupContext.

   Different changes to the group will have different effects on the
   group state.  These effects are described in their respective
   subsections of Section 13.1.  The following general rules apply:

   *  The group_id field is constant.

   *  The epoch field increments by one for each Commit message that is
      processed.

   *  The tree_hash is updated to represent the current tree and
      credentials.

   *  The confirmed_transcript_hash field is updated with the data for
      an MLSPlaintext message encoding a Commit message as described
      below.

   *  The extensions field changes when a GroupContextExtensions
      proposal is committed.

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9.2.  Transcript Hashes

   The transcript hashes computed in MLS represent a running hash over
   all Proposal and Commit messages that have ever been sent in a group.
   Commit messages are included directly.  Proposal messages are
   indirectly included via the Commit that applied them.  Both types of
   message are included by hashing the MLSPlaintext in which they were
   sent.

   The confirmed_transcript_hash is updated with an MLSMessageContent
   and MLSMessageAuth containing a Commit in two steps:

   struct {
       WireFormat wire_format;
       MLSMessageContent content; //with content.content_type == commit
       opaque signature<V>;
   } MLSMessageCommitContent;

   struct {
       MAC confirmation_tag;
   } MLSMessageCommitAuthData;

   interim_transcript_hash_[0] = ""; // zero-length octet string

   confirmed_transcript_hash_[n] =
       Hash(interim_transcript_hash_[n] ||
           MLSMessageCommitContent_[n]);

   interim_transcript_hash_[n+1] =
       Hash(confirmed_transcript_hash_[n] ||
           MLSMessageCommitAuthData_[n]);

   Thus the confirmed_transcript_hash field in a GroupContext object
   represents a transcript over the whole history of MLSMessage Commit
   messages, up to the confirmation_tag field of the most recent Commit.
   The confirmation tag is then included in the transcript for the next
   epoch.  The interim transcript hash is computed by new members using
   the confirmation_tag of the GroupInfo struct, while existing members
   can compute it directly.

   As shown above, when a new group is created, the
   interim_transcript_hash field is set to the zero-length octet string.

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9.3.  External Initialization

   In addition to initializing a new epoch via KDF invocations as
   described above, an MLS group can also initialize a new epoch via an
   asymmetric interaction using the external key pair for the previous
   epoch.  This is done when an new member is joining via an external
   commit.

   In this process, the joiner sends a new init_secret value to the
   group using the HPKE export method.  The joiner then uses that
   init_secret with information provided in the GroupInfo and an
   external Commit to initialize their copy of the key schedule for the
   new epoch.

   kem_output, context = SetupBaseS(external_pub, "")
   init_secret = context.export("MLS 1.0 external init secret", KDF.Nh)

   Members of the group receive the kem_output in an ExternalInit
   proposal and preform the corresponding calculation to retrieve the
   init_secret value.

   context = SetupBaseR(kem_output, external_priv, "")
   init_secret = context.export("MLS 1.0 external init secret", KDF.Nh)

   In both cases, the info input to HPKE is set to the GroupInfo for the
   previous epoch, encoded using the TLS serialization.

9.4.  Pre-Shared Keys

   Groups which already have an out-of-band mechanism to generate shared
   group secrets can inject those into the MLS key schedule to seed the
   MLS group secrets computations by this external entropy.

   Injecting an external PSK can improve security in the case where
   having a full run of updates across members is too expensive, or if
   the external group key establishment mechanism provides stronger
   security against classical or quantum adversaries.

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   Note that, as a PSK may have a different lifetime than an update, it
   does not necessarily provide the same Forward Secrecy (FS) or Post-
   Compromise Security (PCS) guarantees as a Commit message.  Unlike the
   key pairs populated in the tree by an Update or Commit, which are
   always freshly generated, PSKs may be pre-distributed and stored.
   This creates the risk that a PSK may be compromised in the process of
   distribution and storage.  The security that the group gets from
   injecting a PSK thus depends on both the entropy of the PSK and the
   risk of compromise.  These factors are outside of the scope of this
   document, but should be considered by application designers relying
   on PSKs.

   Each PSK in MLS has a type that designates how it was provisioned.
   External PSKs are provided by the application, while resumption PSKs
   are derived from the MLS key schedule and used in cases where it is
   necessary to authenticate a member's participation in a prior epoch.

   The injection of one or more PSKs into the key schedule is signaled
   in two ways: Existing members are informed via PreSharedKey proposals
   covered by a Commit, and new members added in the Commit are informed
   via GroupSecrets object in the Welcome message corresponding to the
   Commit.  To ensure that existing and new members compute the same PSK
   input to the key schedule, the Commit and GroupSecrets objects MUST
   indicate the same set of PSKs, in the same order.

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   enum {
     reserved(0),
     external(1),
     resumption(2),
     (255)
   } PSKType;

   enum {
     reserved(0),
     application(1),
     reinit(2),
     branch(3),
   } ResumptionPSKUsage;

   struct {
     PSKType psktype;
     select (PreSharedKeyID.psktype) {
       case external:
         opaque psk_id<V>;

       case resumption:
         ResumptionPSKUsage usage;
         opaque psk_group_id<V>;
         uint64 psk_epoch;
     }
     opaque psk_nonce<V>;
   } PreSharedKeyID;

   struct {
       PreSharedKeyID psks<V>;
   } PreSharedKeys;

   On receiving a Commit with a PreSharedKey proposal or a GroupSecrets
   object with the psks field set, the receiving Client includes them in
   the key schedule in the order listed in the Commit, or in the psks
   field respectively.  For resumption PSKs, the PSK is defined as the
   resumption_psk of the group and epoch specified in the PreSharedKeyID
   object.  Specifically, psk_secret is computed as follows:

   struct {
       PreSharedKeyID id;
       uint16 index;
       uint16 count;
   } PSKLabel;

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   psk_extracted_[i] = KDF.Extract(0, psk_[i])
   psk_input_[i] = ExpandWithLabel(psk_extracted_[i], "derived psk",
                     PSKLabel, KDF.Nh)

   psk_secret_[0] = 0
   psk_secret_[i] = KDF.Extract(psk_input_[i-1], psk_secret_[i-1])
   psk_secret     = psk_secret_[n]

   Here 0 represents the all-zero vector of length KDF.Nh.  The index
   field in PSKLabel corresponds to the index of the PSK in the psk
   array, while the count field contains the total number of PSKs.  In
   other words, the PSKs are chained together with KDF.Extract
   invocations (labelled "Extract" for brevity in the diagram), as
   follows:

                   0                               0    = psk_secret_[0]
                   |                               |
                   V                               V
  psk_[0]   --> Extract --> ExpandWithLabel --> Extract = psk_secret_[1]
                                                   |
                   0                               |
                   |                               |
                   V                               V
  psk_[1]   --> Extract --> ExpandWithLabel --> Extract = psk_secret_[2]
                                                   |
                   0                              ...
                   |                               |
                   V                               V
  psk_[n-1] --> Extract --> ExpandWithLabel --> Extract = psk_secret_[n]

   In particular, if there are no PreSharedKey proposals in a given
   Commit, then the resulting psk_secret is psk_secret_[0], the all-zero
   vector.

9.5.  Exporters

   The main MLS key schedule provides an exporter_secret which can be
   used by an application as the basis to derive new secrets called
   exported_value outside the MLS layer.

   MLS-Exporter(Label, Context, key_length) =
          ExpandWithLabel(DeriveSecret(exporter_secret, Label),
                            "exporter", Hash(Context), key_length)

   Each application SHOULD provide a unique label to MLS-Exporter that
   identifies its use case.  This is to prevent two exported outputs
   from being generated with the same values and used for different
   functionalities.

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   The exported values are bound to the group epoch from which the
   exporter_secret is derived, hence reflects a particular state of the
   group.

   It is RECOMMENDED for the application generating exported values to
   refresh those values after a Commit is processed.

9.6.  Resumption PSK

   The main MLS key schedule provides a resumption_psk that is used as a
   PSK to inject entropy from one epoch into another.  This
   functionality is used in the reinitialization and branching processes
   described in Section 12.2 and Section 12.3, but may be used by
   applications for other purposes.

   Some uses of resumption PSKs might call for the use of PSKs from
   historical epochs.  The application SHOULD specify an upper limit on
   the number of past epochs for which the resumption_psk may be stored.

9.7.  Epoch Authenticators

   The main MLS key schedule provides a per-epoch epoch_authenticator.
   If one member of the group is being impersonated by an active
   attacker, the epoch_authenticator computed by their client will
   differ from those computed by the other group members.

   This property can be used to construct defenses against impersonation
   attacks that are effective even if members' signature keys are
   compromised.  As a trivial example, if the users of the clients in an
   MLS group were to meet in person and reliably confirm that their
   epoch authenticator values were equal (using some suitable user
   interface), then each user would be assured that the others were not
   being impersonated in the current epoch.  As soon as the epoch
   changed, though, they would need to re-do this confirmation.  The
   state of the group would have changed, possibly introducing an
   attacker.

   More generally, in order for the members of an MLS group to obtain
   concrete authentication protections using the epoch_authenticator,
   they will need to use it in some secondary protocol (such as the
   face-to-face protocol above).  The details of that protocol will then
   determine the specific authentication protections provided to the MLS
   group.

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10.  Secret Tree

   For the generation of encryption keys and nonces, the key schedule
   begins with the encryption_secret at the root and derives a tree of
   secrets with the same structure as the group's ratchet tree.  Each
   leaf in the Secret Tree is associated with the same group member as
   the corresponding leaf in the ratchet tree.

   If N is a parent node in the Secret Tree then the secrets of the
   children of N are defined as follows (where left(N) and right(N)
   denote the children of N):

   tree_node_[N]_secret
           |
           |
           +--> ExpandWithLabel(., "tree", "left", KDF.Nh)
           |    = tree_node_[left(N)]_secret
           |
           +--> ExpandWithLabel(., "tree", "right", KDF.Nh)
                = tree_node_[right(N)]_secret

   The secret in the leaf of the Secret Tree is used to initiate two
   symmetric hash ratchets, from which a sequence of single-use keys and
   nonces are derived, as described in Section 10.1.  The root of each
   ratchet is computed as:

   tree_node_[N]_secret
           |
           |
           +--> ExpandWithLabel(., "handshake", "", KDF.Nh)
           |    = handshake_ratchet_secret_[N]_[0]
           |
           +--> ExpandWithLabel(., "application", "", KDF.Nh)
                = application_ratchet_secret_[N]_[0]

10.1.  Encryption Keys

   As described in Section 7, MLS encrypts three different types of
   information:

   *  Metadata (sender information)

   *  Handshake messages (Proposal and Commit)

   *  Application messages

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   The sender information used to look up the key for content encryption
   is encrypted with an AEAD where the key and nonce are derived from
   both sender_data_secret and a sample of the encrypted message
   content.

   For handshake and application messages, a sequence of keys is derived
   via a "sender ratchet".  Each sender has their own sender ratchet,
   and each step along the ratchet is called a "generation".

   A sender ratchet starts from a per-sender base secret derived from a
   Secret Tree, as described in Section 10.  The base secret initiates a
   symmetric hash ratchet which generates a sequence of keys and nonces.
   The sender uses the j-th key/nonce pair in the sequence to encrypt
   (using the AEAD) the j-th message they send during that epoch.  Each
   key/nonce pair MUST NOT be used to encrypt more than one message.

   Keys, nonces, and the secrets in ratchets are derived using
   DeriveTreeSecret.  The context in a given call consists of the
   current position in the ratchet.

   DeriveTreeSecret(Secret, Label, Generation, Length) =
       ExpandWithLabel(Secret, Label, Generation, Length)

   Where Generation is encoded as a uint32.

   ratchet_secret_[N]_[j]
         |
         +--> DeriveTreeSecret(., "nonce", j, AEAD.Nn)
         |    = ratchet_nonce_[N]_[j]
         |
         +--> DeriveTreeSecret(., "key", j,  AEAD.Nk)
         |    = ratchet_key_[N]_[j]
         |
         V
   DeriveTreeSecret(., "secret", j, KDF.Nh)
   = ratchet_secret_[N]_[j+1]

   Here, AEAD.Nn and AEAD.Nk denote the lengths in bytes of the nonce
   and key for the AEAD scheme defined by the ciphersuite.

10.2.  Deletion Schedule

   It is important to delete all security-sensitive values as soon as
   they are _consumed_. A sensitive value S is said to be _consumed_ if

   *  S was used to encrypt or (successfully) decrypt a message, or if

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   *  a key, nonce, or secret derived from S has been consumed.  (This
      goes for values derived via DeriveSecret as well as
      ExpandWithLabel.)

   Here, S may be the init_secret, commit_secret, epoch_secret,
   encryption_secret as well as any secret in a Secret Tree or one of
   the ratchets.

   As soon as a group member consumes a value they MUST immediately
   delete (all representations of) that value.  This is crucial to
   ensuring forward secrecy for past messages.  Members MAY keep
   unconsumed values around for some reasonable amount of time to handle
   out-of-order message delivery.

   For example, suppose a group member encrypts or (successfully)
   decrypts an application message using the j-th key and nonce in the
   ratchet of leaf node L in some epoch n.  Then, for that member, at
   least the following values have been consumed and MUST be deleted:

   *  the commit_secret, joiner_secret, epoch_secret, encryption_secret
      of that epoch n as well as the init_secret of the previous epoch
      n-1,

   *  all node secrets in the Secret Tree on the path from the root to
      the leaf with node L,

   *  the first j secrets in the application data ratchet of node L and

   *  application_ratchet_nonce_[L]_[j] and
      application_ratchet_key_[L]_[j].

   Concretely, suppose we have the following Secret Tree and ratchet for
   participant D:

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          G
          |
        .-+-.
       /     \
      E       F
     / \     / \
    A   B   C   D
               / \
             HR0  AR0--+--K0
                   |   |
                   |   +--N0
                   |
                  AR1--+--K1
                   |   |
                   |   +--N1
                   |
                  AR2

   Then if a client uses key K1 and nonce N1 during epoch n then it must
   consume (at least) values G, F, D, AR0, AR1, K1, N1 as well as the
   key schedule secrets used to derive G (the encryption_secret), namely
   init_secret of epoch n-1 and commit_secret, joiner_secret,
   epoch_secret of epoch n.  The client MAY retain (not consume) the
   values K0 and N0 to allow for out-of-order delivery, and SHOULD
   retain AR2 for processing future messages.

11.  Key Packages

   In order to facilitate the asynchronous addition of clients to a
   group, key packages are pre-published that provide some public
   information about a user.  A KeyPackage object specifies:

   1.  A protocol version and ciphersuite that the client supports,

   2.  a public key that others can use to encrypt a Welcome message to
       this client, (an "init key") and

   3.  the content of the leaf node that should be added to the tree to
       represent this client.

   KeyPackages are intended to be used only once and SHOULD NOT be
   reused except in case of last resort.  (See Section 17.4).  Clients
   MAY generate and publish multiple KeyPackages to support multiple
   ciphersuites.

   The value for init_key MUST be a public key for the asymmetric
   encryption scheme defined by cipher_suite, and it MUST be unique
   among the set of KeyPackages created by this client.  Likewise, the

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   leaf_node field MUST be valid for the ciphersuite, including both the
   encryption_key and signature_key fields.  The whole structure is
   signed using the client's signature key.  A KeyPackage object with an
   invalid signature field MUST be considered malformed.

   The signature is computed by the function SignWithLabel with a label
   KeyPackage and a content comprising of all of the fields except for
   the signature field.

   struct {
       ProtocolVersion version;
       CipherSuite cipher_suite;
       HPKEPublicKey init_key;
       LeafNode leaf_node;
       Extension extensions<V>;
       // SignWithLabel(., "KeyPackageTBS", KeyPackageTBS)
       opaque signature<V>;
   } KeyPackage;

   struct {
       ProtocolVersion version;
       CipherSuite cipher_suite;
       HPKEPublicKey init_key;
       LeafNode leaf_node;
       Extension extensions<V>;
   } KeyPackageTBS;

   If a client receives a KeyPackage carried within an MLSMessage
   object, then it MUST verify that the version field of the KeyPackage
   has the same value as the version field of the MLSMessage.  The
   version field in the KeyPackage provides an explicit signal of the
   intended version to the other members of group when they receive the
   KeyPackage in an Add proposal.

   The field leaf_node.capabilities indicates what protocol versions,
   ciphersuites, credential types, and non-default proposal/extension
   types are supported by the client.  (Proposal and extension types
   defined in this document are considered "default" and not listed.)
   This information allows MLS session establishment to be safe from
   downgrade attacks on the parameters described (as discussed in
   Section 12), while still only advertising one version / ciphersuite
   per KeyPackage.

   The field leaf_node.leaf_node_source of the LeafNode in a KeyPackage
   MUST be set to key_package.

   Extension included in the extensions or leaf_node.extensions fields
   MUST be included in the leaf_node.capabilities field.

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11.1.  KeyPackage Validation

   The validity of a KeyPackage needs to be verified at a few stages:

   *  When a KeyPackage is downloaded by a group member, before it is
      used to add the client to the group

   *  When a KeyPackage is received by a group member in an Add message

   The client verifies the validity of a KeyPackage using the following
   steps:

   *  Verify that the ciphersuite and protocol version of the KeyPackage
      match those in use in the group.

   *  Verify the leaf_node field of the KeyPackage according to the
      process defined in Section 8.3.

   *  Verify that the signature on the KeyPackage is valid using the
      public key in leaf_node.credential.

   *  Verify that the value of leaf_node.encryption_key is different
      from the value of the init_key field.

11.2.  KeyPackage Identifiers

   Within MLS, a KeyPackage is identified by its hash (see, e.g.,
   Section 13.2.3.2).  The external_key_id extension allows applications
   to add an explicit, application-defined identifier to a KeyPackage.

   opaque external_key_id<V>;

12.  Group Creation

   A group is always created with a single member, the "creator".  The
   other members are added when the creator effectively sends itself an
   Add proposal and commits it, then sends the corresponding Welcome
   message to the new participants.  These processes are described in
   detail in Section 13.1.1, Section 13.2, and Section 13.2.3.2.

   The creator of a group MUST take the following steps to initialize
   the group:

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   *  Fetch KeyPackages for the members to be added, and select a
      version and ciphersuite according to the capabilities of the
      members.  To protect against downgrade attacks, the creator MUST
      use the capabilities information in these KeyPackages to verify
      that the chosen version and ciphersuite is the best option
      supported by all members.

   *  Initialize a one-member group with the following initial values:

      -  Ratchet tree: A tree with a single node, a leaf containing an
         HPKE public key and credential for the creator

      -  Group ID: A value set by the creator

      -  Epoch: 0

      -  Tree hash: The root hash of the above ratchet tree

      -  Confirmed transcript hash: The zero-length octet string

      -  Interim transcript hash: The zero-length octet string

      -  Init secret: A fresh random value of size KDF.Nh

      -  Extensions: Any values of the creator's choosing

   *  For each member, construct an Add proposal from the KeyPackage for
      that member (see Section 13.1.1)

   *  Construct a Commit message that commits all of the Add proposals,
      in any order chosen by the creator (see Section 13.2)

   *  Process the Commit message to obtain a new group state (for the
      epoch in which the new members are added) and a Welcome message

   *  Transmit the Welcome message to the other new members

   The recipient of a Welcome message processes it as described in
   Section 13.2.3.2.  If application context informs the recipient that
   the Welcome should reflect the creation of a new group (for example,
   due to a branch or reinitialization), then the recipient MUST verify
   that the epoch value in the GroupInfo is equal to 1.

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   In principle, the above process could be streamlined by having the
   creator directly create a tree and choose a random value for first
   epoch's epoch secret.  We follow the steps above because it removes
   unnecessary choices, by which, for example, bad randomness could be
   introduced.  The only choices the creator makes here are its own
   KeyPackage and the leaf secret from which the Commit is built.

12.1.  Required Capabilities

   The configuration of a group imposes certain requirements on clients
   in the group.  At a minimum, all members of the group need to support
   the ciphersuite and protocol version in use.  Additional requirements
   can be imposed by including a required_capabilities extension in the
   GroupContext.

   struct {
       ExtensionType extension_types<V>;
       ProposalType proposal_types<V>;
       CredentialType credential_types<V>;
   } RequiredCapabilities;

   This extension lists the extensions, proposals, and credential types
   that must be supported by all members of the group.  The "default"
   proposal and extension types defined in this document are assumed to
   be implemented by all clients, and need not be listed in
   RequiredCapabilities in order to be safely used.

   For new members, support for required capabilities is enforced by
   existing members during the application of Add commits.  Existing
   members should of course be in compliance already.  In order to
   ensure this continues to be the case even as the group's extensions
   can be updated, a GroupContextExtensions proposal is invalid if it
   contains a required_capabilities extension that requires non-default
   capabilities not supported by all current members.

12.2.  Reinitialization

   A group may be reinitialized by creating a new group with the same
   membership and different parameters, and linking it to the old group
   via a resumption PSK.  The members of a group reinitialize it using
   the following steps:

   1.  A member of the old group sends a ReInit proposal (see
       Section 13.1.5)

   2.  A member of the old group sends a Commit covering the ReInit
       proposal

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   3.  A member of the old group sends a Welcome message for the new
       group that matches the ReInit

       *  The group_id, version, and cipher_suite fields in the Welcome
          message MUST be the same as the corresponding fields in the
          ReInit proposal.

       *  The epoch in the Welcome message MUST be 1

       *  The Welcome MUST specify a PreSharedKey of type resumption
          with usage reinit.  The group_id must match the old group, and
          the epoch must indicate the epoch after the Commit covering
          the ReInit.

       *  The psk_nonce included in the PreSharedKeyID of the resumption
          PSK MUST be a randomly sampled nonce of length KDF.Nh, for the
          KDF defined by the new group's ciphersuite.

   Note that these three steps may be done by the same group member or
   different members.  For example, if a group member sends a commit
   with an inline ReInit proposal (steps 1 and 2), but then goes
   offline, another group member may send the corresponding Welcome.
   This flexibility avoids situations where a group gets stuck between
   steps 2 and 3.

   Resumption PSKs with usage reinit MUST NOT be used in other contexts.
   A PreSharedKey proposal with type resumption and usage reinit MUST be
   considered invalid.

12.3.  Sub-group Branching

   A new group can be formed from a subset of an existing group's
   members, using the same parameters as the old group.

   A member can create a sub-group by performing the following steps:

   1.  Determine a subset of existing members that should be a part of
       the sub-group.

   2.  Create a new tree for the sub-group by fetching a new KeyPackage
       for each existing member that should be included in the sub-
       group.

   3.  Create a Welcome message that includes a PreSharedKey of type
       resumption with usage branch.

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   A client receiving a Welcome including a PreSharedKey of type
   resumption with usage branch MUST verify that the new group reflects
   a subgroup branched from the referenced group.

   *  The version and ciphersuite values in the Welcome MUST be the same
      as those used by the old group.

   *  Each LeafNode in a new subgroup MUST match some LeafNode in the
      original group.  In this context, a pair of LeafNodes is said to
      "match" if the identifiers presented by their respective
      credentials are considered equivalent by the application.

   In addition, to avoid key re-use, the psk_nonce included in the
   PreSharedKeyID object MUST be a randomly sampled nonce of length
   KDF.Nh.

   Resumption PSKs with usage branch MUST NOT be used in other contexts.
   A PreSharedKey proposal with type resumption and usage branch MUST be
   considered invalid.

13.  Group Evolution

   Over the lifetime of a group, its membership can change, and existing
   members might want to change their keys in order to achieve post-
   compromise security.  In MLS, each such change is accomplished by a
   two-step process:

   1.  A proposal to make the change is broadcast to the group in a
       Proposal message

   2.  A member of the group or a new member broadcasts a Commit message
       that causes one or more proposed changes to enter into effect

   In cases where the Proposal and Commit are sent by the same member,
   these two steps can be combined by sending the proposals in the
   commit.

   The group thus evolves from one cryptographic state to another each
   time a Commit message is sent and processed.  These states are
   referred to as "epochs" and are uniquely identified among states of
   the group by eight-octet epoch values.  When a new group is
   initialized, its initial state epoch is 0x0000000000000000.  Each
   time a state transition occurs, the epoch number is incremented by
   one.

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13.1.  Proposals

   Proposals are included in an MLSMessageContent by way of a Proposal
   structure that indicates their type:

   // See IANA registry for registered values
   uint16 ProposalType;

   struct {
       ProposalType msg_type;
       select (Proposal.msg_type) {
           case add:                      Add;
           case update:                   Update;
           case remove:                   Remove;
           case psk:                      PreSharedKey;
           case reinit:                   ReInit;
           case external_init:            ExternalInit;
           case app_ack:                  AppAck;
           case group_context_extensions: GroupContextExtensions;
       };
   } Proposal;

   On receiving an MLSMessageContent containing a Proposal, a client
   MUST verify the signature inside MLSMessageAuth.  If the signature
   verifies successfully, then the Proposal should be cached in such a
   way that it can be retrieved by hash (as a ProposalOrRef object) in a
   later Commit message.

13.1.1.  Add

   An Add proposal requests that a client with a specified KeyPackage be
   added to the group.  The proposer of the Add MUST verify the validity
   of the KeyPackage, as specified in Section 11.1.

   struct {
       KeyPackage key_package;
   } Add;

   An Add is applied after being included in a Commit message.  The
   position of the Add in the list of proposals determines the leaf node
   where the new member will be added.  For the first Add in the Commit,
   the corresponding new member will be placed in the leftmost empty
   leaf in the tree, for the second Add, the next empty leaf to the
   right, etc.  If no empty leaf exists, the tree is extended to the
   right.

   *  Validate the KeyPackage as specified in Section 11.1.  The
      leaf_node_source field in the LeafNode MUST be set to key_package.

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   *  Identify the leaf L for the new member: if there are empty leaves
      in the tree, L is the leftmost empty leaf.  Otherwise, the tree is
      extended to the right by one leaf node and L is the new leaf.

   *  For each non-blank intermediate node along the path from the leaf
      L to the root, add L's leaf index to the unmerged_leaves list for
      the node.

   *  Set the leaf node L to a new node containing the LeafNode object
      carried in the leaf_node field of the KeyPackage in the Add.

13.1.2.  Update

   An Update proposal is a similar mechanism to Add with the distinction
   that it replaces the sender's LeafNode in the tree instead of adding
   a new leaf to the tree.

   struct {
       LeafNode leaf_node;
   } Update;

   A member of the group applies an Update message by taking the
   following steps:

   *  Validate the LeafNode as specified in Section 8.3.  The
      leaf_node_source field MUST be set to update.

   *  Verify that the encryption_key value in the LeafNode is different
      from the corresponding field in the LeafNode being replaced.

   *  Replace the sender's LeafNode with the one contained in the Update
      proposal

   *  Blank the intermediate nodes along the path from the sender's leaf
      to the root

13.1.3.  Remove

   A Remove proposal requests that the member with LeafNodeRef removed
   be removed from the group.

   struct {
       LeafNodeRef removed;
   } Remove;

   A member of the group applies a Remove message by taking the
   following steps:

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   *  Identify a leaf node matching removed.  This lookup MUST be done
      on the tree before any non-Remove proposals have been applied (the
      "old" tree in the terminology of Section 13.2), since proposals
      such as Update can change the LeafNode stored at a leaf.  Let L be
      this leaf node.

   *  Replace the leaf node L with a blank node

   *  Blank the intermediate nodes along the path from L to the root

   *  Truncate the tree by removing leaves from the right side of the
      tree until the rightmost leaf node is not blank.

13.1.4.  PreSharedKey

   A PreSharedKey proposal can be used to request that a pre-shared key
   be injected into the key schedule in the process of advancing the
   epoch.

   struct {
       PreSharedKeyID psk;
   } PreSharedKey;

   The psktype of the pre-shared key MUST be external and the psk_nonce
   MUST be a randomly sampled nonce of length KDF.Nh.  When processing a
   Commit message that includes one or more PreSharedKey proposals,
   group members derive psk_secret as described in Section 9.4, where
   the order of the PSKs corresponds to the order of the PreSharedKey
   proposals in the Commit.

13.1.5.  ReInit

   A ReInit proposal represents a request to reinitialize the group with
   different parameters, for example, to increase the version number or
   to change the ciphersuite.  The reinitialization is done by creating
   a completely new group and shutting down the old one.

   struct {
       opaque group_id<V>;
       ProtocolVersion version;
       CipherSuite cipher_suite;
       Extension extensions<V>;
   } ReInit;

   A member of the group applies a ReInit proposal by waiting for the
   committer to send the Welcome message that matches the ReInit,
   according to the criteria in Section 12.2.

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   If a ReInit proposal is included in a Commit, it MUST be the only
   proposal referenced by the Commit.  If other non-ReInit proposals
   have been sent during the epoch, the committer SHOULD prefer them
   over the ReInit proposal, allowing the ReInit to be resent and
   applied in a subsequent epoch.  The version field in the ReInit
   proposal MUST be no less than the version for the current group.

13.1.6.  ExternalInit

   An ExternalInit proposal is used by new members that want to join a
   group by using an external commit.  This proposal can only be used in
   that context.

   struct {
     opaque kem_output<V>;
   } ExternalInit;

   A member of the group applies an ExternalInit message by initializing
   the next epoch using an init secret computed as described in
   Section 9.3.  The kem_output field contains the required KEM output.

13.1.7.  AppAck

   An AppAck proposal is used to acknowledge receipt of application
   messages.  Though this information implies no change to the group, it
   is structured as a Proposal message so that it is included in the
   group's transcript by being included in Commit messages.

   struct {
       LeafNodeRef sender;
       uint32 first_generation;
       uint32 last_generation;
   } MessageRange;

   struct {
       MessageRange received_ranges<V>;
   } AppAck;

   An AppAck proposal represents a set of messages received by the
   sender in the current epoch.  Messages are represented by the sender
   and generation values in the MLSCiphertext for the message.  Each
   MessageRange represents receipt of a span of messages whose
   generation values form a continuous range from first_generation to
   last_generation, inclusive.

   AppAck proposals are sent as a guard against the Delivery Service
   dropping application messages.  The sequential nature of the
   generation field provides a degree of loss detection, since gaps in

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   the generation sequence indicate dropped messages.  AppAck completes
   this story by addressing the scenario where the Delivery Service
   drops all messages after a certain point, so that a later generation
   is never observed.  Obviously, there is a risk that AppAck messages
   could be suppressed as well, but their inclusion in the transcript
   means that if they are suppressed then the group cannot advance at
   all.

   The schedule on which sending AppAck proposals are sent is up to the
   application, and determines which cases of loss/suppression are
   detected.  For example:

   *  The application might have the committer include an AppAck
      proposal whenever a Commit is sent, so that other members could
      know when one of their messages did not reach the committer.

   *  The application could have a client send an AppAck whenever an
      application message is sent, covering all messages received since
      its last AppAck.  This would provide a complete view of any losses
      experienced by active members.

   *  The application could simply have clients send AppAck proposals on
      a timer, so that all participants' state would be known.

   An application using AppAck proposals to guard against loss/
   suppression of application messages also needs to ensure that AppAck
   messages and the Commits that reference them are not dropped.  One
   way to do this is to always encrypt Proposal and Commit messages, to
   make it more difficult for the Delivery Service to recognize which
   messages contain AppAcks.  The application can also have clients
   enforce an AppAck schedule, reporting loss if an AppAck is not
   received at the expected time.

13.1.8.  GroupContextExtensions

   A GroupContextExtensions proposal is used to update the list of
   extensions in the GroupContext for the group.

   struct { Extension extensions<V>; } GroupContextExtensions;

   A member of the group applies a GroupContextExtensions proposal with
   the following steps:

   *  If the new extensions include a required_capabilities extension,
      verify that all members of the group support the required
      capabilities (including those added in the same commit, and
      excluding those removed).

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   *  Remove all of the existing extensions from the GroupContext object
      for the group and replacing them with the list of extensions in
      the proposal.  (This is a wholesale replacement, not a merge.  An
      extension is only carried over if the sender of the proposal
      includes it in the new list.)

   Note that once the GroupContext is updated, its inclusion in the
   confirmation_tag by way of the key schedule will confirm that all
   members of the group agree on the extensions in use.

13.1.9.  External Proposals

   Add and Remove proposals can be constructed and sent to the group by
   a party that is outside the group.  For example, a Delivery Service
   might propose to remove a member of a group who has been inactive for
   a long time, or propose adding a newly-hired staff member to a group
   representing a real-world team.  Proposals originating outside the
   group are identified by a external or new_member SenderType in
   MLSPlaintext.

   ReInit proposals can also be sent to the group by a external sender,
   for example to enforce a changed policy regarding MLS version or
   ciphersuite.

   The new_member SenderType is used for clients proposing that they
   themselves be added.  For this ID type the sender value MUST be zero
   and the Proposal type MUST be Add. The MLSPlaintext MUST be signed
   with the private key corresponding to the KeyPackage in the Add
   message.  Recipients MUST verify that the MLSPlaintext carrying the
   Proposal message is validly signed with this key.

   The external SenderType is reserved for signers that are pre-
   provisioned to the clients within a group.  It can only be used if
   the external_senders extension is present in the group's group
   context extensions.

   An external proposal MUST be sent as an MLSPlaintext object, since
   the sender will not have the keys necessary to construct an
   MLSCiphertext object.

13.1.9.1.  External Senders Extension

   The external_senders extension is a group context extension that
   contains credentials of senders that are permitted to send external
   proposals to the group.

   Credential external_senders<V>;

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13.2.  Commit

   A Commit message initiates a new epoch for the group, based on a
   collection of Proposals.  It instructs group members to update their
   representation of the state of the group by applying the proposals
   and advancing the key schedule.

   Each proposal covered by the Commit is included by a ProposalOrRef
   value, which identifies the proposal to be applied by value or by
   reference.  Commits that refer to new Proposals from the committer
   can be included by value.  Commits for previously sent proposals from
   anyone (including the committer) can be sent by reference.  Proposals
   sent by reference are specified by including the hash of the
   MLSPlaintext in which the proposal was sent (see Section 6.2).

   enum {
     reserved(0),
     proposal(1)
     reference(2),
     (255)
   } ProposalOrRefType;

   struct {
     ProposalOrRefType type;
     select (ProposalOrRef.type) {
       case proposal:  Proposal proposal;
       case reference: ProposalRef reference;
     }
   } ProposalOrRef;

   struct {
       ProposalOrRef proposals<V>;
       optional<UpdatePath> path;
   } Commit;

   A group member that has observed one or more valid proposals within
   an epoch MUST send a Commit message before sending application data.
   This ensures, for example, that any members whose removal was
   proposed during the epoch are actually removed before any application
   data is transmitted.

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   The sender of a Commit MUST include all valid proposals that it has
   received during the current epoch.  Invalid proposals include, for
   example, proposals with an invalid signature or proposals that are
   semantically invalid, such as an Add when the sender does not have
   the application-level permission to add new users.  Proposals with a
   non-default proposal type MUST NOT be included in a commit unless the
   proposal type is supported by all the members of the group that will
   process the Commit (i.e., not including any members being added or
   removed by the Commit).

   If there are multiple proposals that apply to the same leaf, or
   multiple PreSharedKey proposals that reference the same
   PreSharedKeyID, the committer MUST choose one and include only that
   one in the Commit, considering the rest invalid.  The committer MUST
   NOT include any Update proposals generated by the committer, since
   they would be duplicative with the path field in the Commit.  The
   committer MUST prefer any Remove received, or the most recent Update
   for the leaf if there are no Removes.  If there are multiple Add
   proposals containing KeyPackages that the committer considers to
   represent the same client or a client already in the group (for
   example, identical KeyPackages or KeyPackages sharing the same
   Credential), the committer again chooses one to include and considers
   the rest invalid.  The committer MUST consider invalid any Add or
   Update proposal if the Credential in the contained KeyPackage shares
   the same signature key with a Credential in any leaf of the group, or
   if the LeafNode in the KeyPackage shares the same encryption_key with
   another LeafNode in the group.

   The Commit MUST NOT combine proposals sent within different epochs.
   Due to the asynchronous nature of proposals, receivers of a Commit
   SHOULD NOT enforce that all valid proposals sent within the current
   epoch are referenced by the next Commit.  In the event that a valid
   proposal is omitted from the next Commit, and that proposal is still
   valid in the current epoch, the sender of the proposal MAY resend it
   after updating it to reflect the current epoch.

   A member of the group MAY send a Commit that references no proposals
   at all, which would thus have an empty proposals vector.  Such a
   Commit resets the sender's leaf and the nodes along its direct path,
   and provides forward secrecy and post-compromise security with regard
   to the sender of the Commit.  An Update proposal can be regarded as a
   "lazy" version of this operation, where only the leaf changes and
   intermediate nodes are blanked out.

   By default, the path field of a Commit MUST be populated.  The path
   field MAY be omitted if (a) it covers at least one proposal and (b)
   none of the proposals covered by the Commit are of "path required"
   types.  A proposal type requires a path if it cannot change the group

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   membership in a way that requires the forward secrecy and post-
   compromise security guarantees that an UpdatePath provides.  The only
   proposal types defined in this document that do not require a path
   are:

   *  add

   *  psk

   *  app_ack

   *  reinit

   New proposal types MUST state whether they require a path.  If any
   instance of a proposal type requires a path, then the proposal type
   requires a path.  This attribute of a proposal type is reflected in
   the "Path Required" field of the proposal type registry defined in
   Section 18.3.

   Update and Remove proposals are the clearest examples of proposals
   that require a path.  An UpdatePath is required to evict the removed
   member or the old appearance of the updated member.

   In pseudocode, the logic for validating the path field of a Commit is
   as follows:

   pathRequiredTypes = [
       update,
       remove,
       external_init,
       group_context_extensions
   ]

   pathRequired = false

   for i, id in commit.proposals:
       proposal = proposalCache[id]
       assert(proposal != null)

       pathRequired = pathRequired ||
                      (proposal.msg_type in pathRequiredTypes)

   if len(commit.proposals) == 0 || pathRequired:
       assert(commit.path != null)

   To summarize, a Commit can have three different configurations, with
   different uses:

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   1.  An "empty" Commit that references no proposals, which updates the
       committer's contribution to the group and provides PCS with
       regard to the committer.

   2.  A "partial" Commit that references proposals that do not require
       a path, and where the path is empty.  Such a commit doesn't
       provide PCS with regard to the committer.

   3.  A "full" Commit that references proposals of any type, which
       provides FS with regard to any removed members and PCS for the
       committer and any updated members.

13.2.1.  Creating a Commit

   When creating or processing a Commit, three different ratchet trees
   and their associated GroupContexts are used:

   1.  "Old" refers to the ratchet tree and GroupContext for the epoch
       before the commit.  The old GroupContext is used when signing the
       MLSPlainText so that existing group members can verify the
       signature before processing the commit.

   2.  "Provisional" refers to the ratchet tree and GroupContext
       constructed after applying the proposals that are referenced by
       the Commit.  The provisional GroupContext uses the epoch number
       for the new epoch, and the old confirmed transcript hash.  This
       is used when creating the UpdatePath, if the UpdatePath is
       needed.

   3.  "New" refers to the ratchet tree and GroupContext constructed
       after applying the proposals and the UpdatePath (if any).  The
       new GroupContext uses the epoch number for the new epoch, and the
       new confirmed transcript hash.  This is used when deriving the
       new epoch secrets, and is the only GroupContext that newly-added
       members will have.

   A member of the group creates a Commit message and the corresponding
   Welcome message at the same time, by taking the following steps:

   *  Construct an initial Commit object with the proposals field
      populated from Proposals received during the current epoch, and an
      empty path field.

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   *  Generate the provisional ratchet tree and GroupContext by applying
      the proposals referenced in the initial Commit object, as
      described in Section 13.1.  Update proposals are applied first,
      followed by Remove proposals, and then finally Add proposals.  Add
      proposals are applied in the order listed in the proposals vector,
      and always to the leftmost unoccupied leaf in the tree, or the
      right edge of the tree if all leaves are occupied.

      -  Note that the order in which different types of proposals are
         applied should be updated by the implementation to include any
         new proposals added by negotiated group extensions.

      -  PreSharedKey proposals are processed later when deriving the
         psk_secret for the Key Schedule.

   *  Decide whether to populate the path field: If the path field is
      required based on the proposals that are in the commit (see
      above), then it MUST be populated.  Otherwise, the sender MAY omit
      the path field at its discretion.

   *  If populating the path field: Create an UpdatePath using the
      provisional ratchet tree and GroupContext.  Any new member (from
      an add proposal) MUST be excluded from the resolution during the
      computation of the UpdatePath.  The leaf_node for this UpdatePath
      MUST have leaf_node_source set to commit.  Note that the LeafNode
      in the UpdatePath effectively updates an existing LeafNode in the
      group and thus MUST adhere to the same restrictions as LeafNodes
      used in Update proposals (aside from leaf_node_source).

      -  Assign this UpdatePath to the path field in the Commit.

      -  Apply the UpdatePath to the tree, as described in Section 8.6,
         creating the new ratchet tree.  Define commit_secret as the
         value path_secret[n+1] derived from the path_secret[n] value
         assigned to the root node.

   *  If not populating the path field: Set the path field in the Commit
      to the null optional.  Define commit_secret as the all-zero vector
      of length KDF.Nh (the same length as a path_secret value would
      be).  In this case, the new ratchet tree is the same as the
      provisional ratchet tree.

   *  Derive the psk_secret as specified in Section 9.4, where the order
      of PSKs in the derivation corresponds to the order of PreSharedKey
      proposals in the proposals vector.

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   *  Construct an MLSMessageContent object containing the Commit
      object.  Sign the MLSMessageContent using the old GroupContext as
      context.

      -  Use the MLSMessageContent to update the confirmed transcript
         hash and generate the new GroupContext.

      -  Use the init_secret from the previous epoch, the commit_secret
         and the psk_secret as defined in the previous steps, and the
         new GroupContext to compute the new joiner_secret,
         welcome_secret, epoch_secret, and derived secrets for the new
         epoch.

      -  Use the confirmation_key for the new epoch to compute the
         confirmation_tag value, and the membership_key for the old
         epoch to compute the membership_tag value in the MLSPlaintext.

      -  Calculate the interim transcript hash using the new confirmed
         transcript hash and the confirmation_tag from the
         MLSMessageAuth.

   *  Construct a GroupInfo reflecting the new state:

      -  Group ID, epoch, tree, confirmed transcript hash, interim
         transcript hash, and group context extensions from the new
         state

      -  The confirmation_tag from the MLSMessageAuth object

      -  Other extensions as defined by the application

      -  Optionally derive an external keypair as described in Section 9
         (required for External Commits, see Section 13.2.3.1)

      -  Sign the GroupInfo using the member's private signing key

      -  Encrypt the GroupInfo using the key and nonce derived from the
         joiner_secret for the new epoch (see Section 13.2.3.2)

   *  For each new member in the group:

      -  Identify the lowest common ancestor in the tree of the new
         member's leaf node and the member sending the Commit

      -  If the path field was populated above: Compute the path secret
         corresponding to the common ancestor node

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      -  Compute an EncryptedGroupSecrets object that encapsulates the
         init_secret for the current epoch and the path secret (if
         present).

   *  Construct a Welcome message from the encrypted GroupInfo object,
      the encrypted key packages, and any PSKs for which a proposal was
      included in the Commit.  The order of the psks MUST be the same as
      the order of PreSharedKey proposals in the proposals vector.

   *  If a ReInit proposal was part of the Commit, the committer MUST
      create a new group with the parameters specified in the ReInit
      proposal, and with the same members as the original group.  The
      Welcome message MUST include a PreSharedKeyID with the following
      parameters:

      -  psktype: resumption

      -  usage: reinit

      -  group_id: The group ID for the current group

      -  epoch: The epoch that the group will be in after this Commit

13.2.2.  Processing a Commit

   A member of the group applies a Commit message by taking the
   following steps:

   *  Verify that the epoch field of the enclosing MLSMessageContent is
      equal to the epoch field of the current GroupContext object

   *  Verify that the signature on the MLSMessageContent message as
      described in Section Section 7.1.

   *  Verify that all PreSharedKey proposals in the proposals vector
      have unique PreSharedKeyIDs and are available.

   *  Generate the provisional ratchet tree and GroupContext by applying
      the proposals referenced in the initial Commit object, as
      described in Section 13.1.  Update proposals are applied first,
      followed by Remove proposals, and then finally Add proposals.  Add
      proposals are applied in the order listed in the proposals vector,
      and always to the leftmost unoccupied leaf in the tree, or the
      right edge of the tree if all leaves are occupied.

      -  Note that the order in which different types of proposals are
         applied should be updated by the implementation to include any
         new proposals added by negotiated group extensions.

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   *  Verify that the path value is populated if the proposals vector
      contains any Update or Remove proposals, or if it's empty.
      Otherwise, the path value MAY be omitted.

   *  If the path value is populated: Process the path value using the
      provisional ratchet tree and GroupContext, to generate the new
      ratchet tree and the commit_secret:

      -  Validate the LeafNode as specified in Section 8.3.  The
         leaf_node_source field MUST be set to commit.

      -  Verify that the encryption_key value in the LeafNode is
         different from the committer's current leaf node.

      -  Apply the UpdatePath to the tree, as described in Section 8.6,
         and store leaf_node at the committer's leaf.

      -  Verify that the LeafNode has a parent_hash field and that its
         value matches the new parent of the sender's leaf node.

      -  Define commit_secret as the value path_secret[n+1] derived from
         the path_secret[n] value assigned to the root node.

   *  If the path value is not populated: Define commit_secret as the
      all-zero vector of length KDF.Nh (the same length as a path_secret
      value would be).

   *  Update the confirmed and interim transcript hashes using the new
      Commit, and generate the new GroupContext.

   *  Derive the psk_secret as specified in Section 9.4, where the order
      of PSKs in the derivation corresponds to the order of PreSharedKey
      proposals in the proposals vector.

   *  Use the init_secret from the previous epoch, the commit_secret and
      the psk_secret as defined in the previous steps, and the new
      GroupContext to compute the new joiner_secret, welcome_secret,
      epoch_secret, and derived secrets for the new epoch.

   *  Use the confirmation_key for the new epoch to compute the
      confirmation tag for this message, as described below, and verify
      that it is the same as the confirmation_tag field in the
      MLSMessageAuth object.

   *  If the above checks are successful, consider the new GroupContext
      object as the current state of the group.

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   *  If the Commit included a ReInit proposal, the client MUST NOT use
      the group to send messages anymore.  Instead, it MUST wait for a
      Welcome message from the committer meeting the requirements of
      Section 12.2.

13.2.3.  Adding Members to the Group

   New members can join the group in two ways.  Either by being added by
   a group member, or by adding themselves through an external Commit.
   In both cases, the new members need information to bootstrap their
   local group state.

   struct {
       CipherSuite cipher_suite;
       opaque group_id<V>;
       uint64 epoch;
       opaque tree_hash<V>;
       opaque confirmed_transcript_hash<V>;
       Extension group_context_extensions<V>;
       Extension other_extensions<V>;
       MAC confirmation_tag;
       LeafNodeRef signer;
       // SignWithLabel(., "GroupInfoTBS", GroupInfoTBS)
       opaque signature<V>;
   } GroupInfo;

   New members MUST verify the signature using the public key taken from
   the credential in the leaf node of the member with LeafNodeRef
   signer.  The signature covers the following structure, comprising all
   the fields in the GroupInfo above signature:

   struct {
       CipherSuite cipher_suite;
       opaque group_id<V>;
       uint64 epoch;
       opaque tree_hash<V>;
       opaque confirmed_transcript_hash<V>;
       Extension group_context_extensions<V>;
       Extension other_extensions<V>;
       MAC confirmation_tag;
       LeafNodeRef signer;
   } GroupInfoTBS;

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13.2.3.1.  Joining via External Commits

   External Commits are a mechanism for new members (external parties
   that want to become members of the group) to add themselves to a
   group, without requiring that an existing member has to come online
   to issue a Commit that references an Add Proposal.

   Whether existing members of the group will accept or reject an
   External Commit follows the same rules that are applied to other
   handshake messages.

   New members can create and issue an External Commit if they have
   access to the following information for the group's current epoch:

   *  group ID

   *  epoch ID

   *  ciphersuite

   *  public tree hash

   *  confirmed transcript hash

   *  confirmation tag of the most recent Commit

   *  group extensions

   *  external public key

   In other words, to join a group via an External Commit, a new member
   needs a GroupInfo with an ExternalPub extension present in the
   other_extensions.

   struct {
       HPKEPublicKey external_pub;
   } ExternalPub;

   Thus, a member of the group can enable new clients to join by making
   a GroupInfo object available to them.  Note that because a GroupInfo
   object is specific to an epoch, it will need to be updated as the
   group advances.  In particular, each GroupInfo object can be used for
   one external join, since that external join will cause the epoch to
   change.

   Note that the tree_hash field is used the same way as in the Welcome
   message.  The full tree can be included via the ratchet_tree
   extension Section 13.3.

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   The information in a GroupInfo is not generally public information,
   but applications can choose to make it available to new members in
   order to allow External Commits.

   In principle, External Commits work like regular Commits.  However,
   their content has to meet a specific set of requirements:

   *  External Commits MUST contain a path field (and is therefore a
      "full" Commit).  The joiner is added at the leftmost free leaf
      node (just as if they were added with an Add proposal), and the
      path is calculated relative to that leaf node.

   *  The Commit MUST NOT include any proposals by reference, since an
      external joiner cannot determine the validity of proposals sent
      within the group

   *  The proposals included by value in an External Commit MUST meet
      the following conditions:

      -  There MUST be a single ExternalInit proposal.

      -  There MAY be a single Remove proposal, where the LeafNode in
         the path field MUST meet the same criteria as the LeafNode in
         an Update for the removed leaf (see Section 13.1.2).  In
         particular, the credential in the LeafNode MUST present a set
         of identifiers that is acceptable to the application for the
         removed participant.

      -  There MAY be one or more PreSharedKey proposals.

      -  There MUST NOT be any other proposals.

   *  External Commits MUST be signed by the new member.  In particular,
      the signature on the enclosing MLSPlaintext MUST verify using the
      public key for the credential in the leaf_node of the path field.

   *  When processing a Commit, both existing and new members MUST use
      the external init secret as described in Section 9.3.

   *  The sender type for the MLSPlaintext encapsulating the External
      Commit MUST be new_member

   External Commits come in two "flavors" -- a "join" commit that adds
   the sender to the group or a "resync" commit that replaces a member's
   prior appearance with a new one.

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   Note that the "resync" operation allows an attacker that has
   compromised a member's signature private key to introduce themselves
   into the group and remove the prior, legitimate member in a single
   Commit.  Without resync, this can still be done, but requires two
   operations, the external Commit to join and a second Commit to remove
   the old appearance.  Applications for whom this distinction is
   salient can choose to disallow external commits that contain a
   Remove, or to allow such resync commits only if they contain a
   "reinit" PSK proposal that demonstrates the joining member's presence
   in a prior epoch of the group.  With the latter approach, the
   attacker would need to compromise the PSK as well as the signing key,
   but the application will need to ensure that continuing, non-
   resynchronizing members have the required PSK.

13.2.3.2.  Joining via Welcome Message

   The sender of a Commit message is responsible for sending a single
   Welcome message to all the new members added via Add proposals.  The
   Welcome message provides the new members with the current state of
   the group, after the application of the Commit message.  The new
   members will not be able to decrypt or verify the Commit message, but
   will have the secrets they need to participate in the epoch initiated
   by the Commit message.

   In order to allow the same Welcome message to be sent to all new
   members, information describing the group is encrypted with a
   symmetric key and nonce derived from the joiner_secret for the new
   epoch.  The joiner_secret is then encrypted to each new member using
   HPKE.  In the same encrypted package, the committer transmits the
   path secret for the lowest (closest to the leaf) node which is
   contained in the direct paths of both the committer and the new
   member.  This allows the new member to compute private keys for nodes
   in its direct path that are being reset by the corresponding Commit.

   If the sender of the Welcome message wants the receiving member to
   include a PSK in the derivation of the epoch_secret, they can
   populate the psks field indicating which PSK to use.

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   struct {
     opaque path_secret<V>;
   } PathSecret;

   struct {
     opaque joiner_secret<V>;
     optional<PathSecret> path_secret;
     PreSharedKeys psks;
   } GroupSecrets;

   struct {
     KeyPackageRef new_member;
     HPKECiphertext encrypted_group_secrets;
   } EncryptedGroupSecrets;

   struct {
     CipherSuite cipher_suite;
     EncryptedGroupSecrets secrets<V>;
     opaque encrypted_group_info<V>;
   } Welcome;

   The client processing a Welcome message will need to have a copy of
   the group's ratchet tree.  The tree can be provided in the Welcome
   message, in an extension of type ratchet_tree.  If it is sent
   otherwise (e.g., provided by a caching service on the Delivery
   Service), then the client MUST download the tree before processing
   the Welcome.

   On receiving a Welcome message, a client processes it using the
   following steps:

   *  Identify an entry in the secrets array where the new_member value
      corresponds to one of this client's KeyPackages, using the hash
      indicated by the cipher_suite field.  If no such field exists, or
      if the ciphersuite indicated in the KeyPackage does not match the
      one in the Welcome message, return an error.

   *  Decrypt the encrypted_group_secrets using HPKE with the algorithms
      indicated by the ciphersuite and the HPKE private key
      corresponding to the GroupSecrets.  If a PreSharedKeyID is part of
      the GroupSecrets and the client is not in possession of the
      corresponding PSK, return an error.

   *  From the joiner_secret in the decrypted GroupSecrets object and
      the PSKs specified in the GroupSecrets, derive the welcome_secret
      and using that the welcome_key and welcome_nonce.  Use the key and
      nonce to decrypt the encrypted_group_info field.

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   welcome_nonce = KDF.Expand(welcome_secret, "nonce", AEAD.Nn)
   welcome_key = KDF.Expand(welcome_secret, "key", AEAD.Nk)

   *  Verify the signature on the GroupInfo object.  The signature input
      comprises all of the fields in the GroupInfo object except the
      signature field.  The public key and algorithm are taken from the
      credential in the leaf node of the member with LeafNodeRef signer.
      If there is no matching leaf node, or if signature verification
      fails, return an error.

   *  Verify the integrity of the ratchet tree.

      -  Verify that the tree hash of the ratchet tree matches the
         tree_hash field in the GroupInfo.

      -  For each non-empty parent node, verify that exactly one of the
         node's children are non-empty and have the hash of this node
         set as their parent_hash value (if the child is another parent)
         or has a parent_hash field in the LeafNode containing the same
         value (if the child is a leaf).  If either of the node's
         children is empty, and in particular does not have a parent
         hash, then its respective children's parent_hash values have to
         be considered instead.

      -  For each non-empty leaf node, validate the LeafNode as
         described in Section 8.3.

   *  Identify a leaf in the tree array (any even-numbered node) whose
      LeafNode is identical to the one in the KeyPackage.  If no such
      field exists, return an error.  Let my_leaf represent this leaf in
      the tree.

   *  Construct a new group state using the information in the GroupInfo
      object.

      -  The GroupContext contains the group_id, epoch, tree_hash,
         confirmed_transcript_hash, and group_context_extensions fields
         from the GroupInfo object.

      -  The new member's position in the tree is at the leaf my_leaf,
         as defined above.

      -  Update the leaf my_leaf with the private key corresponding to
         the public key in the node.

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      -  If the path_secret value is set in the GroupSecrets object:
         Identify the lowest common ancestor of the leaf node my_leaf
         and of the node of the member with LeafNodeRef
         GroupInfo.signer.  Set the private key for this node to the
         private key derived from the path_secret.

      -  For each parent of the common ancestor, up to the root of the
         tree, derive a new path secret and set the private key for the
         node to the private key derived from the path secret.  The
         private key MUST be the private key that corresponds to the
         public key in the node.

   *  Use the joiner_secret from the GroupSecrets object to generate the
      epoch secret and other derived secrets for the current epoch.

   *  Set the confirmed transcript hash in the new state to the value of
      the confirmed_transcript_hash in the GroupInfo.

   *  Verify the confirmation tag in the GroupInfo using the derived
      confirmation key and the confirmed_transcript_hash from the
      GroupInfo.

   *  Use the confirmed transcript hash and confirmation tag to compute
      the interim transcript hash in the new state.

13.3.  Ratchet Tree Extension

   By default, a GroupInfo message only provides the joiner with a
   commitment to the group's ratchet tree.  In order to process or
   generate handshake messages, the joiner will need to get a copy of
   the ratchet tree from some other source.  (For example, the DS might
   provide a cached copy.)  The inclusion of the tree hash in the
   GroupInfo message means that the source of the ratchet tree need not
   be trusted to maintain the integrity of tree.

   In cases where the application does not wish to provide such an
   external source, the whole public state of the ratchet tree can be
   provided in an extension of type ratchet_tree, containing a
   ratchet_tree object of the following form:

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   enum {
       reserved(0),
       leaf(1),
       parent(2),
       (255)
   } NodeType;

   struct {
       NodeType node_type;
       select (Node.node_type) {
           case leaf:   LeafNode leaf_node;
           case parent: ParentNode parent_node;
       };
   } Node;

   optional<Node> ratchet_tree<V>;

   The nodes are listed in the order specified by a left-to-right in-
   order traversal of the rachet tree.  Each node is listed between its
   left subtree and its right subtree.  (This is the same ordering as
   specified for the array-based trees outlined in Appendix B.)

   The leaves of the tree are stored in even-numbered entries in the
   array (the leaf with index L in array position 2*L).  The root node
   of the tree is at position 2^k - 1 of the array, where k is the
   largest number such that 2^k is smaller than the length of the array.
   Intermediate parent nodes can be identified by performing the same
   calculation to the subarrays to the left and right of the root,
   following something like the following algorithm:

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   # Assuming a class Node that has left and right members
   def subtree_root(nodes):
       # If there is only one node in the array return it
       if len(nodes) == 1:
           return Node(nodes[0])

       # Otherwise, the length of the array MUST be odd
       if len(nodes) % 2 == 0:
           raise Exception("Malformed node array {}", len(nodes))

       # Identify the root of the subtree
       k = 0
       while (2**(k+1)) < len(nodes):
          k += 1
       R = 2**k - 1
       root = Node(nodes[R])
       root.left = subtree_root(nodes[:R])
       root.right = subtree_root(nodes[(R+1):])
       return root

   (Note that this is the same ordering of nodes as in the array-based
   tree representation described in Appendix B.  The algorithms in that
   section may be used to simplify decoding this extension into other
   representations.)

   The example tree in Section 5.1 would be represented as an array of
   nodes in the following form, where R represents the "subtree root"
   for a given subarray of the node array:

                 P
                 |
           .-----+-----.
          /             \
         N               R
         |               |
       .-+-.           .-+
      /     \         /   \
     M       O       Q     |
    / \     / \     / \    |
   A   B   C   D   E   F   G

                       1 1 1
   0 1 2 3 4 5 6 7 8 9 0 1 2
   <-----------> R <------->
   <---> R <--->   <---> R -
   - R -   - R -   - R -

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   The presence of a ratchet_tree extension in a GroupInfo message does
   not result in any changes to the GroupContext extensions for the
   group.  The ratchet tree provided is simply stored by the client and
   used for MLS operations.

   If this extension is not provided in a Welcome message, then the
   client will need to fetch the ratchet tree over some other channel
   before it can generate or process Commit messages.  Applications
   should ensure that this out-of-band channel is provided with security
   protections equivalent to the protections that are afforded to
   Proposal and Commit messages.  For example, an application that
   encrypts Proposal and Commit messages might distribute ratchet trees
   encrypted using a key exchanged over the MLS channel.

   Regardless of how the client obtains the tree, the client MUST verify
   that the root hash of the ratchet tree matches the tree_hash of the
   GroupContext before using the tree for MLS operations.

14.  Extensibility

   The base MLS protocol can be extended in a few ways.  New
   ciphersuites can be added to enable the use of new cryptographic
   algorithms.  New types of proposals can be used to perform new
   actions within an epoch.  Extension fields can be used to add
   additional information to the protocol.  In this section, we discuss
   some constraints on these extensibility mechanisms that are necessary
   to ensure broad interoperability.

14.1.  Ciphersuites

   As discussed in Section 6.1, MLS allows the participants in a group
   to negotiate the cryptographic algorithms used within the group.
   This extensibility is important for maintaining the security of the
   protocol over time [RFC7696].  It also creates a risk of
   interoperability failure due to clients not supporting a common
   ciphersuite.

   The ciphersuite registry defined in Section 18.1 attempts to strike a
   balance on this point.  On the one hand, the base policy for the
   registry is Specification Required, a fairly low bar designed to
   avoid the need for standards work in cases where different ciphers
   are needed for niche applications.  There is a higher bar (Standards
   Action) for ciphers to set the Recommended field in the registry.
   This higher bar is there in part to ensure that the interoperability
   implications of new ciphersuites are considered.

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   MLS ciphersuites are defined independent of MLS versions, so that in
   principle the same ciphersuite can be used across versions.
   Standards work defining new versions of MLS should consider whether
   it is desirable for the new version to be compatible with existing
   ciphersuites, or whether the new version should rule out some
   ciphersuites.  For example, a new version could follow the example of
   HTTP/2, which restricted the set of allowed TLS ciphers (see
   Section 9.2.2 of [RFC7540].

14.2.  Proposals

   Commit messages do not have an extension field because the set of
   protocols is extensible.  As discussed in Section 13.2, Proposals
   with a non-default proposal type MUST NOT be included in a commit
   unless the proposal type is supported by all the members of the group
   that will process the Commit.

14.3.  Credential Extensibility

   In order to ensure that MLS provides meaningful authentication it is
   important that each member is able to authenticate some identity
   information for each other member.  Identity information is encoded
   in Credentials, so this property is assured by ensuring that members
   use compatible credential types.

   The types of credential that may be used in a group is restricted to
   what all members of the group support, as specified by the
   capabilities field of each LeafNode in the RatchetTree.  An
   application can introduce new credential types by choosing an
   unallocated identifier from the registry in Section 18.4 and
   indicating support for the credential type in published LeafNodes,
   whether in Update proposals to existing groups or KeyPackages that
   are added to new groups.  Once all members in a group indicate
   support for the credential type, members can start using LeafNodes
   with the new credential.  Application may enforce that certain
   credential types always remain supported by adding a
   required_capabilities extension to the group's GroupContext, which
   would prevent any member from being added to the group that doesn't
   support them.

   In future extensions to MLS, it may be useful to allow a member to
   present more than one credential.  For example, such credentials
   might present different attributes attested by different authorities.
   To be consistent with the general principle stated at the beginning
   of this section, such an extension would need to ensure that each
   member can authenticate some identity for each other member.  For
   each pair of members (Alice, Bob), Alice would need to present at
   least one credential of a type that Bob supports. ## Extensions

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   This protocol includes a mechanism for negotiating extension
   parameters similar to the one in TLS [RFC8446].  In TLS, extension
   negotiation is one-to-one: The client offers extensions in its
   ClientHello message, and the server expresses its choices for the
   session with extensions in its ServerHello and EncryptedExtensions
   messages.  In MLS, extensions appear in the following places:

   *  In KeyPackages, to describe additional information related to the
      client

   *  In LeafNodes, to describe additional information about the client
      or its participation in the group (once in the ratchet tree)

   *  In the GroupInfo, to tell new members of a group what parameters
      are being used by the group, and to provide any additional details
      required to join the group

   *  In the GroupContext object, to ensure that all members of the
      group have the same view of the parameters in use

   In other words, an application can use GroupContext extensions to
   ensure that all members of the group agree on a set of parameters.
   Clients indicate their support for parameters in the capabilities
   field of their LeafNode.  New members of a group are informed of the
   group's GroupContext extensions via the group_context_extensions
   field in the GroupInfo object.  The other_extensions field in a
   GroupInfo object can be used to provide additional parameters to new
   joiners that are used to join the group.

   This extension mechanism is designed to allow for secure and forward-
   compatible negotiation of extensions.  For this to work,
   implementations MUST correctly handle extensible fields:

   *  A client that posts a KeyPackage MUST support all parameters
      advertised in it.  Otherwise, another client might fail to
      interoperate by selecting one of those parameters.

   *  A client initiating a group MUST ignore all unrecognized
      ciphersuites, extensions, and other parameters.  Otherwise, it may
      fail to interoperate with newer clients.

   *  A client adding a new member to a group MUST verify that the
      LeafNode for the new member is compatible with the group's
      extensions.  The capabilities field MUST indicate support for each
      extension in the GroupContext.

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   *  If any extension in a GroupInfo message is unrecognized (i.e., not
      contained in the capabilities of the corresponding KeyPackage),
      then the client MUST reject the Welcome message and not join the
      group.

   *  The extensions populated into a GroupContext object are drawn from
      those in the GroupInfo object, according to the definitions of
      those extensions.

   *  Any field containing a list of extensions MUST NOT have more than
      one extension of any given type.

   Note that the latter two requirements mean that all MLS extensions
   are mandatory, in the sense that an extension in use by the group
   MUST be supported by all members of the group.

   This document does not define any way for the parameters of the group
   to change once it has been created; such a behavior could be
   implemented as an extension.

15.  Sequencing of State Changes

   Each Commit message is premised on a given starting state, indicated
   by the epoch field of the enclosing MLSMessageContent.  If the
   changes implied by a Commit messages are made starting from a
   different state, the results will be incorrect.

   This need for sequencing is not a problem as long as each time a
   group member sends a Commit message, it is based on the most current
   state of the group.  In practice, however, there is a risk that two
   members will generate Commit messages simultaneously, based on the
   same state.

   When this happens, there is a need for the members of the group to
   deconflict the simultaneous Commit messages.  There are two general
   approaches:

   *  Have the Delivery Service enforce a total order

   *  Have a signal in the message that clients can use to break ties

   As long as Commit messages cannot be merged, there is a risk of
   starvation.  In a sufficiently busy group, a given member may never
   be able to send a Commit message, because he always loses to other
   members.  The degree to which this is a practical problem will depend
   on the dynamics of the application.

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   It might be possible, because of the non-contributivity of
   intermediate nodes, that Commit messages could be applied one after
   the other without the Delivery Service having to reject any Commit
   message, which would make MLS more resilient regarding the
   concurrency of Commit messages.  The Messaging system can decide to
   choose the order for applying the state changes.  Note that there are
   certain cases (if no total ordering is applied by the Delivery
   Service) where the ordering is important for security, ie. all
   updates must be executed before removes.

   Regardless of how messages are kept in sequence, implementations MUST
   only update their cryptographic state when valid Commit messages are
   received.  Generation of Commit messages MUST NOT modify a client's
   state, since the endpoint doesn't know at that time whether the
   changes implied by the Commit message will succeed or not.

15.1.  Server-Enforced Ordering

   With this approach, the Delivery Service ensures that incoming
   messages are added to an ordered queue and outgoing messages are
   dispatched in the same order.  The server is trusted to break ties
   when two members send a Commit message at the same time.

   Messages should have a counter field sent in clear-text that can be
   checked by the server and used for tie-breaking.  The counter starts
   at 0 and is incremented for every new incoming message.  If two group
   members send a message with the same counter, the first message to
   arrive will be accepted by the server and the second one will be
   rejected.  The rejected message needs to be sent again with the
   correct counter number.

   To prevent counter manipulation by the server, the counter's
   integrity can be ensured by including the counter in a signed message
   envelope.

   This applies to all messages, not only state changing messages.

15.2.  Client-Enforced Ordering

   Order enforcement can be implemented on the client as well, one way
   to achieve it is to use a two step update protocol: the first client
   sends a proposal to update and the proposal is accepted when it gets
   50%+ approval from the rest of the group, then it sends the approved
   update.  Clients which didn't get their proposal accepted, will wait
   for the winner to send their update before retrying new proposals.

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   While this seems safer as it doesn't rely on the server, it is more
   complex and harder to implement.  It also could cause starvation for
   some clients if they keep failing to get their proposal accepted.

16.  Application Messages

   The primary purpose of the Handshake protocol is to provide an
   authenticated group key exchange to clients.  In order to protect
   Application messages sent among the members of a group, the
   Application secret provided by the Handshake key schedule is used to
   derive nonces and encryption keys for the Message Protection Layer
   according to the Application Key Schedule.  That is, each epoch is
   equipped with a fresh Application Key Schedule which consist of a
   tree of Application Secrets as well as one symmetric ratchet per
   group member.

   Each client maintains their own local copy of the Application Key
   Schedule for each epoch during which they are a group member.  They
   derive new keys, nonces and secrets as needed while deleting old ones
   as soon as they have been used.

   Application messages MUST be protected with the Authenticated-
   Encryption with Associated-Data (AEAD) encryption scheme associated
   with the MLS ciphersuite using the common framing mechanism.  Note
   that "Authenticated" in this context does not mean messages are known
   to be sent by a specific client but only from a legitimate member of
   the group.  To authenticate a message from a particular member,
   signatures are required.  Handshake messages MUST use asymmetric
   signatures to strongly authenticate the sender of a message.

16.1.  Message Encryption and Decryption

   The group members MUST use the AEAD algorithm associated with the
   negotiated MLS ciphersuite to AEAD encrypt and decrypt their
   Application messages according to the Message Framing section.

   The group identifier and epoch allow a recipient to know which group
   secrets should be used and from which Epoch secret to start computing
   other secrets and keys.  The sender identifier is used to identify
   the member's symmetric ratchet from the initial group Application
   secret.  The application generation field is used to determine how
   far into the ratchet to iterate in order to reproduce the required
   AEAD keys and nonce for performing decryption.

   Application messages SHOULD be padded to provide some resistance
   against traffic analysis techniques over encrypted traffic.  [CLINIC]
   [HCJ16] While MLS might deliver the same payload less frequently
   across a lot of ciphertexts than traditional web servers, it might

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   still provide the attacker enough information to mount an attack.  If
   Alice asks Bob: "When are we going to the movie ?" the answer
   "Wednesday" might be leaked to an adversary by the ciphertext length.
   An attacker expecting Alice to answer Bob with a day of the week
   might find out the plaintext by correlation between the question and
   the length.

   The content and length of the padding field in MLSCiphertextContent
   can be chosen at the time of message encryption by the sender.  It is
   recommended that padding data is comprised of zero-valued bytes and
   follows an established deterministic padding scheme.

16.2.  Restrictions

   During each epoch senders MUST NOT encrypt more data than permitted
   by the security bounds of the AEAD scheme used.

   Note that each change to the Group through a Handshake message will
   also set a new encryption_secret.  Hence this change MUST be applied
   before encrypting any new application message.  This is required both
   to ensure that any users removed from the group can no longer receive
   messages and to (potentially) recover confidentiality and
   authenticity for future messages despite a past state compromise.

16.3.  Delayed and Reordered Application messages

   Since each Application message contains the group identifier, the
   epoch and a message counter, a client can receive messages out of
   order.  If they are able to retrieve or recompute the correct AEAD
   decryption key from currently stored cryptographic material clients
   can decrypt these messages.

   For usability, MLS clients might be required to keep the AEAD key and
   nonce for a certain amount of time to retain the ability to decrypt
   delayed or out of order messages, possibly still in transit while a
   decryption is being done.

17.  Security Considerations

   The security goals of MLS are described in
   [I-D.ietf-mls-architecture].  We describe here how the protocol
   achieves its goals at a high level, though a complete security
   analysis is outside of the scope of this document.

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17.1.  Confidentiality of the Group Secrets

   Group secrets are partly derived from the output of a ratchet tree.
   Ratchet trees work by assigning each member of the group to a leaf in
   the tree and maintaining the following property: the private key of a
   node in the tree is known only to members of the group that are
   assigned a leaf in the node's subtree.  This is called the _ratchet
   tree invariant_ and it makes it possible to encrypt to all group
   members except one, with a number of ciphertexts that's logarithmic
   in the number of group members.

   The ability to efficiently encrypt to all members except one allows
   members to be securely removed from a group.  It also allows a member
   to rotate their keypair such that the old private key can no longer
   be used to decrypt new messages.

17.2.  Authentication

   The first form of authentication we provide is that group members can
   verify a message originated from one of the members of the group.
   For encrypted messages, this is guaranteed because messages are
   encrypted with an AEAD under a key derived from the group secrets.
   For plaintext messages, this is guaranteed by the use of a
   membership_tag which constitutes a MAC over the message, under a key
   derived from the group secrets.

   The second form of authentication is that group members can verify a
   message originated from a particular member of the group.  This is
   guaranteed by a digital signature on each message from the sender's
   signature key.

   The signature keys held by group members are critical to the security
   of MLS against active attacks.  If a member's signature key is
   compromised, then an attacker can create LeafNodes and KeyPackages
   impersonating the member; depending on the application, this can then
   allow the attacker to join the group with the compromised member's
   identity.  For example, if a group has enabled external parties to
   join via external commits, then an attacker that has compromised a
   member's signature key could use an external commit to insert
   themselves into the group -- even using a "resync"-style external
   commit to replace the compromised member in the group.

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   Applications can mitigate the risks of signature key compromise using
   pre-shared keys.  If a group requires joiners to know a PSK in
   addition to authenticating with a credential, then in order to mount
   an impersonation attack, the attacker would need to compromise the
   relevant PSK as well as the victim's signature key.  The cost of this
   mitigation is that the application needs some external arrangement
   that ensures that the legitimate members of the group to have the
   required PSKs.

17.3.  Forward Secrecy and Post-Compromise Security

   Post-compromise security is provided between epochs by members
   regularly updating their leaf key in the ratchet tree.  Updating
   their leaf key prevents group secrets from continuing to be encrypted
   to previously compromised public keys.

   Forward-secrecy between epochs is provided by deleting private keys
   from past version of the ratchet tree, as this prevents old group
   secrets from being re-derived.  Forward secrecy _within_ an epoch is
   provided by deleting message encryption keys once they've been used
   to encrypt or decrypt a message.

   Post-compromise security is also provided for new groups by members
   regularly generating new KeyPackages and uploading them to the
   Delivery Service, such that compromised key material won't be used
   when the member is added to a new group.

17.4.  KeyPackage Reuse

   KeyPackages are intended to be used only once.  That is, once a
   KeyPackage has been used to introduce the corresponding client to a
   group, it SHOULD be deleted from the KeyPackage publication system.
   Reuse of KeyPackages can lead to replay attacks.

   An application MAY allow for reuse of a "last resort" KeyPackage in
   order to prevent denial of service attacks.  Since a KeyPackage is
   needed to add a client to a new group, an attacker could prevent a
   client being added to new groups by exhausting all available
   KeyPackages.  To prevent such a denial of service attack, the
   KeyPackage publication system SHOULD rate limit KeyPackage requests,
   especially if not authenticated.

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17.5.  Group Fragmentation by Malicious Insiders

   It is possible for a malicious member of a group to "fragment" the
   group by crafting an invalid UpdatePath.  Recall that an UpdatePath
   encrypts a sequence of path secrets to different subtrees of the
   group's ratchet trees.  These path secrets should be derived in a
   sequence as described in Section 8.4, but the UpdatePath syntax
   allows the sender to encrypt arbitrary, unrelated secrets.  The
   syntax also does not guarantee that the encrypted path secret
   encrypted for a given node corresponds to the public key provided for
   that node.

   Both of these types of corruption will cause processing of a Commit
   to fail for some members of the group.  If the public key for a node
   does not match the path secret, then the members that decrypt that
   path secret will reject the commit based on this mismatch.  If the
   path secret sequence is incorrect at some point, then members that
   can decrypt nodes before that point will compute a different public
   key for the mismatched node than the one in the UpdatePath, which
   also causes the Commit to fail.  Applications SHOULD provide
   mechanisms for failed commits to be reported, so that group members
   who were not able to recognize the error themselves can reject the
   commit and roll back to a previous state if necessary.

   Even with such an error reporting mechanism in place, however, it is
   still possible for members to get locked out of the group by a
   malformed commit.  Since malformed Commits can only be recognized by
   certain members of the group, in an asynchronous application, it may
   be the case that all members that could detect a fault in a Commit
   are offline.  In such a case, the Commit will be accepted by the
   group, and the resulting state possibly used as the basis for further
   Commits.  When the affected members come back online, they will
   reject the first commit, and thus be unable to catch up with the
   group.

   Applications can address this risk by requiring certain members of
   the group to acknowledge successful processing of a Commit before the
   group regards the Commit as accepted.  The minimum set of
   acknowledgements necessary to verify that a Commit is well-formed
   comprises an acknowledgement from one member per node in the
   UpdatePath, that is, one member from each subtree rooted in the
   copath node corresponding to the node in the UpdatePath.

18.  IANA Considerations

   This document requests the creation of the following new IANA
   registries:

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   *  MLS Ciphersuites (Section 18.1)

   *  MLS Extension Types (Section 18.2)

   *  MLS Proposal Types (Section 18.3)

   *  MLS Credential Types (Section 18.4)

   All of these registries should be under a heading of "Messaging Layer
   Security", and assignments are made via the Specification Required
   policy [RFC8126].  See Section 18.5 for additional information about
   the MLS Designated Experts (DEs).

   RFC EDITOR: Please replace XXXX throughout with the RFC number
   assigned to this document

18.1.  MLS Ciphersuites

   A ciphersuite is a combination of a protocol version and the set of
   cryptographic algorithms that should be used.

   Ciphersuite names follow the naming convention:

   CipherSuite MLS_LVL_KEM_AEAD_HASH_SIG = VALUE;

   Where VALUE is represented as a sixteen-bit integer:

   uint16 CipherSuite;

             +===========+==================================+
             | Component | Contents                         |
             +===========+==================================+
             | LVL       | The security level               |
             +-----------+----------------------------------+
             | KEM       | The KEM algorithm used for HPKE  |
             |           | in ratchet tree operations       |
             +-----------+----------------------------------+
             | AEAD      | The AEAD algorithm used for HPKE |
             |           | and message protection           |
             +-----------+----------------------------------+
             | HASH      | The hash algorithm used for HPKE |
             |           | and the MLS transcript hash      |
             +-----------+----------------------------------+
             | SIG       | The Signature algorithm used for |
             |           | message authentication           |
             +-----------+----------------------------------+

                                 Table 4

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   The columns in the registry are as follows:

   *  Value: The numeric value of the ciphersuite

   *  Name: The name of the ciphersuite

   *  Recommended: Whether support for this ciphersuite is recommended
      by the IETF MLS WG.  Valid values are "Y" and "N".  The
      "Recommended" column is assigned a value of "N" unless explicitly
      requested, and adding a value with a "Recommended" value of "Y"
      requires Standards Action [RFC8126].  IESG Approval is REQUIRED
      for a Y->N transition.

   *  Reference: The document where this ciphersuite is defined

   Initial contents:

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   +======+=====================================================+=+====+
   |Value | Name                                                |R|Ref |
   +======+=====================================================+=+====+
   |0x0000| RESERVED                                            |-|RFC |
   |      |                                                     | |XXXX|
   +------+-----------------------------------------------------+-+----+
   |0x0001| MLS_128_DHKEMX25519_AES128GCM_SHA256_Ed25519        |Y|RFC |
   |      |                                                     | |XXXX|
   +------+-----------------------------------------------------+-+----+
   |0x0002| MLS_128_DHKEMP256_AES128GCM_SHA256_P256             |Y|RFC |
   |      |                                                     | |XXXX|
   +------+-----------------------------------------------------+-+----+
   |0x0003| MLS_128_DHKEMX25519_CHACHA20POLY1305_SHA256_Ed25519 |Y|RFC |
   |      |                                                     | |XXXX|
   +------+-----------------------------------------------------+-+----+
   |0x0004| MLS_256_DHKEMX448_AES256GCM_SHA512_Ed448            |Y|RFC |
   |      |                                                     | |XXXX|
   +------+-----------------------------------------------------+-+----+
   |0x0005| MLS_256_DHKEMP521_AES256GCM_SHA512_P521             |Y|RFC |
   |      |                                                     | |XXXX|
   +------+-----------------------------------------------------+-+----+
   |0x0006| MLS_256_DHKEMX448_CHACHA20POLY1305_SHA512_Ed448     |Y|RFC |
   |      |                                                     | |XXXX|
   +------+-----------------------------------------------------+-+----+
   |0x0007| MLS_256_DHKEMP384_AES256GCM_SHA384_P384.            |Y|RFC |
   |      |                                                     | |XXXX|
   +------+-----------------------------------------------------+-+----+
   |0xff00| Reserved for Private Use                            |-|RFC |
   |-     |                                                     | |XXXX|
   |0xffff|                                                     | |    |
   +------+-----------------------------------------------------+-+----+

                                  Table 5

   All of these ciphersuites use HMAC [RFC2104] as their MAC function,
   with different hashes per ciphersuite.  The mapping of ciphersuites
   to HPKE primitives, HMAC hash functions, and TLS signature schemes is
   as follows [RFC9180] [RFC8446]:

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   +======+========+========+========+========+========================+
   |Value | KEM    | KDF    | AEAD   | Hash   | Signature              |
   +======+========+========+========+========+========================+
   |0x0001| 0x0020 | 0x0001 | 0x0001 | SHA256 | ed25519                |
   +------+--------+--------+--------+--------+------------------------+
   |0x0002| 0x0010 | 0x0001 | 0x0001 | SHA256 | ecdsa_secp256r1_sha256 |
   +------+--------+--------+--------+--------+------------------------+
   |0x0003| 0x0020 | 0x0001 | 0x0003 | SHA256 | ed25519                |
   +------+--------+--------+--------+--------+------------------------+
   |0x0004| 0x0021 | 0x0003 | 0x0002 | SHA512 | ed448                  |
   +------+--------+--------+--------+--------+------------------------+
   |0x0005| 0x0012 | 0x0003 | 0x0002 | SHA512 | ecdsa_secp521r1_sha512 |
   +------+--------+--------+--------+--------+------------------------+
   |0x0006| 0x0021 | 0x0003 | 0x0003 | SHA512 | ed448                  |
   +------+--------+--------+--------+--------+------------------------+
   |0x0007| 0x0011 | 0x0002 | 0x0002 | SHA384 | ecdsa_secp384r1_sha384 |
   +------+--------+--------+--------+--------+------------------------+

                                  Table 6

   The hash used for the MLS transcript hash is the one referenced in
   the ciphersuite name.  In the ciphersuites defined above, "SHA256",
   "SHA384", and "SHA512" refer to the SHA-256, SHA-384, and SHA-512
   functions defined in [SHS].

   It is advisable to keep the number of ciphersuites low to increase
   the chances clients can interoperate in a federated environment,
   therefore the ciphersuites only include modern, yet well-established
   algorithms.  Depending on their requirements, clients can choose
   between two security levels (roughly 128-bit and 256-bit).  Within
   the security levels clients can choose between faster X25519/X448
   curves and FIPS 140-2 compliant curves for Diffie-Hellman key
   negotiations.  Additionally clients that run predominantly on mobile
   processors can choose ChaCha20Poly1305 over AES-GCM for performance
   reasons.  Since ChaCha20Poly1305 is not listed by FIPS 140-2 it is
   not paired with FIPS 140-2 compliant curves.  The security level of
   symmetric encryption algorithms and hash functions is paired with the
   security level of the curves.

   The mandatory-to-implement ciphersuite for MLS 1.0 is
   MLS_128_DHKEMX25519_AES128GCM_SHA256_Ed25519 which uses Curve25519
   for key exchange, AES-128-GCM for HPKE, HKDF over SHA2-256, and
   Ed25519 for signatures.

   Values with the first byte 255 (decimal) are reserved for Private
   Use.

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   New ciphersuite values are assigned by IANA as described in
   Section 18.

18.2.  MLS Extension Types

   This registry lists identifiers for extensions to the MLS protocol.
   The extension type field is two bytes wide, so valid extension type
   values are in the range 0x0000 to 0xffff.

   Template:

   *  Value: The numeric value of the extension type

   *  Name: The name of the extension type

   *  Message(s): The messages in which the extension may appear, drawn
      from the following list:

      -  KP: KeyPackage objects

      -  LN: LeafNode objects

      -  GC: GroupContext objects (and the group_context_extensions
         field of GroupInfo objects)

      -  GI: The other_extensions field of GroupInfo objects

   *  Recommended: Whether support for this extension is recommended by
      the IETF MLS WG.  Valid values are "Y" and "N".  The "Recommended"
      column is assigned a value of "N" unless explicitly requested, and
      adding a value with a "Recommended" value of "Y" requires
      Standards Action [RFC8126].  IESG Approval is REQUIRED for a Y->N
      transition.

   *  Reference: The document where this extension is defined

   Initial contents:

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   +======+=======================+============+=============+=========+
   |Value | Name                  | Message(s) | Recommended |Reference|
   +======+=======================+============+=============+=========+
   |0x0000| RESERVED              | N/A        | N/A         |RFC XXXX |
   +------+-----------------------+------------+-------------+---------+
   |0x0001| external_key_id       | KP         | Y           |RFC XXXX |
   +------+-----------------------+------------+-------------+---------+
   |0x0002| ratchet_tree          | GI         | Y           |RFC XXXX |
   +------+-----------------------+------------+-------------+---------+
   |0x0003| required_capabilities | GC         | Y           |RFC XXXX |
   +------+-----------------------+------------+-------------+---------+
   |0x0004| external_pub          | GI         | Y           |RFC XXXX |
   +------+-----------------------+------------+-------------+---------+
   |0xff00| Reserved for Private  | N/A        | N/A         |RFC XXXX |
   |-     | Use                   |            |             |         |
   |0xffff|                       |            |             |         |
   +------+-----------------------+------------+-------------+---------+

                                  Table 7

18.3.  MLS Proposal Types

   This registry lists identifiers for types of proposals that can be
   made for changes to an MLS group.  The extension type field is two
   bytes wide, so valid extension type values are in the range 0x0000 to
   0xffff.

   Template:

   *  Value: The numeric value of the proposal type

   *  Name: The name of the proposal type

   *  Recommended: Whether support for this extension is recommended by
      the IETF MLS WG.  Valid values are "Y" and "N".  The "Recommended"
      column is assigned a value of "N" unless explicitly requested, and
      adding a value with a "Recommended" value of "Y" requires
      Standards Action [RFC8126].  IESG Approval is REQUIRED for a Y->N
      transition.

   *  Path Required: Whether a Commit covering a proposal of this type
      is required to have its path field populated (see Section 13.2).

   *  Reference: The document where this extension is defined

   Initial contents:

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   +======+==========================+=============+========+==========+
   |Value | Name                     | Recommended |Path    |Reference |
   |      |                          |             |Required|          |
   +======+==========================+=============+========+==========+
   |0x0000| RESERVED                 | N/A         |N/A     |RFC XXXX  |
   +------+--------------------------+-------------+--------+----------+
   |0x0001| add                      | Y           |N       |RFC XXXX  |
   +------+--------------------------+-------------+--------+----------+
   |0x0002| update                   | Y           |Y       |RFC XXXX  |
   +------+--------------------------+-------------+--------+----------+
   |0x0003| remove                   | Y           |Y       |RFC XXXX  |
   +------+--------------------------+-------------+--------+----------+
   |0x0004| psk                      | Y           |N       |RFC XXXX  |
   +------+--------------------------+-------------+--------+----------+
   |0x0005| reinit                   | Y           |N       |RFC XXXX  |
   +------+--------------------------+-------------+--------+----------+
   |0x0006| external_init            | Y           |Y       |RFC XXXX  |
   +------+--------------------------+-------------+--------+----------+
   |0x0007| app_ack                  | Y           |N       |RFC XXXX  |
   +------+--------------------------+-------------+--------+----------+
   |0x0008| group_context_extensions | Y           |Y       |RFC XXXX  |
   +------+--------------------------+-------------+--------+----------+
   |0xff00| Reserved for Private Use | N/A         |N/A     |RFC XXXX  |
   |-     |                          |             |        |          |
   |0xffff|                          |             |        |          |
   +------+--------------------------+-------------+--------+----------+

                                  Table 8

18.4.  MLS Credential Types

   This registry lists identifiers for types of credentials that can be
   used for authentication in the MLS protocol.  The credential type
   field is two bytes wide, so valid credential type values are in the
   range 0x0000 to 0xffff.

   Template:

   *  Value: The numeric value of the credential type

   *  Name: The name of the credential type

   *  Recommended: Whether support for this credential is recommended by
      the IETF MLS WG.  Valid values are "Y" and "N".  The "Recommended"
      column is assigned a value of "N" unless explicitly requested, and
      adding a value with a "Recommended" value of "Y" requires
      Standards Action [RFC8126].  IESG Approval is REQUIRED for a Y->N
      transition.

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   *  Reference: The document where this credential is defined

   Initial contents:

       +=================+==============+=============+===========+
       | Value           | Name         | Recommended | Reference |
       +=================+==============+=============+===========+
       | 0x0000          | RESERVED     | N/A         | RFC XXXX  |
       +-----------------+--------------+-------------+-----------+
       | 0x0001          | basic        | Y           | RFC XXXX  |
       +-----------------+--------------+-------------+-----------+
       | 0x0002          | x509         | Y           | RFC XXXX  |
       +-----------------+--------------+-------------+-----------+
       | 0xff00 - 0xffff | Reserved for | N/A         | RFC XXXX  |
       |                 | Private Use  |             |           |
       +-----------------+--------------+-------------+-----------+

                                 Table 9

18.5.  MLS Designated Expert Pool

   Specification Required [RFC8126] registry requests are registered
   after a three-week review period on the MLS DEs' mailing list: mls-
   reg-review@ietf.org (mailto:mls-reg-review@ietf.org), on the advice
   of one or more of the MLS DEs.  However, to allow for the allocation
   of values prior to publication, the MLS DEs may approve registration
   once they are satisfied that such a specification will be published.

   Registration requests sent to the MLS DEs mailing list for review
   SHOULD use an appropriate subject (e.g., "Request to register value
   in MLS Bar registry").

   Within the review period, the MLS DEs will either approve or deny the
   registration request, communicating this decision to the MLS DEs
   mailing list and IANA.  Denials SHOULD include an explanation and, if
   applicable, suggestions as to how to make the request successful.
   Registration requests that are undetermined for a period longer than
   21 days can be brought to the IESG's attention for resolution using
   the iesg@ietf.org (mailto:iesg@ietf.org) mailing list.

   Criteria that SHOULD be applied by the MLS DEs includes determining
   whether the proposed registration duplicates existing functionality,
   whether it is likely to be of general applicability or useful only
   for a single application, and whether the registration description is
   clear.  For example, the MLS DEs will apply the ciphersuite-related
   advisory found in Section 6.1.

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   IANA MUST only accept registry updates from the MLS DEs and SHOULD
   direct all requests for registration to the MLS DEs' mailing list.

   It is suggested that multiple MLS DEs be appointed who are able to
   represent the perspectives of different applications using this
   specification, in order to enable broadly informed review of
   registration decisions.  In cases where a registration decision could
   be perceived as creating a conflict of interest for a particular MLS
   DE, that MLS DE SHOULD defer to the judgment of the other MLS DEs.

18.6.  The "message/mls" MIME Type

   This document registers the "message/mls" MIME media type in order to
   allow other protocols (ex: HTTP [RFC7540]) to convey MLS messages.

   Media type name:  message

   Media subtype name:  mls

   Required parameters:  none

   Optional parameters:  version

                         version:  The MLS protocol version expressed as
         a string <major>.<minor>.  If omitted the version is "1.0",
         which corresponds to MLS ProtocolVersion mls10.  If for some
         reason the version number in the MIME type parameter differs
         from the ProtocolVersion embedded in the protocol, the protocol
         takes precedence.

   Encoding scheme:  MLS messages are represented using the TLS
      presentation language [RFC8446].  Therefore MLS messages need to
      be treated as binary data.

   Security considerations:  MLS is an encrypted messaging layer
      designed to be transmitted over arbitrary lower layer protocols.
      The security considerations in this document (RFC XXXX) also
      apply.

19.  Contributors

   *  Joel Alwen
      Wickr
      joel.alwen@wickr.com

   *  Karthikeyan Bhargavan
      INRIA
      karthikeyan.bhargavan@inria.fr

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   *  Cas Cremers
      University of Oxford
      cremers@cispa.de

   *  Alan Duric
      Wire
      alan@wire.com

   *  Britta Hale
      Naval Postgraduate School
      britta.hale@nps.edu

   *  Srinivas Inguva
      Twitter
      singuva@twitter.com

   *  Konrad Kohbrok
      Aalto University
      konrad.kohbrok@datashrine.de

   *  Albert Kwon
      MIT
      kwonal@mit.edu

   *  Brendan McMillion
      Cloudflare
      brendan@cloudflare.com

   *  Eric Rescorla
      Mozilla
      ekr@rtfm.com

   *  Michael Rosenberg
      Trail of Bits
      michael.rosenberg@trailofbits.com

   *  Thyla van der Merwe
      Royal Holloway, University of London
      thyla.van.der@merwe.tech

20.  References

20.1.  Normative References

   [RFC2104]  Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed-
              Hashing for Message Authentication", RFC 2104,
              DOI 10.17487/RFC2104, February 1997,
              <https://www.rfc-editor.org/rfc/rfc2104>.

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   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <https://www.rfc-editor.org/rfc/rfc2119>.

   [RFC7540]  Belshe, M., Peon, R., and M. Thomson, Ed., "Hypertext
              Transfer Protocol Version 2 (HTTP/2)", RFC 7540,
              DOI 10.17487/RFC7540, May 2015,
              <https://www.rfc-editor.org/rfc/rfc7540>.

   [RFC8126]  Cotton, M., Leiba, B., and T. Narten, "Guidelines for
              Writing an IANA Considerations Section in RFCs", BCP 26,
              RFC 8126, DOI 10.17487/RFC8126, June 2017,
              <https://www.rfc-editor.org/rfc/rfc8126>.

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <https://www.rfc-editor.org/rfc/rfc8174>.

   [RFC8446]  Rescorla, E., "The Transport Layer Security (TLS) Protocol
              Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,
              <https://www.rfc-editor.org/rfc/rfc8446>.

   [RFC9180]  Barnes, R., Bhargavan, K., Lipp, B., and C. Wood, "Hybrid
              Public Key Encryption", RFC 9180, DOI 10.17487/RFC9180,
              February 2022, <https://www.rfc-editor.org/rfc/rfc9180>.

20.2.  Informative References

   [art]      Cohn-Gordon, K., Cremers, C., Garratt, L., Millican, J.,
              and K. Milner, "On Ends-to-Ends Encryption: Asynchronous
              Group Messaging with Strong Security Guarantees", 18
              January 2018, <https://eprint.iacr.org/2017/666.pdf>.

   [CLINIC]   Miller, B., Huang, L., Joseph, A., and J. Tygar, "I Know
              Why You Went to the Clinic: Risks and Realization of HTTPS
              Traffic Analysis", Privacy Enhancing Technologies pp.
              143-163, DOI 10.1007/978-3-319-08506-7_8, 2014,
              <https://doi.org/10.1007/978-3-319-08506-7_8>.

   [doubleratchet]
              Cohn-Gordon, K., Cremers, C., Dowling, B., Garratt, L.,
              and D. Stebila, "A Formal Security Analysis of the Signal
              Messaging Protocol", 2017 IEEE European Symposium on
              Security and Privacy (EuroS&P),
              DOI 10.1109/eurosp.2017.27, April 2017,
              <https://doi.org/10.1109/eurosp.2017.27>.

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   [HCJ16]    Husák, M., Čermák, M., Jirsík, T., and P. Čeleda, "HTTPS
              traffic analysis and client identification using passive
              SSL/TLS fingerprinting", EURASIP Journal on Information
              Security Vol. 2016, DOI 10.1186/s13635-016-0030-7,
              February 2016,
              <https://doi.org/10.1186/s13635-016-0030-7>.

   [I-D.ietf-mls-architecture]
              Beurdouche, B., Rescorla, E., Omara, E., Inguva, S., Kwon,
              A., and A. Duric, "The Messaging Layer Security (MLS)
              Architecture", Work in Progress, Internet-Draft, draft-
              ietf-mls-architecture-07, 4 October 2021,
              <https://datatracker.ietf.org/doc/html/draft-ietf-mls-
              architecture-07>.

   [I-D.ietf-trans-rfc6962-bis]
              Laurie, B., Langley, A., Kasper, E., Messeri, E., and R.
              Stradling, "Certificate Transparency Version 2.0", Work in
              Progress, Internet-Draft, draft-ietf-trans-rfc6962-bis-42,
              31 August 2021, <https://datatracker.ietf.org/doc/html/
              draft-ietf-trans-rfc6962-bis-42>.

   [RFC5116]  McGrew, D., "An Interface and Algorithms for Authenticated
              Encryption", RFC 5116, DOI 10.17487/RFC5116, January 2008,
              <https://www.rfc-editor.org/rfc/rfc5116>.

   [RFC6125]  Saint-Andre, P. and J. Hodges, "Representation and
              Verification of Domain-Based Application Service Identity
              within Internet Public Key Infrastructure Using X.509
              (PKIX) Certificates in the Context of Transport Layer
              Security (TLS)", RFC 6125, DOI 10.17487/RFC6125, March
              2011, <https://www.rfc-editor.org/rfc/rfc6125>.

   [RFC7696]  Housley, R., "Guidelines for Cryptographic Algorithm
              Agility and Selecting Mandatory-to-Implement Algorithms",
              BCP 201, RFC 7696, DOI 10.17487/RFC7696, November 2015,
              <https://www.rfc-editor.org/rfc/rfc7696>.

   [RFC8032]  Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
              Signature Algorithm (EdDSA)", RFC 8032,
              DOI 10.17487/RFC8032, January 2017,
              <https://www.rfc-editor.org/rfc/rfc8032>.

   [RFC9000]  Iyengar, J., Ed. and M. Thomson, Ed., "QUIC: A UDP-Based
              Multiplexed and Secure Transport", RFC 9000,
              DOI 10.17487/RFC9000, May 2021,
              <https://www.rfc-editor.org/rfc/rfc9000>.

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   [SECG]     "Elliptic Curve Cryptography, Standards for Efficient
              Cryptography Group, ver. 2", 2009,
              <https://secg.org/sec1-v2.pdf>.

   [SHS]      Dang, Q., "Secure Hash Standard", National Institute of
              Standards and Technology report,
              DOI 10.6028/nist.fips.180-4, July 2015,
              <https://doi.org/10.6028/nist.fips.180-4>.

   [signal]   Perrin(ed), T. and M. Marlinspike, "The Double Ratchet
              Algorithm", 20 November 2016,
              <https://www.signal.org/docs/specifications/
              doubleratchet/>.

Appendix A.  Protocol Origins of Example Trees

   Protocol operations in MLS give rise to specific forms of ratchet
   tree, typically affecting a whole direct path at once.  In this
   section, we describe the protocol operations that could have given
   rise to the various example trees in this document.

   To construct the tree in Figure 9:

   *  A creates a group with B, ..., G

   *  Each member sends an empty Commit setting its direct path

   To construct the tree in Figure 10:

   *  A creates a group with B, C, D, as well as some members outside
      this subtree

   *  D removes C, setting Z and the top node (as well as any further
      nodes in the direct path)

   *  A member outside this subtree removes B, blanking B's direct path

   *  A adds a new member at C with a partial Commit, adding it as
      unmerged at Z

   To construct the tree in Figure 11:

   *  A creates a group with B, C, D

   *  B sends a full Commit, setting X and Y

   *  D removes C, setting Z and Y

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   *  B adds a new member at C with a full Commit

      -  The Add proposal adds C as unmerged at Z and Y

      -  The path in the Commit resets X and Y, clearing Y's unmerged
         leaves

   To construct the tree in Figure 12:

   *  A creates a group with B, ..., G

   *  A removes F in a full Commit, setting T, U, and W

   *  E sends an empty Commit, setting Y and W

   *  A adds a new member at F in a partial Commit, adding F as unmerged
      at Y and W

Appendix B.  Array-Based Trees

   One benefit of using left-balanced trees is that they admit a simple
   flat array representation.  In this representation, leaf nodes are
   even-numbered nodes, with the n-th leaf at 2*n.  Intermediate nodes
   are held in odd-numbered nodes.  For example, tree with 11 leaves has
   the following structure:

                                                      X
                                                      |
                                .---------------------+---------.
                               /                                 \
                              X                                   |
                              |                                   |
                    .---------+---------.                         |
                   /                     \                        |
                  X                       X                       X
                  |                       |                       |
              .---+---.               .---+---.               .---+.
             /         \             /         \             /      \
            X           X           X           X           X        |
           / \         / \         / \         / \         / \       |
          /   \       /   \       /   \       /   \       /   \      |
         X     X     X     X     X     X     X     X     X     X     X

   Node: 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20

   Leaf: 0     1     2     3     4     5     6     7     8     9    10

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   This allows us to compute relationships between tree nodes simply by
   manipulating indices, rather than having to maintain complicated
   structures in memory.  The basic rule is that the high-order bits of
   parent and child nodes indices have the following relation (where x
   is an arbitrary bit string):

   parent=01x => left=00x, right=10x

   Since node relationships are implicit, the algorithms for adding and
   removing nodes at the right edge of the tree are quite simple:

   *  Add: Append a blank parent node to the array of nodes, then append
      the new leaf node

   *  Remove: Remove the rightmost two nodes from the array of nodes

   The following python code demonstrates the tree computations
   necessary to use an array-based tree for MLS.

   # The exponent of the largest power of 2 less than x. Equivalent to:
   #   int(math.floor(math.log(x, 2)))
   def log2(x):
       if x == 0:
           return 0

       k = 0
       while (x >> k) > 0:
           k += 1
       return k-1

   # The level of a node in the tree. Leaves are level 0, their parents
   # are level 1, etc. If a node's children are at different levels,
   # then its level is the max level of its children plus one.
   def level(x):
       if x & 0x01 == 0:
           return 0

       k = 0
       while ((x >> k) & 0x01) == 1:
           k += 1
       return k

   # The number of nodes needed to represent a tree with n leaves.
   def node_width(n):
       if n == 0:
           return 0
       else:
           return 2*(n - 1) + 1

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   # The index of the root node of a tree with n leaves.
   def root(n):
       w = node_width(n)
       return (1 << log2(w)) - 1

   # The left child of an intermediate node. Note that because the tree
   # is left-balanced, there is no dependency on the size of the tree.
   def left(x):
       k = level(x)
       if k == 0:
           raise Exception('leaf node has no children')

       return x ^ (0x01 << (k - 1))

   # The right child of an intermediate node. Depends on the number of
   # leaves because the straightforward calculation can take you beyond
   # the edge of the tree.
   def right(x, n):
       k = level(x)
       if k == 0:
           raise Exception('leaf node has no children')

       r = x ^ (0x03 << (k - 1))
       while r >= node_width(n):
           r = left(r)
       return r

   # The immediate parent of a node. May be beyond the right edge of the
   # tree.
   def parent_step(x):
       k = level(x)
       b = (x >> (k + 1)) & 0x01
       return (x | (1 << k)) ^ (b << (k + 1))

   # The parent of a node. As with the right child calculation, we have
   # to walk back until the parent is within the range of the tree.
   def parent(x, n):
       if x == root(n):
           raise Exception('root node has no parent')

       p = parent_step(x)
       while p >= node_width(n):
           p = parent_step(p)
       return p

   # The other child of the node's parent.
   def sibling(x, n):
       p = parent(x, n)

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       if x < p:
           return right(p, n)
       else:
           return left(p)

   # The direct path of a node, ordered from leaf to root.
   def direct_path(x, n):
       r = root(n)
       if x == r:
           return []

       d = []
       while x != r:
           x = parent(x, n)
           d.append(x)
       return d

   # The copath of a node, ordered from leaf to root.
   def copath(x, n):
       if x == root(n):
           return []

       d = direct_path(x, n)
       d.insert(0, x)
       d.pop()
       return [sibling(y, n) for y in d]

   # The common ancestor of two nodes is the lowest node that is in the
   # direct paths of both leaves.
   def common_ancestor_semantic(x, y, n):
       dx = set([x]) | set(direct_path(x, n))
       dy = set([y]) | set(direct_path(y, n))
       dxy = dx & dy
       if len(dxy) == 0:
           raise Exception('failed to find common ancestor')

       return min(dxy, key=level)

   # The common ancestor of two nodes is the lowest node that is in the
   # direct paths of both leaves.
   def common_ancestor_direct(x, y, _):
       # Handle cases where one is an ancestor of the other
       lx, ly = level(x)+1, level(y)+1
       if (lx <= ly) and (x>>ly == y>>ly):
         return y
       elif (ly <= lx) and (x>>lx == y>>lx):
         return x

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       # Handle other cases
       xn, yn = x, y
       k = 0
       while xn != yn:
          xn, yn = xn >> 1, yn >> 1
          k += 1
       return (xn << k) + (1 << (k-1)) - 1

Appendix C.  Link-Based Trees

   An implementation may choose to store ratchet trees in a "link-based"
   representation, where each node stores references to its parents and/
   or children.  (As opposed to the array-based representation suggested
   above, where these relationships are computed from relationships
   between nodes' indices in the array.)  Such an implementation needs
   to update these links to maintain the left-balanced structure of the
   tree as the tree is extended to add new members, or truncated when
   memebers are removed.

   The following code snippet shows how these algorithms could be
   implemented in Python.

   class Node:
       def __init__(self, value, parent=None, left=None, right=None):
           self.value = value    # Value of the node
           self.parent = parent  # Parent node
           self.left = left      # Left child node
           self.right = right    # Right child node

       def leaf(self):
           return self.left == None and self.right == None

       def span(self):
           if self.leaf():
               return 1
           return self.left.span() + self.right.span()

       def full(self):
           span = self.span()
           while span % 2 == 0:
               span >>= 1
           return span == 1

       def rightmost_leaf(self):
           X = self
           while X.right != None:
               X = X.right
           return X

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   class Tree:
       def __init__(self):
           self.root = None  # Root node of the tree, initially empty

       def extend(self, N):
           if self.root == None:
               self.root = N
               return

           # Identify the proper point to insert the new parent node
           X = self.root.rightmost_leaf()
           while X.full() and X != self.root:
               X = X.parent

           # If X is not full, insert the new parent under X
           P = Node("_", right=N)
           N.parent = P
           if not X.full():
               P.parent = X
               P.left = X.right
               X.right.parent = P
               X.right = P
               return

           # If X is full, then X is the root, so P replaces the root
           P.left = self.root
           self.root.parent = P
           self.root = P
           return

       def truncate(self):
           X = self.root.rightmost_leaf()
           if X == self.root:
               self.root = None
               return

           # If X's parent is the root, then shift the root to the left
           if X.parent == self.root:
               self.root = self.root.left
               self.root.parent = None
               return

           # Otherwise, reassign the right child of the parent's parent
           Q = X.parent.parent
           Q.right = X.parent.left
           Q.right.parent = Q
           return

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Authors' Addresses

   Richard Barnes
   Cisco
   Email: rlb@ipv.sx

   Benjamin Beurdouche
   Inria & Mozilla
   Email: ietf@beurdouche.com

   Raphael Robert
   Email: ietf@raphaelrobert.com

   Jon Millican
   Facebook
   Email: jmillican@fb.com

   Emad Omara
   Google
   Email: emadomara@google.com

   Katriel Cohn-Gordon
   University of Oxford
   Email: me@katriel.co.uk

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