Network Working Group                                          R. Barnes
Internet-Draft                                                     Cisco
Intended status: Informational                             B. Beurdouche
Expires: 4 May 2021                                                Inria
                                                             J. Millican
                                                                E. Omara
                                                          K. Cohn-Gordon
                                                    University of Oxford
                                                               R. Robert
                                                         31 October 2020

              The Messaging Layer Security (MLS) Protocol


   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.

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

   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 May 2021.

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Copyright Notice

   Copyright (c) 2020 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 (
   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 Simplified BSD License text
   as described in Section 4.e of the Trust Legal Provisions and are
   provided without warranty as described in the Simplified BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   4
     1.1.  Change Log  . . . . . . . . . . . . . . . . . . . . . . .   4
   2.  Terminology . . . . . . . . . . . . . . . . . . . . . . . . .   8
   3.  Basic Assumptions . . . . . . . . . . . . . . . . . . . . . .   9
   4.  Protocol Overview . . . . . . . . . . . . . . . . . . . . . .  10
   5.  Ratchet Trees . . . . . . . . . . . . . . . . . . . . . . . .  13
     5.1.  Tree Computation Terminology  . . . . . . . . . . . . . .  14
     5.2.  Ratchet Tree Nodes  . . . . . . . . . . . . . . . . . . .  16
     5.3.  Views of a Ratchet Tree . . . . . . . . . . . . . . . . .  17
     5.4.  Ratchet Tree Evolution  . . . . . . . . . . . . . . . . .  18
     5.5.  Synchronizing Views of the Tree . . . . . . . . . . . . .  20
   6.  Cryptographic Objects . . . . . . . . . . . . . . . . . . . .  21
     6.1.  Ciphersuites  . . . . . . . . . . . . . . . . . . . . . .  21
     6.2.  Credentials . . . . . . . . . . . . . . . . . . . . . . .  22
   7.  Key Packages  . . . . . . . . . . . . . . . . . . . . . . . .  24
     7.1.  Client Capabilities . . . . . . . . . . . . . . . . . . .  25
     7.2.  Lifetime  . . . . . . . . . . . . . . . . . . . . . . . .  26
     7.3.  KeyPackage Identifiers  . . . . . . . . . . . . . . . . .  26
     7.4.  Parent Hash . . . . . . . . . . . . . . . . . . . . . . .  26
     7.5.  Tree Hashes . . . . . . . . . . . . . . . . . . . . . . .  27
     7.6.  Group State . . . . . . . . . . . . . . . . . . . . . . .  28
     7.7.  Update Paths  . . . . . . . . . . . . . . . . . . . . . .  29
   8.  Key Schedule  . . . . . . . . . . . . . . . . . . . . . . . .  30
     8.1.  External Initialization . . . . . . . . . . . . . . . . .  33
     8.2.  Pre-Shared Keys . . . . . . . . . . . . . . . . . . . . .  34
     8.3.  Secret Tree . . . . . . . . . . . . . . . . . . . . . . .  36
     8.4.  Encryption Keys . . . . . . . . . . . . . . . . . . . . .  37
     8.5.  Deletion Schedule . . . . . . . . . . . . . . . . . . . .  38
     8.6.  Exporters . . . . . . . . . . . . . . . . . . . . . . . .  39
     8.7.  Resumption Secret . . . . . . . . . . . . . . . . . . . .  40
     8.8.  State Authentication Keys . . . . . . . . . . . . . . . .  40
   9.  Message Framing . . . . . . . . . . . . . . . . . . . . . . .  40

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     9.1.  Content Authentication  . . . . . . . . . . . . . . . . .  43
     9.2.  Content Encryption  . . . . . . . . . . . . . . . . . . .  44
     9.3.  Sender Data Encryption  . . . . . . . . . . . . . . . . .  45
   10. Group Creation  . . . . . . . . . . . . . . . . . . . . . . .  46
     10.1.  Linking a New Group to an Existing Group . . . . . . . .  47
       10.1.1.  Sub-group Branching  . . . . . . . . . . . . . . . .  48
   11. Group Evolution . . . . . . . . . . . . . . . . . . . . . . .  48
     11.1.  Proposals  . . . . . . . . . . . . . . . . . . . . . . .  48
       11.1.1.  Add  . . . . . . . . . . . . . . . . . . . . . . . .  49
       11.1.2.  Update . . . . . . . . . . . . . . . . . . . . . . .  50
       11.1.3.  Remove . . . . . . . . . . . . . . . . . . . . . . .  50
       11.1.4.  PreSharedKey . . . . . . . . . . . . . . . . . . . .  51
       11.1.5.  ReInit . . . . . . . . . . . . . . . . . . . . . . .  51
       11.1.6.  ExternalInit . . . . . . . . . . . . . . . . . . . .  52
       11.1.7.  External Proposals . . . . . . . . . . . . . . . . .  52
     11.2.  Commit . . . . . . . . . . . . . . . . . . . . . . . . .  53
       11.2.1.  External Commits . . . . . . . . . . . . . . . . . .  59
       11.2.2.  Welcoming New Members  . . . . . . . . . . . . . . .  60
     11.3.  Ratchet Tree Extension . . . . . . . . . . . . . . . . .  63
   12. Extensibility . . . . . . . . . . . . . . . . . . . . . . . .  64
   13. Sequencing of State Changes . . . . . . . . . . . . . . . . .  66
     13.1.  Server-Enforced Ordering . . . . . . . . . . . . . . . .  67
     13.2.  Client-Enforced Ordering . . . . . . . . . . . . . . . .  67
   14. Application Messages  . . . . . . . . . . . . . . . . . . . .  67
     14.1.  Message Encryption and Decryption  . . . . . . . . . . .  68
     14.2.  Restrictions . . . . . . . . . . . . . . . . . . . . . .  69
     14.3.  Delayed and Reordered Application messages . . . . . . .  69
   15. Security Considerations . . . . . . . . . . . . . . . . . . .  69
     15.1.  Confidentiality of the Group Secrets . . . . . . . . . .  69
     15.2.  Authentication . . . . . . . . . . . . . . . . . . . . .  70
     15.3.  Forward Secrecy and Post-Compromise Security . . . . . .  70
     15.4.  InitKey Reuse  . . . . . . . . . . . . . . . . . . . . .  70
   16. IANA Considerations . . . . . . . . . . . . . . . . . . . . .  71
     16.1.  MLS Ciphersuites . . . . . . . . . . . . . . . . . . . .  71
     16.2.  MLS Extension Types  . . . . . . . . . . . . . . . . . .  74
     16.3.  MLS Credential Types . . . . . . . . . . . . . . . . . .  75
     16.4.  MLS Designated Expert Pool . . . . . . . . . . . . . . .  76
   17. Contributors  . . . . . . . . . . . . . . . . . . . . . . . .  77
   18. References  . . . . . . . . . . . . . . . . . . . . . . . . .  78
     18.1.  Normative References . . . . . . . . . . . . . . . . . .  78
     18.2.  Informative References . . . . . . . . . . . . . . . . .  79
   Appendix A.  Tree Math  . . . . . . . . . . . . . . . . . . . . .  81
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  84

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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.

   draft is maintained in GitHub.  Suggested changes should be submitted
   as pull requests at
   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
   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



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   *  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 (*)

   *  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 uneeded interim transcript hash from GroupInfo (*)

   *  IANA considerations

   *  Derive an authentication secret

   *  Use Extract/Expand from HPKE KDF

   *  Clarify that application messages MUST be encrypted


   *  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 (*)

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   *  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 (*)

   *  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


   *  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


   *  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 (*)

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

   *  Rename the GroupState structure to GroupContext (*)

   *  Rename UserInitKey to ClientInitKey

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

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


   *  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


   *  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 (*)


   *  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 (*)


   *  Removed ART (*)

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

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   *  Added explicit key confirmation (*)


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

   *  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


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

2.  Terminology

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "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

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      cryptographic state with other clients.  A client is defined by
      the cryptographic keys it holds.

   Group:  A collection of clients with 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
      [I-D.irtf-cfrg-hpke] ) public key that can be used to encrypt to
      that client.

   Initialization Key (InitKey):  A key package that is prepublished by
      a client, 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.

   Terminology specific to tree computations is described in Section 5.

   We use the TLS presentation language [RFC8446] to describe the
   structure of protocol messages.

3.  Basic Assumptions

   This protocol is designed to execute in the context of a Service
   Provider (SP) as described in [I-D.ietf-mls-architecture].  In
   particular, we assume the SP provides the following services:

   *  A signature key provider which allows clients to authenticate
      protocol messages in a group.

   *  A broadcast channel, for each group, which will relay a message to
      all members of a group.  For the most part, we assume that this
      channel delivers messages in the same order to all participants.
      (See Section 13 for further considerations.)

   *  A directory to which clients can publish key packages and download
      key packages for other participants.

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4.  Protocol Overview

   The goal of this protocol is to allow a group of clients to exchange
   confidential and authenticated messages.  It does so by deriving a
   sequence of secrets and keys known only to members.  Those should be
   secret against an active network adversary and should have both
   forward secrecy and post-compromise security with respect to
   compromise of any members.

   We describe the information stored by each client as _state_, which
   includes both public and private data.  An initial state is set up by
   a group creator, which is a group containing only itself.  The
   creator then sends _Add_ proposals for each client in the initial set
   of members, followed by a _Commit_ message which incorporates all of
   the _Adds_ into the group state.  Finally, the group creator
   generates a _Welcome_ message corresponding to the Commit and sends
   this directly to all the new members, who can use the information it
   contains to set up their own group state and derive a shared secret.
   Members exchange Commit messages for post-compromise security, to add
   new members, and to remove existing members.  These messages produce
   new shared secrets which are causally linked to their predecessors,
   forming a logical Directed Acyclic Graph (DAG) of states.

   The protocol algorithms we specify here follow.  Each algorithm
   specifies both (i) how a client performs the operation and (ii) how
   other clients update their state based on it.

   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
   application might gather several proposals before committing them all
   at once.

   Before the initialization of a group, clients publish InitKeys (as
   KeyPackage objects) to a directory provided by the Service Provider.

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

   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 and the corresponding Commit, the
   new member will be able to read and send new messages to the group.
   Messages received before the client has joined the group are ignored.

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   A              B              C          Directory            Channel
   |              |              |              |                   |
   |         KeyPackageB, KeyPackageC           |                   |
   |<-------------------------------------------|                   |
   |state.init()  |              |              |                   |
   |              |              |              |                   |
   |              |              |              | Add(A->AB)        |
   |              |              |              | Commit(Add)       |
   |              |              |              |                   |
   |  Welcome(B)  |              |              |                   |
   |------------->|state.init()  |              |                   |
   |              |              |              |                   |
   |              |              |              | Add(A->AB)        |
   |              |              |              | Commit(Add)       |
   |state.add(B)  |<------------------------------------------------|
   |              |state.join()  |              |                   |
   |              |              |              |                   |
   |              |              |              | Add(AB->ABC)      |
   |              |              |              | Commit(Add)       |
   |              |              |              |                   |
   |              |  Welcome(C)  |              |                   |
   |---------------------------->|state.init()  |                   |
   |              |              |              |                   |
   |              |              |              | Add(AB->ABC)      |
   |              |              |              | Commit(Add)       |
   |state.add(C)  |<------------------------------------------------|
   |              |state.add(C)  |<---------------------------------|
   |              |              |state.join()  |                   |

   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 an Add message that the current group can use to update
   their state, and a Welcome message that the new client can use to
   initialize its state.

   To enforce the forward secrecy and post-compromise security of
   messages, each member periodically updates their leaf secret.  Any
   member can update this information at any time by generating a fresh
   KeyPackage and sending an Update message followed by a Commit
   message.  Once all members have processed both, 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.

   A              B     ...      Z          Directory        Channel
   |              |              |              |              |
   |              | Update(B)    |              |              |
   |              |------------------------------------------->|
   | Commit(Upd)  |              |              |              |
   |              |              |              |              |
   |              |              |              | Update(B)    |
   |              |              |              | Commit(Upd)  |
   |state.upd(B)  |<-------------------------------------------|
   |              |state.upd(B)  |<----------------------------|
   |              |              |state.upd(B)  |              |
   |              |              |              |              |

   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,
   which adds new entropy to the group state that's known to all except
   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.

   A              B     ...      Z          Directory       Channel
   |              |              |              |              |
   |              |              | Remove(B)    |              |
   |              |              | Commit(Rem)  |              |
   |              |              |---------------------------->|
   |              |              |              |              |
   |              |              |              | Remove(B)    |
   |              |              |              | Commit(Rem)  |
   |state.rem(B)  |              |<----------------------------|
   |              |              |state.rem(B)  |              |
   |              |              |              |              |
   |              |              |              |              |

5.  Ratchet Trees

   The protocol uses "ratchet trees" for deriving shared secrets among a
   group of clients.

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5.1.  Tree Computation 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, its 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.  Nodes are _siblings_ if they share the same parent.

   A _subtree_ of a tree is the tree given by the descendants of any
   node, the _head_ of the subtree.  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

   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 (CD, ABCD, ABCDEFG)

   *  The copath of C is (D, AB, EFG)

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              /      \
             /        \
            /          \
        ABCD            EFG
       /    \          /  \
      /      \        /    \
     AB      CD      EF    |
    / \     / \     / \    |
   A   B   C   D   E   F   G

                       1 1 1
   0 1 2 3 4 5 6 7 8 9 0 1 2

   Each node in the tree is assigned a _node index_, starting at zero
   and running from left to right.  A node is a leaf node if and only if
   it has an even index.  The node indices for the nodes in the above
   tree are as follows:

   *  0 = A

   *  1 = AB

   *  2 = B

   *  3 = ABCD

   *  4 = C

   *  5 = CD

   *  6 = D

   *  7 = ABCDEFG

   *  8 = E

   *  9 = EF

   *  10 = F

   *  11 = EFG

   *  12 = G

   The leaves of the tree are indexed separately, using a _leaf index_,
   since the protocol messages only need to refer to leaves in the tree.
   Like nodes, leaves are numbered left to right.  The node with leaf

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   index "k" is also called the "k-th" leaf.  Note that given the above
   numbering, a node is a leaf node if and only if it has an even node
   index, and a leaf node's leaf index is half its node index.  The leaf
   indices in the above tree are as follows:

   *  0 = A

   *  1 = B

   *  2 = C

   *  3 = D

   *  4 = E

   *  5 = F

   *  6 = G

5.2.  Ratchet Tree Nodes

   A particular instance of a ratchet tree is defined by 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

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

   *  An AEAD encryption scheme

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

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

   *  A public key

   *  An ordered list of leaf indices for "unmerged" leaves (see
      Section 5.3)

   *  A credential (only for leaf nodes)

   *  A hash of the node's parent, as of the last time the node was

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   The conditions under which each of these values must or must not be
   present are laid out in Section 5.3.

   A node in the tree may also be _blank_, indicating that no value is
   present at that node.  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 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 tree, where the "_" character
   represents a blank node:

       /   \
      /     \
     _       CD[C]
    / \     / \
   A   _   C   D

   0 1 2 3 4 5 6

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

   *  The resolution of node 5 is the list [CD, C]

   *  The resolution of node 2 is the empty list []

   *  The resolution of node 3 is the list [A, CD, 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 7.5.

5.3.  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 that member's leaf is a descendant of
   the node.

   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 5.4.

5.4.  Ratchet Tree Evolution

   A member of an MLS group advances the key schedule to provide forward
   secrecy and post-compromise security by providing the group with
   fresh key material to be added into the group's shared secret.  To do
   so, one member of the group generates fresh key material, applies it
   to their local tree state, and then sends this key material to other
   members in the group via an UpdatePath message (see Section 7.7) .
   All other group members then apply the key material in the UpdatePath
   to their own local tree state to derive the group's now-updated
   shared secret.

   To begin, the generator of the UpdatePath updates its leaf KeyPackage
   and its direct path to the root with new secret values.  The HPKE
   leaf public key within the KeyPackage MUST be derived from a freshly
   generated HPKE secret key to provide post-compromise security.

   The generator of the UpdatePath starts by sampling a fresh random
   value called "leaf_secret", and uses the leaf_secret to generate
   their leaf HPKE key pair (see Section 7) and to seed a sequence of
   "path secrets", one for each ancestor of its leaf.  In this setting,
   path_secret[0] refers to the node directly above the leaf,

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   path_secret[1] for its parent, and so on.  At each step, the path
   secret is used to derive a new secret value for the corresponding
   node, from which the node's key pair is derived.

   leaf_node_secret = DeriveSecret(leaf_secret, "node")
   path_secret[0] = DeriveSecret(leaf_secret, "path")

   path_secret[n] = DeriveSecret(path_secret[n-1], "path")
   node_secret[n] = DeriveSecret(path_secret[n], "node")

   leaf_priv, leaf_pub = KEM.DeriveKeyPair(leaf_node_secret)
   node_priv[n], node_pub[n] = KEM.DeriveKeyPair(node_secret[n])

   For example, suppose there is a group with four members:

        / \
       /   \
      /     \
     E       _
    / \     / \
   A   B   C   D

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

   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_key_package

   After applying the UpdatePath, the tree will have the following
   structure, where "np[i]" represents the node_priv values generated as
   described above:

            /     \
        np[0]      _
        /  \      / \
       A    B    C   D

   After performing these operations, the generator of the UpdatePath
   MUST delete the leaf_secret.

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5.5.  Synchronizing Views of the Tree

   After generating fresh key material and applying it to ratchet
   forward their local tree state as described in the prior section, 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 message containing a set of public and encrypted
   private values for intermediate nodes in the direct path of a 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.

   To transmit this update, the sender broadcasts to the group the
   following information for each node in the direct path of the 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 leaf node.  There is one encrypted path secret
   for each public key in the resolution of the non-updated child.

   The recipient of a path update processes it with the following steps:

   1.  Compute the updated path secrets.

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

       *  Identify a node in the resolution of the copath node for which
          this node 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

   2.  Merge the updated path secrets into the tree.

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       *  For all updated nodes,

          -  Replace the public key for each node with the received
             public key.

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

          -  Store the updated hash of the node's parent (represented as
             a ParentNode struct), going from root to leaf, so that each
             hash incorporates all the nodes above it.  The root node
             always has a zero-length hash for this value.

       *  For nodes where an updated path secret was computed in step 1,
          compute the corresponding node key pair and replace the values
          stored at the node with the computed values.

   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)                    |
             | pk(ns[1])  | E(pk(C), ps[1]), E(pk(D), ps[1]) |
             | pk(ns[0])  | E(pk(A), ps[0])                  |

                                  Table 1

   In this table, the value pk(ns[X]) represents the public key derived
   from the node secret X, whereas pk(X) represents the public leaf key
   for user X.  The value E(K, S) represents the public-key encryption
   of the path secret S to the public key K.

   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.

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:

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   *  HPKE parameters:

      -  A Key Encapsulation Mechanism (KEM)

      -  A Key Derivation Function (KDF)

      -  An AEAD encryption algorithm

   *  A hash algorithm

   *  A signature algorithm

   The HPKE parameters are used to instantiate HPKE [I-D.irtf-cfrg-hpke]
   for the purpose of public-key encryption.  The "DeriveKeyPair"
   function associated to the KEM for the ciphersuite maps octet strings
   to HPKE key pairs.

   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<1..2^16-1>;

   The signature algorithm specified in the ciphersuite is the mandatory
   algorithm to be used for signatures in MLSPlaintext and the tree
   signatures.  It MUST be the same as the signature algorithm specified
   in the credential field of the KeyPackage objects in the leaves of
   the tree (including the InitKeys used to add new members).

   The ciphersuites are defined in section Section 16.1.

6.2.  Credentials

   A member of a group authenticates the identities of other
   participants by means of credentials issued by some authentication
   system, like a PKI.  Each type of credential MUST express the
   following data in the context of the group it is used with:

   *  The public key of a signature key pair matching the
      SignatureScheme specified by the CipherSuite of the group

   *  The identity of the holder of the private key

   Credentials MAY also include information that allows a relying party
   to verify the identity / signing key binding.

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   Additionally, Credentials SHOULD specify the signature scheme
   corresponding to each contained public key.

   // See RFC 8446 and the IANA TLS SignatureScheme registry
   uint16 SignatureScheme;

   // See IANA registry for registered values
   uint16 CredentialType;

   struct {
       opaque identity<0..2^16-1>;
       SignatureScheme signature_scheme;
       opaque signature_key<0..2^16-1>;
   } BasicCredential;

   struct {
       opaque cert_data<0..2^16-1>;
   } Certificate;

   struct {
       CredentialType credential_type;
       select (Credential.credential_type) {
           case basic:

           case x509:
               Certificate chain<1..2^32-1>;
   } Credential;

   A BasicCredential is a raw, unauthenticated assertion of an identity/
   key binding.  The format of the key in the "public_key" field is
   defined by the relevant ciphersuite: the group ciphersuite for a
   credential in a ratchet tree, the KeyPackage ciphersuite for a
   credential in a KeyPackage object.

   For X509Credential, each entry in the chain represents a single DER-
   encoded X509 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 algorithm for the "public_key" in the end-entity
   certificate MUST match the relevant ciphersuite.

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

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   Note that each new credential that has not already been validated by
   the application MUST be validated against the Authentication Service.

7.  Key Packages

   In order to facilitate asynchronous addition of clients to a group,
   it is possible to pre-publish key packages that provide some public
   information about a user.  KeyPackage structures provide information
   about a client that any existing member can use to add this client to
   the group asynchronously.

   A KeyPackage object specifies a ciphersuite that the client supports,
   as well as providing a public key that others can use for key
   agreement.  The client's signature key can be updated throughout the
   lifetime of the group by sending a new KeyPackage with a new
   Credential.  However, the identity MUST be the same in both
   Credentials and the new Credential MUST be validated by the
   authentication service.

   When used as InitKeys, KeyPackages are intended to be used only once
   and SHOULD NOT be reused except in case of last resort.  (See
   Section 15.4).  Clients MAY generate and publish multiple InitKeys to
   support multiple ciphersuites.

   KeyPackages contain a public key chosen by the client, which the
   client MUST ensure uniquely identifies a given KeyPackage object
   among the set of KeyPackages created by this client.

   The value for hpke_init_key MUST be a public key for the asymmetric
   encryption scheme defined by cipher_suite.  The whole structure is
   signed using the client's signature key.  A KeyPackage object with an
   invalid signature field MUST be considered malformed.  The input to
   the signature computation comprises all of the fields except for the
   signature field.

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   enum {
   } ProtocolVersion;

   // See IANA registry for registered values
   uint16 ExtensionType;

   struct {
       ExtensionType extension_type;
       opaque extension_data<0..2^16-1>;
   } Extension;

   struct {
       ProtocolVersion version;
       CipherSuite cipher_suite;
       HPKEPublicKey hpke_init_key;
       Credential credential;
       Extension extensions<8..2^32-1>;
       opaque signature<0..2^16-1>;
   } KeyPackage;

   KeyPackage objects MUST contain at least two extensions, one of type
   "capabilities", and one of type "lifetime".  The "capabilities"
   extension allow MLS session establishment to be safe from downgrade
   attacks on the parameters described (as discussed in Section 10),
   while still only advertising one version / ciphersuite per

   As the "KeyPackage" is a structure which is stored in the Ratchet
   Tree and updated depending on the evolution of this tree, each
   modification of its content MUST be reflected by a change of its
   signature.  This allow other members to control the validity of the
   KeyPackage at any time and in particular in the case of a newcomer
   joining the group.

7.1.  Client Capabilities

   The "capabilities" extension indicates what protocol versions,
   ciphersuites, and protocol extensions are supported by a client.

   struct {
       ProtocolVersion versions<0..255>;
       CipherSuite ciphersuites<0..255>;
       ExtensionType extensions<0..255>;
   } Capabilities;

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   This extension MUST be always present in a KeyPackage.  Extensions
   that appear in the "extensions" field of a KeyPackage MUST be
   included in the "extensions" field of the "capabilities" extension.

7.2.  Lifetime

   The "lifetime" extension represents the times between which clients
   will consider a KeyPackage valid.  This time is represented as an
   absolute time, measured in seconds since the Unix epoch
   (1970-01-01T00:00:00Z).  A client MUST NOT use the data in a
   KeyPackage for any processing before the "not_before" date, or after
   the "not_after" date.

   uint64 not_before;
   uint64 not_after;

   Applications MUST define a maximum total lifetime that is acceptable
   for a KeyPackage, and reject any KeyPackage where the total lifetime
   is longer than this duration.

   This extension MUST always be present in a KeyPackage.

7.3.  KeyPackage Identifiers

   Within MLS, a KeyPackage is identified by its hash (see, e.g.,
   Section 11.2.2).  The "key_id" extension allows applications to add
   an explicit, application-defined identifier to a KeyPackage.

   opaque key_id<0..2^16-1>;

7.4.  Parent Hash

   The "parent_hash" extension serves to bind a KeyPackage to all the
   nodes above it in the group's ratchet tree.  This enforces the tree
   invariant, meaning that malicious members can't lie about the state
   of the ratchet tree when they send Welcome messages to new members.

   opaque parent_hash<0..255>;

   This extension MUST be present in all Updates that are sent as part
   of a Commit message.  If the extension is present, clients MUST
   verify that "parent_hash" matches the hash of the leaf's parent node
   when represented as a ParentNode struct.

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7.5.  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 that represents the contents of the group's
   ratchet tree and the members' KeyPackages.

   The hash of a tree is the hash of its root node, which we define
   recursively, starting with the leaves.

   Elements of the ratchet tree are called "Node" objects and the leaves
   contain an optional "KeyPackage", while the parents contain an
   optional "ParentNode".

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

   struct {
       HPKEPublicKey public_key;
       uint32 unmerged_leaves<0..2^32-1>;
       opaque parent_hash<0..255>;
   } ParentNode;

   When computing the hash of a parent node, the "ParentNodeHashInput"
   structure is used:

   struct {
       uint32 node_index;
       optional<ParentNode> parent_node;
       opaque left_hash<0..255>;
       opaque right_hash<0..255>;
   } ParentNodeHashInput;

   The "left_hash" and "right_hash" fields hold the hashes of the node's
   left and right children, respectively.  When computing the hash of a
   leaf node, the hash of a "LeafNodeHashInput" object is used:

   struct {
       uint32 node_index;
       optional<KeyPackage> key_package;
   } LeafNodeHashInput;

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   Note that the "node_index" field contains the index of the leaf among
   the nodes in the tree, not its index among the leaves; "node_index =
   2 * leaf_index".

7.6.  Group State

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

   struct {
       opaque group_id<0..255>;
       uint64 epoch;
       opaque tree_hash<0..255>;
       opaque confirmed_transcript_hash<0..255>;
       Extension extensions<0..2^32-1>;
   } GroupContext;

   The fields in this state have the following semantics:

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

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

   *  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 7.5.

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

   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 11.1.  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

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   *  The "confirmed_transcript_hash" is updated with the data for an
      MLSPlaintext message encoding a Commit message in two parts:

   struct {
       opaque group_id<0..255>;
       uint64 epoch;
       Sender sender;
       ContentType content_type = commit;
       Commit commit;
       opaque signature<0..2^16-1>;
   } MLSPlaintextCommitContent;

   struct {
       MAC confirmation_tag;
   } MLSPlaintextCommitAuthData;

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

   confirmed_transcript_hash_[n] =
       Hash(interim_transcript_hash_[n] ||

   interim_transcript_hash_[n+1] =
       Hash(confirmed_transcript_hash_[n] ||

   Thus the "confirmed_transcript_hash" field in a GroupContext object
   represents a transcript over the whole history of MLSPlaintext Commit
   messages, up to the confirmation tag field in the current
   MLSPlaintext message.  The confirmation tag is then included in the
   transcript for the next epoch.  The interim transcript hash is passed
   to new members in the GroupInfo struct, and enables existing members
   to incorporate a Commit message into the transcript without having to
   store the whole MLSPlaintextCommitAuthData structure.

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

7.7.  Update Paths

   As described in Section 11.2, each MLS Commit message may optionally
   transmit a KeyPackage leaf and node values along its direct path.
   The path contains a public key and encrypted secret value for all
   intermediate nodes in the path above the leaf.  The path is ordered
   from the closest node to the leaf to the root; each node MUST be the
   parent of its predecessor.

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   struct {
       opaque kem_output<0..2^16-1>;
       opaque ciphertext<0..2^16-1>;
   } HPKECiphertext;

   struct {
       HPKEPublicKey public_key;
       HPKECiphertext encrypted_path_secret<0..2^32-1>;
   } UpdatePathNode;

   struct {
       KeyPackage leaf_key_package;
       UpdatePathNode nodes<0..2^32-1>;
   } 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, "")
   ciphertext = context.Seal(group_context, path_secret)

   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 [I-D.irtf-cfrg-hpke].

   Decryption is performed in the corresponding way, using the private
   key of the resolution node and the ephemeral public key transmitted
   in the message.

8.  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:

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

   Where KDFLabel is specified as:

   struct {
       uint16 length = Length;
       opaque label<7..255> = "mls10 " + Label;
       opaque context<0..2^32-1> = Context;
   } KDFLabel;

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

   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 IKM
      argument from the left

   *  DeriveSecret takes its Secret argument from the incoming arrow

   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

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

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    commit_secret -> KDF.Extract = joiner_secret
                   Derive-Secret(., "member")
psk_secret (or 0) -> KDF.Extract = member_secret
                         +--> Derive-Secret(., "welcome")
                         |    = welcome_secret
                   ExpandWithLabel(., "epoch", GroupContext_[n], KDF.Nh)
                         +--> Derive-Secret(., <label>)
                         |    = <secret>
                   Derive-Secret(., "init")

   A number of secrets are derived from the epoch secret for different

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               | Secret                | Label            |
               | sender_data_secret    | "sender data"    |
               | encryption_secret     | "encryption"     |
               | exporter_secret       | "exporter"       |
               | authentication_secret | "authentication" |
               | external_secret       | "external"       |
               | confirmation_key      | "confirm"        |
               | membership_key        | "membership"     |
               | resumption_secret     | "resumption"     |

                                 Table 2

   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
   "PublicGroupState" struct in order to allow non-members to join the
   group using an external commit.

8.1.  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

   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 PublicGroupState and
   an external Commit to initialize their copy of the key schedule for
   the new epoch.

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

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

   In both cases, the "info" input to HPKE is set to the
   PublicGroupState for the previous epoch, encoded using the TLS

8.2.  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.

   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.

   Each PSK in MLS has a type that designates how it was provisioned.
   External PSKs are provided by the application, while recovery and re-
   init 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 group state.  In particular, in addition to external PSK types,
   a PSK derived from within MLS may be used in the following cases:

   *  Re-Initialization: If during the lifetime of a group, the group
      members decide to switch to a more secure ciphersuite or newer
      protocol version, a PSK can be used to carry entropy from the old
      group forward into a new group with the desired parameters.

   *  Branching: A PSK may be used to bootstrap a subset of current
      group members into a new group.  This applies if a subset of
      current group members wish to branch based on the current group

   The injection of one or more PSKs into the key schedule is signaled
   in two ways: 1) as a "PreSharedKey" proposal, and 2) in the
   "GroupSecrets" object of a Welcome message sent to new members added
   in that epoch.

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   enum {
   } PSKType;

   struct {
     PSKType psktype;
     select (PreSharedKeyID.psktype) {
       case external:
         opaque psk_id<0..255>;

       case reinit:
         opaque psk_group_id<0..255>;
         uint64 psk_epoch;

       case branch:
         opaque psk_group_id<0..255>;
         uint64 psk_epoch;
     opaque psk_nonce<0..255>;
   } PreSharedKeyID;

   struct {
       PreSharedKeyID psks<0..2^16-1>;
   } 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_secret" 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;

psk_input_[i] = KDF.Extract(0, psk_[i])
psk_secret_[i] = ExpandWithLabel(psk_input_[i], "derived psk", PSKLabel, KDF.Nh)
psk_secret     = psk_secret_[0] || ... || psk_secret_[n-1]

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   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.

8.3.  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.  Nodes are also assigned
   an index according to their position in the array representation of
   the tree (described in Appendix A).  If N is a node index in the
   Secret Tree then left(N) and right(N) denote the children of N (if
   they exist).

   The secret of any other node in the tree is derived from its parent's
   secret using a call to DeriveTreeSecret:

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

   Where TreeContext is specified as:

   struct {
       uint32 node = Node;
       uint32 generation = Generation;
   } TreeContext;

   If N is a node index in the Secret Tree then the secrets of the
   children of N are defined to be:

           +--> DeriveTreeSecret(., "tree", left(N), 0, KDF.Nh)
           |    = tree_node_[left(N)]_secret
           +--> DeriveTreeSecret(., "tree", right(N), 0, 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 8.4.  The root of each
   ratchet is computed as:

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

8.4.  Encryption Keys

   As described in Section 9, MLS encrypts three different types of

   *  Metadata (sender information)

   *  Handshake messages (Proposal and Commit)

   *  Application messages

   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

   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 8.3.  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

   Keys, nonces, and the secrets in ratchets are derived using
   DeriveTreeSecret.  The context in a given call consists of the index
   of the sender's leaf in the ratchet tree and the current position in
   the ratchet.  In particular, the node index of the sender's leaf in
   the ratchet tree is the same as the node index of the leaf in the
   Secret Tree used to initialize the sender's ratchet.

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

8.5.  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

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

   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 node index N 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", "member_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 index N,

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

   *  "application_ratchet_nonce_[N]_[j]" and

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

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

   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", "member_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

8.6.  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)

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   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

   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.

8.7.  Resumption Secret

   The main MLS key schedule provides a "resumption_secret" which can
   provide extra security in some cross-group operations.

   The application SHOULD specify an upper limit on the number of past
   epochs for which the "resumption_secret" may be stored.

   There are two ways in which a "resumption_secret" can be used: to re-
   initialize the group with different parameters, or to create a sub-
   group of an existing group as detailed in Section 8.2.

   Resumption keys are distinguished from exporter keys in that they
   have specific use inside the MLS protocol, whereas the use of
   exporter secrets may be decided by an external application.  They are
   thus derived separately to avoid key material reuse.

8.8.  State Authentication Keys

   The main MLS key schedule provides a per-epoch
   "authentication_secret".  If one of the parties is being actively
   impersonated by an attacker, their "authentication_secret" will
   differ from that of the other group members.  Thus, members of a
   group MAY use their "authentication_secrets" within an out-of-band
   authentication protocol to ensure that they share the same view of
   the group.

9.  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

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   The two main structures involved are MLSPlaintext and MLSCiphertext.
   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 {
   } ContentType;

   enum {
   } SenderType;

   struct {
       SenderType sender_type;
       uint32 sender;
   } Sender;

   struct {
       opaque mac_value<0..255>;
   } MAC;

   struct {
       opaque group_id<0..255>;
       uint64 epoch;
       Sender sender;
       opaque authenticated_data<0..2^32-1>;

       ContentType content_type;
       select (MLSPlaintext.content_type) {
           case application:
             opaque application_data<0..2^32-1>;

           case proposal:
             Proposal proposal;

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           case commit:
             Commit commit;

       opaque signature<0..2^16-1>;
       optional<MAC> confirmation_tag;
       optional<MAC> membership_tag;
   } MLSPlaintext;

   struct {
       opaque group_id<0..255>;
       uint64 epoch;
       ContentType content_type;
       opaque authenticated_data<0..2^32-1>;
       opaque encrypted_sender_data<0..255>;
       opaque ciphertext<0..2^32-1>;
   } MLSCiphertext;

   The field "confirmation_tag" MUST be present if "content_type" equals
   commit.  Otherwise, it MUST NOT be present.

   External sender types are sent as MLSPlaintext, see Section 11.1.7
   for their use.

   The remainder of this section describes how to compute the signature
   of an MLSPlaintext object and how to convert it to an MLSCiphertext
   object for "member" sender types.  The steps are:

   *  Set group_id, epoch, content_type and authenticated_data fields
      from the MLSPlaintext object directly

   *  Identify the key and key generation depending on the content type

   *  Encrypt an MLSCiphertextContent for the ciphertext field using the
      key identified and MLSPlaintext object

   *  Encrypt the sender data using a key and nonce derived from the
      "sender_data_secret" for the epoch and a sample of the encrypted

   Decryption is done by decrypting the sender data, then the message,
   and then verifying the content signature.

   The following sections describe the encryption and signing processes
   in detail.

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9.1.  Content Authentication

   The "signature" field in an MLSPlaintext object is computed using the
   signing private key corresponding to the credential at the leaf of
   the tree indicated by the sender field.  The signature covers the
   plaintext metadata and message content, which is all of MLSPlaintext
   except for the "signature", the "confirmation_tag" and
   "membership_tag" fields.  If the sender is a member of the group, the
   signature also covers the GroupContext for the current epoch, so that
   signatures are specific to a given group and epoch.

   struct {
       select (MLSPlaintextTBS.sender.sender_type) {
           case member:
               GroupContext context;

       opaque group_id<0..255>;
       uint64 epoch;
       Sender sender;
       opaque authenticated_data<0..2^32-1>;

       ContentType content_type;
       select (MLSPlaintextTBS.content_type) {
           case application:
             opaque application_data<0..2^32-1>;

           case proposal:
             Proposal proposal;

           case commit:
             Commit commit;
   } MLSPlaintextTBS;

   The "membership_tag" field in the MLSPlaintext object authenticates
   the sender's membership in the group.  For an MLSPlaintext with a
   sender type other than "member", this field MUST be omitted.  For
   messages sent by members, it MUST be present and set to the following

   struct {
     MLSPlaintextTBS tbs;
     opaque signature<0..2^16-1>;
     optional<MAC> confirmation_tag;
   } MLSPlaintextTBM;

   membership_tag = MAC(membership_key, MLSPlaintextTBM);

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   Note that the "membership_tag" only needs to be computed for
   MLSPlaintext messages that will be sent over the wire, and isn't
   needed for those that will be encrypted and transmitted as
   MLSCiphertext messages.

9.2.  Content Encryption

   The "ciphertext" field of the MLSCiphertext object is produced by
   supplying the inputs described below to the AEAD function specified
   by the ciphersuite in use.  The plaintext input contains the content
   and signature of the MLSPlaintext, plus optional padding.  These
   values are encoded in the following form:

   struct {
       select (MLSCiphertext.content_type) {
           case application:
             opaque application_data<0..2^32-1>;

           case proposal:
             Proposal proposal;

           case commit:
             Commit commit;

       opaque signature<0..2^16-1>;
       optional<MAC> confirmation_tag;
       opaque padding<0..2^16-1>;
   } 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

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   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

   |   Key Schedule Nonce  |
   | 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<0..255>;
       uint64 epoch;
       ContentType content_type;
       opaque authenticated_data<0..2^32-1>;
   } MLSCiphertextContentAAD;

9.3.  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 {
       uint32 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

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   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 9.2.

   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 all the fields of MLSCiphertext excluding "encrypted_sender_data":

   struct {
       opaque group_id<0..255>;
       uint64 epoch;
       ContentType content_type;
   } MLSSenderDataAAD;

   When parsing a SenderData struct as part of message decryption, the
   recipient MUST verify that the sender field represents an occupied
   leaf in the ratchet tree.  In particular, the sender index value MUST
   be less than the number of leaves in the tree.

10.  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 11.1.1, Section 11.2, and Section 11.2.2.

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

   *  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" extensions in these KeyPackages to verify
      that the chosen version and ciphersuite is the best option
      supported by all members.

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   *  Initialize a one-member group with the following initial values
      (where "0" represents an all-zero vector of size KDF.Nh):

      -  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: 0

      -  Interim transcript hash: 0

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

   *  For each member, construct an Add proposal from the KeyPackage for
      that member (see Section 11.1.1)

   *  Construct a Commit message that commits all of the Add proposals,
      in any order chosen by the creator (see Section 11.2)

   *  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 11.2.2.

   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, the leaf secret from which the Commit is built, and the
   intermediate key pairs along the direct path to the root.

10.1.  Linking a New Group to an Existing Group

   A new group may be tied to an already existing group for the purpose
   of re-initializing the existing group, or to branch into a sub-group.
   Re-initializing an existing group may be used, for example, to
   restart the group with a different ciphersuite or protocol version.
   Branching may be used to bootstrap a new group consisting of a subset
   of current group members, based on the current group state.

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

10.1.1.  Sub-group Branching

   If a client wants to create a subgroup of an existing group, they MAY
   choose to include a "PreSharedKeyID" in the "GroupSecrets" object of
   the Welcome message choosing the "psktype" "branch", the "group_id"
   of the group from which a subgroup is to be branched, as well as an
   epoch within the number of epochs for which a "resumption_secret" is

11.  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

   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

11.1.  Proposals

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

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   enum {
   } 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;
   } Proposal;

   On receiving an MLSPlaintext containing a Proposal, a client MUST
   verify the signature on the enclosing MLSPlaintext.  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.

11.1.1.  Add

   An Add proposal requests that a client with a specified KeyPackage be
   added to the group.

   struct {
       KeyPackage key_package;
   } Add;

   The proposer of the Add does not control where in the group's ratchet
   tree the new member is added.  Instead, the sender of the Commit
   message chooses a location for each added member and states it in the
   Commit message.

   An Add is applied after being included in a Commit message.  The
   position of the Add in the list of proposals determines the leaf
   index "index" where the new member will be added.  For the first Add
   in the Commit, "index" is the leftmost empty leaf in the tree, for
   the second Add, the next empty leaf to the right, etc.

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   *  If necessary, extend the tree to the right until it has at least
      index + 1 leaves

   *  For each non-blank intermediate node along the path from the leaf
      at position "index" to the root, add "index" to the
      "unmerged_leaves" list for the node.

   *  Set the leaf node in the tree at position "index" to a new node
      containing the public key from the KeyPackage in the Add, as well
      as the credential under which the KeyPackage was signed

11.1.2.  Update

   An Update proposal is a similar mechanism to Add with the distinction
   that it is the sender's leaf KeyPackage in the tree which would be
   updated with a new KeyPackage.

   struct {
       KeyPackage key_package;
   } Update;

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

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

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

11.1.3.  Remove

   A Remove proposal requests that the client at a specified index in
   the tree be removed from the group.

   struct {
       uint32 removed;
   } Remove;

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

   *  Replace the leaf node at position "removed" with a blank node

   *  Blank the intermediate nodes along the path from the removed leaf
      to the root

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11.1.4.  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

   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 8.2, where the order of the PSKs corresponds to
   the order of the "PreSharedKey" proposals in the Commit.

11.1.5.  ReInit

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

   struct {
       opaque group_id<0..255>;
       ProtocolVersion version;
       CipherSuite cipher_suite;
       Extension extensions<0..2^32-1>;
   } ReInit;

   A member of the group applies a ReInit proposal by waiting for the
   committer to send the Welcome message and by checking that the
   "group_id" and the parameters of the new group corresponds to the
   ones specified in the proposal.  The Welcome message MUST specify
   exactly one pre-shared key with "psktype = reinit", and with
   "psk_group_id" and "psk_epoch" equal to the "group_id" and "epoch" of
   the existing group after the Commit containing the "reinit" Proposal
   was processed.  The Welcome message may specify the inclusion of
   other pre-shared keys with a "psktype" different from "reinit".

   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.

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11.1.6.  ExternalInit

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

   struct {
     opaque kem_output<0..2^16-1>;
   } ExternalInit;

   A member of the group applies an ExternalInit message by initializing
   the next epoch using an init secret computed as described in
   Section 8.1.  The "kem_output" field contains the required KEM

11.1.7.  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 "preconfigured" or "new_member" SenderType
   in MLSPlaintext.

   ReInit proposals can also be sent to the group by a "preconfigured"
   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 "preconfigured" SenderType is reserved for signers that are pre-
   provisioned to the clients within a group.  If proposals with these
   sender IDs are to be accepted within a group, the members of the
   group MUST be provisioned by the application with a mapping between
   these IDs and authorized signing keys.  Recipients MUST verify that
   the MLSPlaintext carrying the Proposal message is validly signed with
   the corresponding key.  To ensure consistent handling of external
   proposals, the application MUST ensure that the members of a group
   have the same mapping and apply the same policies to external

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

11.2.  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.  Proposals supplied by value are included directly in the
   Commit object.  Proposals supplied by reference are specified by
   including the hash of the MLSPlaintext in which the Proposal was
   sent, using the hash function from the group's ciphersuite.  For
   proposals supplied by value, the sender of the proposal is the same
   as the sender of the Commit.  Conversely, proposals sent by people
   other than the committer MUST be included by reference.

   enum {
   } ProposalOrRefType;

   struct {
     ProposalOrRefType type;
     select (ProposalOrRef.type) {
       case proposal:  Proposal proposal;
       case reference: opaque hash<0..255>;
   } ProposalOrRef;

   struct {
       ProposalOrRef proposals<0..2^32-1>;
       optional<UpdatePath> path;
   } Commit;

   A group member that has observed one or more 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.  If there are
   multiple proposals that apply to the same leaf, the committer chooses
   one and includes only that one in the Commit, considering the rest
   invalid.  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 for the same client, the committer again
   chooses one to include and considers the rest invalid.

   The Commit MUST NOT combine proposals sent within different epochs.
   In the event that a valid proposal is omitted from the next Commit,
   the sender of the proposal SHOULD retransmit it in the new 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.

   The "path" field of a Commit message MUST be populated if the Commit
   covers at least one Update or Remove proposal.  The "path" field MUST
   also be populated if the Commit covers no proposals at all (i.e., if
   the proposals vector is empty).  The "path" field MAY be omitted if
   the Commit covers only Add proposals.  In pseudocode, the logic for
   validating a Commit is as follows:

   hasUpdates = false
   hasRemoves = false

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

       hasUpdates = hasUpdates || proposal.msg_type == update
       hasRemoves = hasRemoves || proposal.msg_type == remove

   if len(commit.proposals) == 0 || hasUpdates || hasRemoves:
     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.  An "add-only" Commit that references only Add proposals, in which
       the path is optional.  Such a commit provides PCS with regard to
       the committer only if the path field is present.

   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.

   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
      empty "key_package" and "path" fields.

   *  Generate a provisional GroupContext object by applying the
      proposals referenced in the initial Commit object, as described in
      Section 11.1.  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 a UpdatePath using the new
      tree.  Any new member (from an add proposal) MUST be exluded from
      the resolution during the computation of the UpdatePath.  The
      GroupContext for this operation uses the "group_id", "epoch",
      "tree_hash", and "confirmed_transcript_hash" values in the initial
      GroupContext object.

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

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      -  Apply the UpdatePath to the tree, as described in Section 5.5.
         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 the same length as a "path_secret" value would be.

   *  Generate a new KeyPackage for the Committer's own leaf, with a
      "parent_hash" extension.  Store it in the ratchet tree and assign
      it to the "key_package" field in the Commit object.

   *  If one or more PreSharedKey proposals are part of the commit,
      derive the "psk_secret" as specified in Section 8.2, where the
      order of PSKs in the derivation corresponds to the order of
      PreSharedKey proposals in the "proposals" vector.  Otherwise, set
      "psk_secret" to a zero-length octet string.

   *  Construct an MLSPlaintext object containing the Commit object.
      Sign the MLSPlaintext using the current epoch's GroupContext as
      context.  Use the signature, the "commit_secret" and the
      "psk_secret" to advance the key schedule and compute the
      "confirmation_tag" value in the MLSPlaintext.

   *  Update the tree in the provisional state by applying the direct

   *  Construct a GroupInfo reflecting the new state:

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

      -  The confirmation_tag from the MLSPlaintext object

      -  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 11.2.2)

   *  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

   *  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 "psktype"
      "reinit" and with "psk_group_id" and "psk_epoch" corresponding to
      the current group and the epoch after the commit was processed.

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

   *  Verify that the "epoch" field of the enclosing MLSPlaintext
      message is equal to the "epoch" field of the current GroupContext

   *  Verify that the signature on the MLSPlaintext message verifies
      using the public key from the credential stored at the leaf in the
      tree indicated by the "sender" field.

   *  Verify that all PSKs specified in any PreSharedKey proposals in
      the "proposals" vector are available.

   *  Generate a provisional GroupContext object by applying the
      proposals referenced in the initial Commit object, as described in
      Section 11.1.  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.

   *  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.

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

      -  Apply the UpdatePath to the tree, as described in Section 5.5,
         and store "key_package" at the Committer's leaf.

      -  Verify that the KeyPackage has a "parent_hash" extension and
         that its value matches the new parent of the sender's leaf

      -  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 the same length as a "path_secret" value
      would be.

   *  Update the new GroupContext's confirmed and interim transcript
      hashes using the new Commit.

   *  If the "proposals" vector contains any PreSharedKey proposals,
      derive the "psk_secret" as specified in Section 8.2, where the
      order of PSKs in the derivation corresponds to the order of
      PreSharedKey proposals in the "proposals" vector.  Otherwise, set
      "psk_secret" to 0.

   *  Use the "commit_secret", the "psk_secret", the provisional
      GroupContext, and the init secret from the previous epoch to
      compute the 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
      MLSPlaintext object.

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

   *  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 and check that

      -  The "version", "cipher_suite" and "extensions" fields of the
         new group corresponds to the ones in the "ReInit" proposal, and
         that the "version" is greater than or equal to that of the
         original group.

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      -  The "psks" field in the Welcome message includes a
         "PreSharedKeyID" with "psktype" = "reinit", and "psk_epoch" and
         "psk_group_id" equal to the epoch and group ID of the original
         group after processing the Commit.

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

   MLSPlaintext.confirmation_tag =
       MAC(confirmation_key, GroupContext.confirmed_transcript_hash)

11.2.1.  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

   *  interim transcript hash

   *  group extensions

   *  external public key

   This information is aggregated in a "PublicGroupState" object as

   "struct { CipherSuite cipher_suite; opaque group_id<0..255>; uint64
   epoch; opaque tree_hash<0..255>; opaque
   interim_transcript_hash<0..255>; Extension extensions<0..2^32-1>;
   HPKEPublicKey external_pub; } PublicGroupState;"

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   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 11.3.

   The information above are not deemed public data in general, but
   applications can choose to make them available to new members in
   order to allow External Commits.

   External Commits work like regular Commits, with a few differences:

   *  External Commits MUST reference an Add Proposal that adds the
      issuing new member to the group

   *  External Commits MUST contain a "path" field (and is therefore a
      "full" Commit)

   *  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_key_package" of the
      "path" field.

   *  An external commit MUST reference no more than one ExternalInit
      proposal, and the ExternalInit proposal MUST be supplied by value,
      not by reference.  When processing a Commit, both existing and new
      members MUST use the external init secret as described in
      Section 8.1.

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

   *  If the Add Proposal is also issued by the new member, its member
      SenderType MUST be "new_member"

11.2.2.  Welcoming New Members

   The sender of a Commit message is responsible for sending a Welcome
   message to any 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 randomly chosen by the sender.  This key and
   nonce are then encrypted to each new member using HPKE.  In the same
   encrypted package, the committer transmits the path secret for the

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   lowest node 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.

   struct {
     opaque group_id<0..255>;
     uint64 epoch;
     opaque tree_hash<0..255>;
     opaque confirmed_transcript_hash<0..255>;
     Extension extensions<0..2^32-1>;
     MAC confirmation_tag;
     uint32 signer_index;
     opaque signature<0..2^16-1>;
   } GroupInfo;

   struct {
     opaque path_secret<1..255>;
   } PathSecret;

   struct {
     opaque joiner_secret<1..255>;
     optional<PathSecret> path_secret;
     optional<PreSharedKeys> psks;
   } GroupSecrets;

   struct {
     opaque key_package_hash<1..255>;
     HPKECiphertext encrypted_group_secrets;
   } EncryptedGroupSecrets;

   struct {
     ProtocolVersion version = mls10;
     CipherSuite cipher_suite;
     EncryptedGroupSecrets secrets<0..2^32-1>;
     opaque encrypted_group_info<1..2^32-1>;
   } 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.

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

   *  Identify an entry in the "secrets" array where the
      "key_package_hash" 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
      "member_secret" and using that the "welcome_secret",
      "welcome_key", and "welcome_nonce".  Use the key and nonce to
      decrypt the "encrypted_group_info" field.

   welcome_secret = Derive-Secret(member_secret, "welcome")
   welcome_nonce = KDF.Expand(welcome_secret, "nonce", nonce_length)
   welcome_key = KDF.Expand(welcome_secret, "key", key_length)

   *  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 at position "signer_index".  If this
      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" extension in the KeyPackage
         containing the same value (if the child is a leaf).

      -  For each non-empty leaf node, verify the signature on the

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   *  Identify a leaf in the "tree" array (any even-numbered node) whose
      "key_package" field is identical to the the KeyPackage.  If no
      such field exists, return an error.  Let "index" represent the
      index of this node among the leaves in the tree, namely the index
      of the node in the "tree" array divided by two.

   *  Construct a new group state using the information in the GroupInfo
      object.  The new member's position in the tree is "index", as
      defined above.  In particular, the confirmed transcript hash for
      the new state is the "prior_confirmed_transcript_hash" in the
      GroupInfo object.

      -  Update the leaf at index "index" with the private key
         corresponding to the public key in the node.

      -  If the "path_secret" value is set in the GroupSecrets object:
         Identify the lowest common ancestor of the leaves at "index"
         and at "GroupInfo.signer_index".  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

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

11.3.  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.

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   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:

   enum {
   } NodeType;

   struct {
       NodeType node_type;
       select (Node.node_type) {
           case leaf:   KeyPackage key_package;
           case parent: ParentNode node;
   } Node;

   optional<Node> ratchet_tree<1..2^32-1>;

   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.

12.  Extensibility

   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 client capabilities and aspects of
      their participation in the group (once in the ratchet tree)

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   *  In the Welcome message, to tell new members of a group what
      parameters are being used by 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, clients advertise their capabilities in KeyPackage
   extensions, the creator of the group expresses its choices for the
   group in Welcome extensions, and the GroupContext confirms that all
   members of the group have the same view of the group's extensions.

   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
      KeyPackage for the new member contains extensions that are
      consistent with the group's extensions.  For each extension in the
      GroupContext, the KeyPackage MUST have an extension of the same
      type, and the contents of the extension MUST be consistent with
      the value of the extension in the GroupContext, according to the
      semantics of the specific extension.

   *  If any extension in a GroupInfo message is unrecognized (i.e., not
      contained in 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.

   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.

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13.  Sequencing of State Changes

   Each Commit message is premised on a given starting state, indicated
   by the "epoch" field of the enclosing MLSPlaintext message.  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

   *  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.

   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.

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13.1.  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

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

13.2.  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.

   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.

14.  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.

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   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.

14.1.  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
   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.

   Similarly to TLS 1.3, if padding is used, the MLS messages MUST be
   padded with zero-valued bytes before AEAD encryption.  Upon AEAD
   decryption, the length field of the plaintext is used to compute the
   number of bytes to be removed from the plaintext to get the correct
   data.  As the padding mechanism is used to improve protection against
   traffic analysis, removal of the padding SHOULD be implemented in a
   "constant-time" manner at the MLS layer and above layers to prevent
   timing side-channels that would provide attackers with information on
   the size of the plaintext.  The padding length length_of_padding can

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   be chosen at the time of the message encryption by the sender.
   Recipients can calculate the padding size from knowing the total size
   of the ApplicationPlaintext and the length of the content.

14.2.  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.

14.3.  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.

15.  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.

15.1.  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.

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   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.

15.2.  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
   identity key.

15.3.  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 InitKeys 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.

15.4.  InitKey Reuse

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

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   An application MAY allow for reuse of a "last resort" InitKey in
   order to prevent denial of service attacks.  Since an InitKey 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

16.  IANA Considerations

   This document requests the creation of the following new IANA

   *  MLS Ciphersuites (Section 16.1)

   *  MLS Extension Types (Section 16.2)

   *  MLS Credential Types (Section 16.3)

   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 16.4 for additional information about
   the MLS Designated Experts (DEs).

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

16.1.  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:


   Where VALUE is represented as a sixteen-bit integer:

   uint16 CipherSuite;

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          | Component | Contents                               |
          | MLS       | The string "MLS" followed by the major |
          |           | and minor version, e.g.  "MLS10"       |
          | LVL       | The security level                     |
          | KEM       | The KEM algorithm used for HPKE in     |
          |           | TreeKEM group 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 3

   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 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

   *  Reference: The document where this ciphersuite is defined

   Initial contents:

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   |Value |Name                                                 |Recommended|Reference|
   |0x0000|RESERVED                                             |N/A        |RFC XXXX |
   |0x0001|MLS10_128_DHKEMX25519_AES128GCM_SHA256_Ed25519       |Y          |RFC XXXX |
   |0x0002|MLS10_128_DHKEMP256_AES128GCM_SHA256_P256            |Y          |RFC XXXX |
   |0x0003|MLS10_128_DHKEMX25519_CHACHA20POLY1305_SHA256_Ed25519|Y          |RFC XXXX |
   |0x0004|MLS10_256_DHKEMX448_AES256GCM_SHA512_Ed448           |Y          |RFC XXXX |
   |0x0005|MLS10_256_DHKEMP521_AES256GCM_SHA512_P521            |Y          |RFC XXXX |
   |0x0006|MLS10_256_DHKEMX448_CHACHA20POLY1305_SHA512_Ed448    |Y          |RFC XXXX |
   |0xff00|Reserved for Private Use                             |N/A        |RFC XXXX |
   |-     |                                                     |           |         |
   |0xffff|                                                     |           |         |

                                  Table 4

   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 [I-D.irtf-cfrg-hpke] [RFC8446]:

   |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                  |

                                  Table 5

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   The hash used for the MLS transcript hash is the one referenced in
   the ciphersuite name.  In the ciphersuites defined above, "SHA256"
   and "SHA512" refer to the SHA-256 and SHA-512 functions defined in

   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 inlcude 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
   "MLS10\_128\_HPKE25519\_AES128GCM\_SHA256\_Ed25519" which uses
   Curve25519 for key exchange, AES-128-GCM for HPKE, HKDF over
   SHA2-256, AES for metadata masking, and Ed25519 for signatures.

   Values with the first byte 255 (decimal) are reserved for Private

   New ciphersuite values are assigned by IANA as described in
   Section 16.

16.2.  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.


   *  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 messages

      -  GI: GroupInfo objects

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   *  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

   *  Reference: The document where this extension is defined

   Initial contents:

    | Value    | Name         | Message(s) | Recommended | Reference |
    | 0x0000   | RESERVED     | N/A        | N/A         | RFC XXXX  |
    | 0x0001   | capabilities | KP         | Y           | RFC XXXX  |
    | 0x0002   | lifetime     | KP         | Y           | RFC XXXX  |
    | 0x0003   | key_id       | KP         | Y           | RFC XXXX  |
    | 0x0004   | parent_hash  | KP         | Y           | RFC XXXX  |
    | 0x0005   | ratchet_tree | GI         | Y           | RFC XXXX  |
    | 0xff00 - | Reserved for | N/A        | N/A         | RFC XXXX  |
    | 0xffff   | Private Use  |            |             |           |

                                 Table 6

16.3.  MLS Credential Types

   This registry lists identifiers for types of credentials that can be
   used for authentication in the MLS protocol.  The extension type
   field is two bytes wide, so valid extension type values are in the
   range 0x0000 to 0xffff.


   *  Value: The numeric value of the credential type

   *  Name: The name of the credential 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

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      adding a value with a "Recommended" value of "Y" requires
      Standards Action [RFC8126].  IESG Approval is REQUIRED for a Y->N

   *  Reference: The document where this extension 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 7

16.4.  MLS Designated Expert Pool

   Specification Required [RFC8126] registry requests are registered
   after a three-week review period on the MLS DEs' mailing list: mls- (, 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 ( mailing list.

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   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.

   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.

17.  Contributors

   *  Joel Alwen


   *  Karthikeyan Bhargavan


   *  Cas Cremers

      University of Oxford

   *  Alan Duric


   *  Britta Hale

      Naval Postgraduate School

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   *  Srinivas Inguva


   *  Konrad Kohbrok

      Aalto University

   *  Albert Kwon


   *  Brendan McMillion


   *  Eric Rescorla


   *  Michael Rosenberg

      Trail of Bits

   *  Thyla van der Merwe

      Royal Holloway, University of London

18.  References

18.1.  Normative References

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              Barnes, R., Bhargavan, K., Lipp, B., and C. Wood, "Hybrid
              Public Key Encryption", Work in Progress, Internet-Draft,
              draft-irtf-cfrg-hpke-06, 23 October 2020,

   [RFC2104]  Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed-
              Hashing for Message Authentication", RFC 2104,
              DOI 10.17487/RFC2104, February 1997,

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,

   [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,

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <>.

   [RFC8446]  Rescorla, E., "The Transport Layer Security (TLS) Protocol
              Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,

18.2.  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, <>.

   [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,

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              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,

   [HCJ16]    Husak, M., Cermak, M., Jirsik, T., and P. Celeda, "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,

              Omara, E., Beurdouche, B., Rescorla, E., Inguva, S., Kwon,
              A., and A. Duric, "The Messaging Layer Security (MLS)
              Architecture", Work in Progress, Internet-Draft, draft-
              ietf-mls-architecture-05, 26 July 2020,

              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-34,
              4 November 2019, <

   [RFC8032]  Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
              Signature Algorithm (EdDSA)", RFC 8032,
              DOI 10.17487/RFC8032, January 2017,

   [SECG]     "Elliptic Curve Cryptography, Standards for Efficient
              Cryptography Group, ver. 2", 2009,

   [SHS]      Dang, Q., "Secure Hash Standard", National Institute of
              Standards and Technology report,
              DOI 10.6028/nist.fips.180-4, July 2015,

   [signal]   Perrin(ed), T. and M. Marlinspike, "The Double Ratchet
              Algorithm", 20 November 2016,

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Appendix A.  Tree Math

   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, an 11-element tree has
   the following structure:

            X                       X                       X
      X           X           X           X           X
   X     X     X     X     X     X     X     X     X     X     X
   0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20

   This allows us to compute relationships between tree nodes simply by
   manipulating indices, rather than having to maintain complicated
   structures in memory, even for partial trees.  The basic rule is that
   the high-order bits of parent and child nodes have the following
   relation (where "x" is an arbitrary bit string):

   parent=01x => left=00x, right=10x

   The following python code demonstrates the tree computations
   necessary for MLS.  Test vectors can be derived from the diagram

# 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

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    return k

# The number of nodes needed to represent a tree with n leaves.
def node_width(n):
    if n == 0:
        return 0
        return 2*(n - 1) + 1

# 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')

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    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)
    if x < p:
        return right(p, n)
        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)
    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)
    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.

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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

    # 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

Authors' Addresses

   Richard Barnes


   Benjamin Beurdouche


   Jon Millican


   Emad Omara


   Katriel Cohn-Gordon
   University of Oxford


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   Raphael Robert


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