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Merkle Tree Certificates
draft-davidben-tls-merkle-tree-certs-08

Document Type Active Internet-Draft (individual)
Authors David Benjamin , Devon O'Brien , Bas Westerbaan , Luke Valenta , Filippo Valsorda
Last updated 2025-10-19
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draft-davidben-tls-merkle-tree-certs-08
Network Working Group                                        D. Benjamin
Internet-Draft                                                Google LLC
Intended status: Experimental                                 D. O'Brien
Expires: 22 April 2026                                                  
                                                        B. E. Westerbaan
                                                              L. Valenta
                                                              Cloudflare
                                                             F. Valsorda
                                                                  Geomys
                                                         19 October 2025

                        Merkle Tree Certificates
                draft-davidben-tls-merkle-tree-certs-08

Abstract

   This document describes Merkle Tree certificates, a new form of X.509
   certificates which integrate public logging of the certificate, in
   the style of Certificate Transparency.  The integrated design reduces
   logging overhead in the face of both shorter-lived certificates and
   large post-quantum signature algorithms, while still achieving
   comparable security properties to traditional X.509 and Certificate
   Transparency.  Merkle Tree certificates additionally admit an
   optional signatureless optimization, which decreases the message size
   by avoiding signatures altogether, at the cost of only applying to
   up-to-date relying parties and older certificates.

About This Document

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

   The latest revision of this draft can be found at
   https://davidben.github.io/merkle-tree-certs/draft-davidben-tls-
   merkle-tree-certs.html.  Status information for this document may be
   found at https://datatracker.ietf.org/doc/draft-davidben-tls-merkle-
   tree-certs/.

   Source for this draft and an issue tracker can be found at
   https://github.com/davidben/merkle-tree-certs.

Status of This Memo

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

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   This Internet-Draft will expire on 22 April 2026.

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   Please review these documents carefully, as they describe your rights
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Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   4
   2.  Conventions and Definitions . . . . . . . . . . . . . . . . .   5
     2.1.  Terminology and Roles . . . . . . . . . . . . . . . . . .   6
   3.  Overview  . . . . . . . . . . . . . . . . . . . . . . . . . .   8
   4.  Subtrees  . . . . . . . . . . . . . . . . . . . . . . . . . .  11
     4.1.  Definition of a Subtree . . . . . . . . . . . . . . . . .  11
     4.2.  Example Subtrees  . . . . . . . . . . . . . . . . . . . .  12
     4.3.  Subtree Inclusion Proofs  . . . . . . . . . . . . . . . .  14
       4.3.1.  Example Subtree Inclusion Proofs  . . . . . . . . . .  14
       4.3.2.  Evaluating a Subtree Inclusion Proof  . . . . . . . .  14
       4.3.3.  Verifying a Subtree Inclusion Proof . . . . . . . . .  15
     4.4.  Subtree Consistency Proofs  . . . . . . . . . . . . . . .  16
       4.4.1.  Generating a Subtree Consistency Proof  . . . . . . .  16
       4.4.2.  Example Subtree Consistency Proofs  . . . . . . . . .  17
       4.4.3.  Verifying a Subtree Consistency Proof . . . . . . . .  19
     4.5.  Arbitrary Intervals . . . . . . . . . . . . . . . . . . .  21
   5.  Issuance Logs . . . . . . . . . . . . . . . . . . . . . . . .  24
     5.1.  Log Parameters  . . . . . . . . . . . . . . . . . . . . .  24
     5.2.  Log IDs . . . . . . . . . . . . . . . . . . . . . . . . .  25
     5.3.  Log Entries . . . . . . . . . . . . . . . . . . . . . . .  25

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     5.4.  Cosigners . . . . . . . . . . . . . . . . . . . . . . . .  27
       5.4.1.  Signature Format  . . . . . . . . . . . . . . . . . .  27
       5.4.2.  Signature Algorithms  . . . . . . . . . . . . . . . .  29
     5.5.  Certification Authority Cosigners . . . . . . . . . . . .  29
     5.6.  Publishing Logs . . . . . . . . . . . . . . . . . . . . .  30
       5.6.1.  Log Pruning . . . . . . . . . . . . . . . . . . . . .  31
   6.  Certificates  . . . . . . . . . . . . . . . . . . . . . . . .  34
     6.1.  Certificate Format  . . . . . . . . . . . . . . . . . . .  34
     6.2.  Full Certificates . . . . . . . . . . . . . . . . . . . .  36
     6.3.  Signatureless Certificates  . . . . . . . . . . . . . . .  37
       6.3.1.  Landmarks . . . . . . . . . . . . . . . . . . . . . .  37
       6.3.2.  Allocating Landmarks  . . . . . . . . . . . . . . . .  39
       6.3.3.  Constructing Signatureless Certificates . . . . . . .  39
     6.4.  Size Estimates  . . . . . . . . . . . . . . . . . . . . .  40
   7.  Relying Parties . . . . . . . . . . . . . . . . . . . . . . .  41
     7.1.  Trust Anchors . . . . . . . . . . . . . . . . . . . . . .  41
     7.2.  Verifying Certificate Signatures  . . . . . . . . . . . .  41
     7.3.  Trusted Cosigners . . . . . . . . . . . . . . . . . . . .  43
     7.4.  Trusted Subtrees  . . . . . . . . . . . . . . . . . . . .  44
     7.5.  Revocation by Index . . . . . . . . . . . . . . . . . . .  45
   8.  Use in TLS  . . . . . . . . . . . . . . . . . . . . . . . . .  46
   9.  ACME Extensions . . . . . . . . . . . . . . . . . . . . . . .  47
   10. Deployment Considerations . . . . . . . . . . . . . . . . . .  48
     10.1.  Operational Costs  . . . . . . . . . . . . . . . . . . .  48
       10.1.1.  Certification Authority Costs  . . . . . . . . . . .  48
       10.1.2.  Cosigner Costs . . . . . . . . . . . . . . . . . . .  48
       10.1.3.  Monitor Costs  . . . . . . . . . . . . . . . . . . .  49
     10.2.  Choosing Cosigners . . . . . . . . . . . . . . . . . . .  49
     10.3.  Log Availability . . . . . . . . . . . . . . . . . . . .  50
     10.4.  Certificate Renewal  . . . . . . . . . . . . . . . . . .  52
     10.5.  Multiple CA Keys . . . . . . . . . . . . . . . . . . . .  52
   11. Privacy Considerations  . . . . . . . . . . . . . . . . . . .  53
   12. Security Considerations . . . . . . . . . . . . . . . . . . .  53
     12.1.  Authenticity . . . . . . . . . . . . . . . . . . . . . .  53
     12.2.  Transparency . . . . . . . . . . . . . . . . . . . . . .  54
     12.3.  Public Key Hashes  . . . . . . . . . . . . . . . . . . .  55
     12.4.  Non-Repudiation  . . . . . . . . . . . . . . . . . . . .  55
     12.5.  New Log Entry Types  . . . . . . . . . . . . . . . . . .  56
     12.6.  Certificate Malleability . . . . . . . . . . . . . . . .  56
   13. IANA Considerations . . . . . . . . . . . . . . . . . . . . .  58
     13.1.  Module Identifier  . . . . . . . . . . . . . . . . . . .  58
     13.2.  Algorithm  . . . . . . . . . . . . . . . . . . . . . . .  58
     13.3.  Relative Distinguished Name Attribute  . . . . . . . . .  59
   14. References  . . . . . . . . . . . . . . . . . . . . . . . . .  59
     14.1.  Normative References . . . . . . . . . . . . . . . . . .  59
     14.2.  Informative References . . . . . . . . . . . . . . . . .  60
   Appendix A.  ASN.1 Module . . . . . . . . . . . . . . . . . . . .  63
   Appendix B.  Merkle Tree Structure  . . . . . . . . . . . . . . .  64

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     B.1.  Binary Representations  . . . . . . . . . . . . . . . . .  64
     B.2.  Inclusion Proof Evaluation  . . . . . . . . . . . . . . .  66
     B.3.  Consistency Proof Structure . . . . . . . . . . . . . . .  67
     B.4.  Consistency Proof Verification  . . . . . . . . . . . . .  69
   Appendix C.  Extensions to Tiled Transparency Logs (To Be
           Removed)  . . . . . . . . . . . . . . . . . . . . . . . .  71
     C.1.  Subtree Signed Note Format  . . . . . . . . . . . . . . .  71
     C.2.  Requesting Subtree Signatures . . . . . . . . . . . . . .  71
   Acknowledgements  . . . . . . . . . . . . . . . . . . . . . . . .  73
   Change log  . . . . . . . . . . . . . . . . . . . . . . . . . . .  73
     Since draft-davidben-tls-merkle-tree-certs-00 . . . . . . . . .  74
     Since draft-davidben-tls-merkle-tree-certs-01 . . . . . . . . .  74
     Since draft-davidben-tls-merkle-tree-certs-02 . . . . . . . . .  74
     Since draft-davidben-tls-merkle-tree-certs-03 . . . . . . . . .  74
     Since draft-davidben-tls-merkle-tree-certs-04 . . . . . . . . .  75
     Since draft-davidben-tls-merkle-tree-certs-05 . . . . . . . . .  75
     Since draft-davidben-tls-merkle-tree-certs-06 . . . . . . . . .  76
     Since draft-davidben-tls-merkle-tree-certs-07 . . . . . . . . .  76
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  76

1.  Introduction

   Authors' Note: This is an early draft of a proposal with many parts.
   We expect most details will change as the proposal evolves.  This
   document has a concrete specification of these details, but this is
   only intended as a starting point, and to help convey the overall
   idea.  The name of the draft says "tls" to keep continuity with
   earlier iterations of this work, but the protocol itself is not TLS-
   specific.

   In Public Key Infrastructures (PKIs) that use Certificate
   Transparency (CT) [RFC6962] for a public logging requirement, an
   authenticating party must present Signed Certificate Timestamps
   (SCTs) alongside certificates.  CT policies often require two or more
   SCTs per certificate [APPLE-CT] [CHROME-CT], each of which carries a
   signature.  These signatures are in addition to those in the
   certificate chain itself.

   Current signature schemes can use as few as 32 bytes per key and 64
   bytes per signature [RFC8032], but post-quantum replacements are much
   larger.  For example, ML-DSA-44 [FIPS204] uses 1,312 bytes per public
   key and 2,420 bytes per signature.  ML-DSA-65 uses 1,952 bytes per
   public key and 3,309 bytes per signature.  Even with a directly-
   trusted intermediate (Section 7.5 of
   [I-D.ietf-tls-trust-anchor-ids]), two SCTs and a leaf certificate
   signature adds 7,260 bytes of authentication overhead with ML-DSA-44
   and 9,927 bytes with ML-DSA-65.

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   This increased overhead additionally impacts CT logs themselves.
   Most of a log's costs scale with the total storage size of the log.
   Each log entry contains both a public key, and a signature from the
   CA.  With larger public keys and signatures, the size of each log
   entry will grow.

   Additionally, as PKIs transition to shorter-lived certificates
   [CABF-153] [CABF-SC081], the number of entries in the log will grow.

   This document introduces Merkle Tree certificates, a new form of
   X.509 certificate that integrates logging with certificate issuance.
   Each CA maintains a log of everything it issues, signing views of the
   log to assert it has issued the contents.  The CA signature is
   combined with cosignatures from other parties who verify correct
   operation and optionally mirror the log.  These signatures, together
   with an inclusion proof for an individual entry, constitute a
   certificate.

   This achieves the following:

   *  Log entries do not scale with public key and signature sizes.
      Entries replace public keys with hashes and do not contain
      signatures, while preserving non-repudiability (Section 12.4).

   *  To bound growth, long-expired entries can be pruned from logs and
      mirrors without interrupting existing clients.  This allows log
      sizes to scale by retention policies, not the lifetime of the log,
      even as certificate lifetimes decrease.

   *  After a processing delay, authenticating parties can obtain a
      second "signatureless" certificate for the same log entry.  This
      second certificate is an optional size optimization that avoids
      the need for any signatures, assuming an up-to-date client that
      has some predistributed log information.

   Section 3 gives an overview of the system.  Section 4 describes a
   Merkle Tree primitive used by this system.  Section 5 describes the
   log structure.  Finally, Section 6 and Section 7 describe how to
   construct and consume a Merkle Tree certificate.

2.  Conventions and Definitions

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

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   This document additionally uses the TLS presentation language defined
   in Section 3 of [RFC8446], as well as the notation defined in
   Section 2.1.1 of [RFC9162].

   U+ followed by four hexadecimal characters denotes a Unicode
   codepoint, to be encoded in UTF-8 [RFC3629]. 0x followed by two
   hexadecimal characters denotes a byte value in the 0-255 range.

   [start, end), where start <= end, denotes the half-open interval
   containing integers x such that start <= x < end.

   Given a non-negative integer n,

   *  LSB(n) refers to the least-significant bit of n's binary
      representation.  Equivalently, it is the remainder when n is
      divided by 2.

   *  BIT_WIDTH(n) refers to the smallest number of bits needed to
      represent n.  BIT_WIDTH(0) is zero.

   *  POPCOUNT(n) refers to the number of set bits in n's binary
      representation`.

   To _right-shift_ a non-negative integer n is to shift each bit in its
   binary representation to one lower position, discarding the least-
   significant bit.  Equivalently, it is the floor of n divided by 2.
   Given non-negative integers a and b, a >> b refers to a right-shifted
   b times.

2.1.  Terminology and Roles

   This document discusses the following roles:

   Authenticating party:  The party that authenticates itself in the
      protocol.  In TLS, this is the side sending the Certificate and
      CertificateVerify message.

   Certification authority (CA):  The service that issues certificates
      to the authenticating party, after performing some validation
      process on the certificate contents.

   Relying party:  The party to whom the authenticating party presents
      its identity.  In TLS, this is the side receiving the Certificate
      and CertificateVerify message.

   Monitor:  Parties who watch logs for certificates of interest,
      analogous to the role in Section 8.2 of [RFC9162].

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   Issuance log:  A log, maintained by the CA, of everything issued by
      that CA.

   Cosigner:  A service that signs views of an issuance log, to assert
      correct operation and other properties about the entries.

   Additionally, there are several terms used throughout this document
   to describe this proposal.  This section provides an overview.  They
   will be further defined and discussed in detail throughout the
   document.

   Checkpoint:  A description of the complete state of the log at some
      time.

   Entry:  An individual element of the log, describing information
      which the CA has validated and certified.

   Subtree:  A smaller Merkle Tree over a portion of the log, defined by
      an interior node of some snapshot of the log.  Subtrees can be
      efficiently shown to be consistent with the whole log.

   Inclusion proof:  A sequence of hashes that efficiently proves some
      entry is contained in some checkpoint or subtree.

   Consistency proof:  A sequence of hashes that efficiently proves a
      checkpoint or subtree is contained within another checkpoint.

   Cosignature:  A signature from either the CA or other cosigner, over
      some checkpoint or subtree.

   Landmark:  One of an infrequent subset of tree sizes that can be used
      to predistribute trusted subtrees to relying parties for
      signatureless certificates.

   Landmark subtree:  A subtree determined by a landmark.  Landmark
      subtrees are common points of reference between relying parties
      and signatureless certificates.

   Full certificate:  A certificate containing an inclusion proof to
      some subtree, and several cosignatures over that subtree.

   Signatureless certificate:  An optimized certificate containing an
      inclusion proof to a landmark subtree, and no signatures.

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3.  Overview

   In Certificate Transparency, a CA first certifies information by
   signing it, then submits the resulting certificate (or
   precertificate) to logs for logging.  Merkle Tree Certificates
   inverts this process: the CA certifies information by logging it,
   then submits the log to cosigners to verify log operation.  A
   certificate is assembled from the result and proves the information
   is in the CA's log.

   +-- Certificate Authority -----+    +--  Authenticating Party ----+
   |                              |    |                             |
   |  2. Validate request     <---+----+--  1. Request certificate   |
   |       |                      |    |                             |
   |       |                      |    |                             |
   |       V                      |    |                             |
   |                              |    |                             |
   |  3. Add to issuance log      |    |                             |
   |       +---[ CA cosign ]      |    |                             |
   |      / \                 ----+----+->  5. Download certificates |
   |     /   \                    |    |                             |
   |    /     \                   |    |          *  tbscert         |
   |   +-------+                  |    |      = = =  inclusion proof |
   |    * * * *  tbscert entries  |    |     [ CA ]  cosignatures    |
   |                              |    | [ mirror ]                  |
   +------------------------------+    +-----------------------------+
              /   |   \
             /    |    \    4. Submit log to cosigners
            V     V     V      for cosignatures

   +-- Mirrors, other cosigners --+    +-- Monitors -----------------+
   |                              |    |                             |
   |       +---[ CA cosign ]      +-+  |                             |
   |      / \  [ mirror cosign ]  | |  |                             |
   |     /   \                    | |  |                             |
   |    /     \                 <-+-+--+--  6. Monitor CA operation  |
   |   +-------+                  | |  |                             |
   |    * * * *                   | |  +-----------------------------+
   +-+----------------------------+ |
     |  ...quorum of cosigners...   |
     +------------------------------+

      Figure 1: A diagram of the issuance architecture, detailed below

   Merkle Tree Certificates are issued as follows.  Figure 1 depicts
   this process.

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   1.  The authenticating party requests a certificate, e.g. over ACME
       [RFC8555]

   2.  The CA validates each incoming issuance request, e.g. with ACME
       challenges.  From there, the process differs.

   3.  The CA operates an append-only _issuance log_ (Section 5).
       Unlike a CT log, this log only contains entries added by the CA:

       1.  The CA adds a TBSCertificateLogEntry (Section 5.3) to its
           log, describing the information it is certifying.

       2.  The CA signs a _checkpoint_, which describes the current
           state of the log.  A signed checkpoint certifies that the CA
           issued _every_ entry in the Merkle Tree (Section 5.5).

       3.  The CA additionally signs _subtrees_ (Section 4) that
           together contain certificates added since the last checkpoint
           (Section 4.5).  This is an optimization to reduce inclusion
           proof sizes.  A signed subtree certifies that the CA has
           issued _every_ entry in the subtree.

   4.  The CA submits the new log state to _cosigners_. Cosigners
       validate the log is append-only and optionally provide additional
       services, such as mirroring its contents.  They cosign the CA's
       checkpoints and subtrees.

   5.  The CA now has enough information to construct a certificate and
       give it to the authenticating party.  A certificate contains:

       *  The TBSCertificate being certified

       *  An inclusion proof from the TBSCertificate to some subtree

       *  Cosignatures from the CA and cosigners on the subtree

   6.  As in Certificate Transparency, monitors observe the issuance log
       to ensure the CA is operated correctly.

   A certificate with cosignatures is known as a _full certificate_.
   Analogous to X.509 trust anchors and trusted CT logs, relying parties
   are configured with trusted cosigners (Section 7.3) that allow them
   to accept Merkle Tree certificates.  The inclusion proof proves the
   TBSCertificate is part of some subtree, and cosignatures from trusted
   cosigners prove the subtree was certified by the CA and available to
   monitors.  Where CT logs entire certificates, the issuance log's
   entries are smaller TBSCertificateLogEntry (Section 5.3) structures,
   which do not scale with public key or signature size.

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   This same issuance process also produces a _signatureless
   certificate_. This is an optional, optimized certificate that avoids
   all cosignatures, including the CA signature.  Signatureless
   certificates are available after a short period of time and usable
   with up-to-date relying parties.

   +-- Certificate Authority -------+
   |                                |  +-- Update Channel --+
   |    /\                          |  |                    |
   |   /  \  1. Allocate landmarks -+--+----------------+   |
   |  +----+                  |     |  |                |   |
   +--------------------------+-----+  +----------------+---+
                              |                         |
       2. Make signatureless  |          3. Distribute  |
          cert from landmark  |              landmarks  |
                              V                         |
   +-- Authenticating Party --------+                   |
   |                                |                   |
   | signatureless cert             |                   V
   |   tbscert                      |  +-- Up-to-date RP -----+
   |   inclusion proof to landmark -+->| landmark hashes      |
   |                                |  | trusted cosigners    |
   |                                |  +----------------------+
   | full cert                      |
   |   tbscert                      |  +-- Unupdated RP ------+
   |   inclusion proof              |  | (stale or no hashes) |
   |   cosignatures     ------------+->| trusted cosigners    |
   |                                |  +----------------------+
   +--------------------------------+
                        4. Select certificate by RP

     Figure 2: A diagram of signatureless certificate construction and
                           usage, detailed below

   Signatureless certificates are constructed and used as follows.
   Figure 2 depicts this process.

   1.  Periodically, the tree size of the CA's most recent checkpoint is
       designated as a _landmark_. This determines _landmark subtrees_,
       which are common points of reference between relying parties and
       signatureless certificates.

   2.  Once some landmark includes the TBSCertificate, the signatureless
       certificate is constructed with:

       *  The TBSCertificate being certified

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       *  An inclusion proof from the TBSCertificate to a landmark
          subtree

   3.  In the background, landmark subtrees are predistributed to
       relying parties, with cosignatures checked against relying party
       requirements.  This occurs periodically in the background,
       separate from the application protocol.

   4.  During the application protocol, such as TLS [RFC8446], if the
       relying party already supports the landmark subtree, the
       authenticating party can present the signatureless certificate.
       Otherwise, it presents a full certificate.  The authenticating
       party may also select between several signatureless certificates,
       as described in Section 10.4.

4.  Subtrees

   This section extends the Merkle Tree definition in Section 2.1 of
   [RFC9162] by defining a _subtree_ of a Merkle Tree.  A subtree is an
   interior node of a Merkle Tree, which can be efficiently shown
   consistent with the original Merkle Tree and any Merkle Tree with
   additional elements appended.  This specification uses subtrees to
   reduce the size of inclusion proofs.

4.1.  Definition of a Subtree

   Given an ordered list of n inputs, D_n = {d[0], d[1], ..., d[n-1]},
   Section 2.1.1 of [RFC9162] defines the Merkle Tree via the Merkle
   Tree Hash MTH(D_n).

   A _subtree_ of this Merkle Tree is defined by two integers, start and
   end, such that:

   *  0 <= start < end <= n

   *  If s is the smallest power of two that is greater than or equal to
      end - start, start must be a multiple of s

   Note that, if start is zero, the second condition is always true.

   A subtree is itself a Merkle Tree, defined by MTH(D[start:end]).  In
   the context of a single Merkle Tree, the subtree defined by start and
   end is denoted by half-open interval [start, end).  It contains the
   entries whose indices are in that half-open interval.

   The _size_ of the subtree is end - start.  If the subtree's size is a
   power of two, it is said to be _full_, otherwise it is said to be
   _partial_.

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   If a subtree is full, then it is directly contained in the tree of
   hash operations in MTH(D_n) for n >= end.

   If a subtree is partial, it is directly contained in MTH(D_n) only if
   n = end.

4.2.  Example Subtrees

   Figure 3 shows the subtrees [4, 8) and [8, 13):

      +--------+
      | [4, 8) |
      +--------+
       /      \
   +-----+ +-----+
   |[4,6)| |[6,8)|
   +-----+ +-----+
     / \     / \
   +-+ +-+ +-+ +-+
   |4| |5| |6| |7|
   +-+ +-+ +-+ +-+

         +----------------+
         |     [8, 13)    |
         +----------------+
            /          |
      +---------+      |
      | [8, 12) |      |
      +---------+      |
        /      \       |
   +------+ +-------+  |
   |[8,10)| |[10,12)|  |
   +------+ +-------+  |
     / \      / \      |
   +-+ +-+ +--+ +--+ +--+
   |8| |9| |10| |11| |12|
   +-+ +-+ +--+ +--+ +--+

          Figure 3: Two example subtrees, one full and one partial

   Both subtrees are directly contained in a Merkle Tree of size 13,
   depicted in Figure 4. [4, 8) is contained because, although n (13) is
   not end (8), the subtree is full. [8, 13) is contained because n (13)
   is end (13).

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                   +-----------------------------+
                   |            [0, 13)          |
                   +-----------------------------+
                      /                       \
          +----------------+             +~~~~~~~~~~~~~~~~+
          |     [0, 8)     |             |     [8, 13)    |
          +----------------+             +~~~~~~~~~~~~~~~~+
           /              \                 /          |
      +--------+      +========+      +~~~~~~~~~+      |
      | [0, 4) |      | [4, 8) |      | [8, 12) |      |
      +--------+      +========+      +~~~~~~~~~+      |
       /      \        /      \         /      \       |
   +-----+ +-----+ +=====+ +=====+ +~~~~~~+ +~~~~~~~+  |
   |[0,2)| |[2,4)| |[4,6)| |[6,8)| |[8,10)| |[10,12)|  |
   +-----+ +-----+ +=====+ +=====+ +~~~~~~+ +~~~~~~~+  |
     / \     / \     / \     / \     / \      / \      |
   +-+ +-+ +-+ +-+ +=+ +=+ +=+ +=+ +~+ +~+ +~~+ +~~+ +~~+
   |0| |1| |2| |3| |4| |5| |6| |7| |8| |9| |10| |11| |12|
   +-+ +-+ +-+ +-+ +=+ +=+ +=+ +=+ +~+ +~+ +~~+ +~~+ +~~+

                     Figure 4: A Merkle Tree of size 13

   In contrast, [8, 13) is not directly contained in a Merkle Tree of
   size 14, depicted in Figure 5.  However, the subtree is still
   computed over consistent elements.

                   +-----------------------------+
                   |            [0, 14)          |
                   +-----------------------------+
                      /                       \
          +----------------+             +----------------+
          |     [0, 8)     |             |     [8, 14)    |
          +----------------+             +----------------+
           /              \                 /           |
      +--------+      +--------+      +---------+       |
      | [0, 4) |      | [4, 8) |      | [8, 12) |       |
      +--------+      +--------+      +---------+       |
       /      \        /      \         /      \        |
   +-----+ +-----+ +-----+ +-----+ +------+ +-------+ +-------+
   |[0,2)| |[2,4)| |[4,6)| |[6,8)| |[8,10)| |[10,12)| |[12,14)|
   +-----+ +-----+ +-----+ +-----+ +------+ +-------+ +-------+
     / \     / \     / \     / \     / \      / \       / \
   +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +~+ +~+ +~~+ +~~+ +~~+ +--+
   |0| |1| |2| |3| |4| |5| |6| |7| |8| |9| |10| |11| |12| |13|
   +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +~+ +~+ +~~+ +~~+ +~~+ +--+

                     Figure 5: A Merkle Tree of size 14

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4.3.  Subtree Inclusion Proofs

   Subtrees are Merkle Trees, so entries can be proven to be contained
   in the subtree.  A subtree inclusion proof for entry index of the
   subtree [start, end) is a Merkle inclusion proof, as defined in
   Section 2.1.3.1 of [RFC9162], where m is index - start and the tree
   inputs are D[start:end].

   Subtree inclusion proofs contain a sequence of nodes that are
   sufficient to reconstruct the subtree hash, MTH(D[start:end]), out of
   the hash for entry index, MTH({d[index]}), thus demonstrating that
   the subtree hash contains the entry's hash.

4.3.1.  Example Subtree Inclusion Proofs

   The inclusion proof for entry 10 of subtree [8, 13) contains the
   hashes MTH({d[11]}), MTH(D[8:10]), and MTH({d[12]}), depicted in
   Figure 6.  MTH({d[10]}) is not part of the proof because the verifier
   is assumed to already know its value.

         +----------------+
         |     [8, 13)    |
         +----------------+
            /          |
      +---------+      |
      | [8, 12) |      |
      +---------+      |
        /      \       |
   +======+ +-------+  |
   |[8,10)| |[10,12)|  |
   +======+ +-------+  |
     / \      / \      |
   +-+ +-+ +~~+ +==+ +==+
   |8| |9| |10| |11| |12|
   +-+ +-+ +~~+ +==+ +==+

                Figure 6: An example subtree inclusion proof

4.3.2.  Evaluating a Subtree Inclusion Proof

   Given a subtree inclusion proof, inclusion_proof, for entry index,
   with hash entry_hash, of a subtree [start, end), the subtree
   inclusion proof can be _evaluated_ to compute the expected subtree
   hash:

   1.  Check that [start, end) is a valid subtree (Section 4.1), and
       that start <= index < end.  If either do not hold, fail proof
       evaluation.

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   2.  Set fn to index - start and sn to end - start - 1.

   3.  Set r to entry_hash.

   4.  For each value p in the inclusion_proof array:

       1.  If sn is 0, then stop the iteration and fail proof
           evaluation.

       2.  If LSB(fn) is set, or if fn is equal to sn, then:

           1.  Set r to HASH(0x01 || p || r).

           2.  Until LSB(fn) is set, right-shift fn and sn equally.

           Otherwise:

           1.  Set r to HASH(0x01 || r || p).

       3.  Finally, right-shift both fn and sn one time.

   5.  If sn is not zero, fail proof evaluation.

   6.  Return r as the expected subtree hash.

   This is the same as the procedure in Section 2.1.3.2 of [RFC9162],
   where leaf_index is index - start, tree_size is end - start, and r is
   returned instead of compared with root_hash.

   Appendix B.2 explains this procedure in more detail.

4.3.3.  Verifying a Subtree Inclusion Proof

   Given a subtree inclusion proof, inclusion_proof, for entry index,
   with hash entry_hash, of a subtree [start, end) with hash
   subtree_hash, the subtree inclusion proof can be _verified_ to verify
   the described entry is contained in the subtree:

   1.  Let expected_subtree_hash be the result of evaluating the
       inclusion proof as described Section 4.3.2.  If evaluation fails,
       fail the proof verification.

   2.  If subtree_hash is equal to expected_subtree_hash, the entry is
       contained in the subtree.  Otherwise, fail the proof
       verification.

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4.4.  Subtree Consistency Proofs

   A subtree [start, end) can be efficiently proven to be consistent
   with the full Merkle Tree.  That is, given MTH(D[start:end]) and
   MTH(D_n), the proof demonstrates that the input D[start:end] to the
   subtree hash was equal to the corresponding elements of the input D_n
   to the Merkle Tree hash.

   Subtree consistency proofs contain sufficient nodes to reconstruct
   both the subtree hash, MTH(D[start:end]), and the full tree hash,
   MTH(D_n), in such a way that every input to the subtree hash was also
   incorporated into the full tree hash.

4.4.1.  Generating a Subtree Consistency Proof

   The subtree consistency proof, SUBTREE_PROOF(start, end, D_n) is
   defined similarly to Section 2.1.4.1 of [RFC9162], in terms of a
   helper function that tracks whether the subtree hash is known:

   SUBTREE_PROOF(start, end, D_n) =
       SUBTREE_SUBPROOF(start, end, D_n, true)

   If start = 0 and end = n, the subtree is the root:

   SUBTREE_SUBPROOF(0, n, D_n, true) = {}
   SUBTREE_SUBPROOF(0, n, D_n, false) = {MTH(D_n)}

   Otherwise, n > 1.  Let k be the largest power of two smaller than n.
   The consistency proof is defined recursively as:

   *  If end <= k, the subtree is on the left of k.  The proof proves
      consistency with the left child and includes the right child:

      SUBTREE_SUBPROOF(start, end, D_n, b) =
          SUBTREE_SUBPROOF(start, end, D[0:k], b) : MTH(D[k:n])

   *  If k <= start, the subtree is on the right of k.  The proof proves
      consistency with the right child and includes the left child.

      SUBTREE_SUBPROOF(start, end, D_n, b) =
          SUBTREE_SUBPROOF(start - k, end - k, D[k:n], b) : MTH(D[0:k])

   *  Otherwise, start < k < end, which implies start = 0.  The proof
      proves consistency with the right child and includes the left
      child.

      SUBTREE_SUBPROOF(0, end, D_n, b) =
          SUBTREE_SUBPROOF(0, end - k, D[k:n], false) : MTH(D[0:k])

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   When start is zero, this computes a Merkle consistency proof:

   SUBTREE_PROOF(0, end, D_n) = PROOF(end, D_n)

   When end = start + 1, this computes a Merkle inclusion proof:

   SUBTREE_PROOF(start, start + 1, D_n) = PATH(start, D_n)

   Appendix B.3 explains the structure of a subtree consistency proof in
   more detail.

4.4.2.  Example Subtree Consistency Proofs

   The subtree consistency proof for [4, 8) and a tree of size 14
   contains MTH(D[0:4]) and MTH(D[8:14]), depicted in Figure 7.  The
   verifier is assumed to know the subtree hash, so there is no need to
   include MTH(D[4:8]) itself in the consistency proof.

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      +~~~~~~~~+
      | [4, 8) |
      +~~~~~~~~+
       /      \
   +-----+ +-----+
   |[4,6)| |[6,8)|
   +-----+ +-----+
     / \     / \
   +-+ +-+ +-+ +-+
   |4| |5| |6| |7|
   +-+ +-+ +-+ +-+

                   +-----------------------------+
                   |            [0, 14)          |
                   +-----------------------------+
                      /                       \
          +----------------+             +================+
          |     [0, 8)     |             |     [8, 14)    |
          +----------------+             +================+
           /              \                 /           |
      +========+      +~~~~~~~~+      +---------+       |
      | [0, 4) |      | [4, 8) |      | [8, 12) |       |
      +========+      +~~~~~~~~+      +---------+       |
       /      \        /      \         /      \        |
   +-----+ +-----+ +-----+ +-----+ +------+ +-------+ +-------+
   |[0,2)| |[2,4)| |[4,6)| |[6,8)| |[8,10)| |[10,12)| |[12,14)|
   +-----+ +-----+ +-----+ +-----+ +------+ +-------+ +-------+
     / \     / \     / \     / \     / \      / \       / \
   +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +--+ +--+ +--+ +--+
   |0| |1| |2| |3| |4| |5| |6| |7| |8| |9| |10| |11| |12| |13|
   +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +--+ +--+ +--+ +--+

     Figure 7: An example subtree consistency proof for a subtree that
                   is directly contained in the full tree

   The subtree consistency proof for [8, 13) and a tree of size 14
   contains MTH({d[12]}), MTH({d[13]}), MTH(D[8:12]), and MTH(D[0:8]),
   depicted in Figure 8. [8, 13) is not directly contained in the tree,
   so the proof must include sufficient nodes to reconstruct both
   hashes.

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         +----------------+
         |     [8, 13)    |
         +----------------+
            /          |
      +=========+      |
      | [8, 12) |      |
      +=========+      |
        /      \       |
   +------+ +-------+  |
   |[8,10)| |[10,12)|  |
   +------+ +-------+  |
     / \      / \      |
   +-+ +-+ +--+ +--+ +==+
   |8| |9| |10| |11| |12|
   +-+ +-+ +--+ +--+ +==+

                   +-----------------------------+
                   |            [0, 14)          |
                   +-----------------------------+
                      /                       \
          +================+             +----------------+
          |     [0, 8)     |             |     [8, 14)    |
          +================+             +----------------+
           /              \                 /           |
      +--------+      +--------+      +=========+       |
      | [0, 4) |      | [4, 8) |      | [8, 12) |       |
      +--------+      +--------+      +=========+       |
       /      \        /      \         /      \        |
   +-----+ +-----+ +-----+ +-----+ +------+ +-------+ +-------+
   |[0,2)| |[2,4)| |[4,6)| |[6,8)| |[8,10)| |[10,12)| |[12,14)|
   +-----+ +-----+ +-----+ +-----+ +------+ +-------+ +-------+
     / \     / \     / \     / \     / \      / \       / \
   +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +--+ +--+ +==+ +==+
   |0| |1| |2| |3| |4| |5| |6| |7| |8| |9| |10| |11| |12| |13|
   +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +--+ +--+ +==+ +==+

     Figure 8: An example subtree consistency proof for a subtree that
                 is not directly contained in the full tree

4.4.3.  Verifying a Subtree Consistency Proof

   The following procedure can be used to verify a subtree consistency
   proof.

   Given a Merkle Tree over n elements, a subtree defined by [start,
   end), a consistency proof proof, a subtree hash node_hash, and a root
   hash root_hash:

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   1.  Check that [start, end) is a valid subtree (Section 4.1), and
       that end <= n.  If either do not hold, fail proof verification.
       These checks imply 0 <= start < end <= n.

   2.  Set fn to start, sn to end - 1, and tn to n - 1.

   3.  If sn is tn, then:

       1.  Until fn is sn, right-shift fn, sn, and tn equally.

   4.  Otherwise:

       1.  Until fn is sn or LSB(sn) is not set, right-shift fn, sn, and
           tn equally.

   5.  If fn is sn, set fr and sr to node_hash.

   6.  Otherwise:

       1.  If proof is an empty array, stop and fail verification.

       2.  Remove the first value of the proof array and set fr and sr
           to the removed value.

   7.  For each value c in the proof array:

       1.  If tn is 0, then stop the iteration and fail the proof
           verification.

       2.  If LSB(sn) is set, or if sn is equal to tn, then:

           1.  If fn < sn, set fr to HASH(0x01 || c || fr).

           2.  Set sr to HASH(0x01 || c || sr).

           3.  Until LSB(sn) is set, right-shift fn, sn, and tn equally.

       3.  Otherwise:

           1.  Set sr to HASH(0x01 || sr || c).

       4.  Right-shift fn, sn, and tn once more.

   8.  Compare tn to 0, fr to node_hash, and sr to root_hash.  If any
       are not equal, fail the proof verification.  If all are equal,
       accept the proof.

   Appendix B.4 explains this procedure in more detail.

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4.5.  Arbitrary Intervals

   Not all [start, end) intervals of a Merkle Tree are valid subtrees.
   This section describes how, for any start < end, to determine up to
   two subtrees that efficiently cover the interval.  The subtrees are
   determined by the following procedure:

   1.  If end - start is one, return a single subtree, [start, end).

   2.  Otherwise, run the following to return a pair of subtrees:

       1.  Let last be end - 1, the last index in [start, end).

       2.  Let split be the bit index of the most significant bit where
           start and last differ.  Bits are numbered from the least
           significant bit, starting at zero. split is the height at
           which start and last's paths in the tree diverge.

       3.  Let mid be last with the least significant split bits set to
           zero. mid is the leftmost leaf node in the above divergence
           point's right branch.

       4.  Within the least significant split bits of left, let b be the
           bit index of the most significant bit with value zero, if
           any:

           1.  If there is such a bit, let left_split be b + 1.

           2.  Otherwise, let left_split be zero.

           left_split is the height of the lowest common ancestor of the
           nodes in [start, mid).

       5.  Let left_start be start with the least significant left_split
           bits set to zero. left_start is the above lowest common
           ancestor's leftmost leaf node.

       6.  Return the subtrees [left_start, mid) and [mid, end).

   When the procedure returns a single subtree, the subtree is [start,
   start+1).  When it returns two subtrees, left and right, the subtrees
   satisfy the following properties:

   *  left.end = right.start.  That is, the two subtrees cover adjacent
      intervals.

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   *  left.start <= start and end = right.end.  That is, the two
      subtrees together cover the entire target interval, possibly with
      some extra entries before start left, but not after end.

   *  left.end - left.start < 2 * (end - start) and right.end -
      right.start <= end - start.  That is, the two subtrees efficiently
      cover the interval.

   *  left is full, while right may be partial.

   The following Python code implements this procedure:

   def find_subtrees(start, end):
       """ Returns a list of one or two subtrees that efficiently
       cover [start, end). """
       assert start < end
       if end - start == 1:
           return [(start, end),]
       last = end - 1
       # Find where start and last's tree paths diverge. The two
       # subtrees will be on either side of the split.
       split = (start ^ last).bit_length() - 1
       mask = (1 << split) - 1
       mid = last & ~mask
       # Maximize the left endpoint. This is just before start's
       # path leaves the right edge of its new subtree.
       left_split = (~start & mask).bit_length()
       left_start = start & ~((1 << left_split) - 1)
       return [(left_start, mid), (mid, end)]

   Figure 9 shows the subtrees which cover [5, 13) in a Merkle Tree of
   13 elements.  The two subtrees selected are [4, 8) and [8, 13).  Note
   that the subtrees cover a slightly larger interval than [5, 13).

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                   +-----------------------------+
                   |            [0, 13)          |
                   +-----------------------------+
                      /                       \
          +----------------+             +================+
          |     [0, 8)     |             |     [8, 13)    |
          +----------------+             +================+
           /              \                 /          |
      +--------+      +========+      +---------+      |
      | [0, 4) |      | [4, 8) |      | [8, 12) |      |
      +--------+      +========+      +---------+      |
       /      \        /      \         /      \       |
   +-----+ +-----+ +-----+ +-----+ +------+ +-------+  |
   |[0,2)| |[2,4)| |[4,6)| |[6,8)| |[8,10)| |[10,12)|  |
   +-----+ +-----+ +-----+ +-----+ +------+ +-------+  |
     / \     / \     / \     / \     / \      / \      |
   +-+ +-+ +-+ +-+ +-+ +=+ +=+ +=+ +=+ +=+ +==+ +==+ +==+
   |0| |1| |2| |3| |4| |5| |6| |7| |8| |9| |10| |11| |12|
   +-+ +-+ +-+ +-+ +-+ +=+ +=+ +=+ +=+ +=+ +==+ +==+ +==+

      Figure 9: An example selection of subtrees to cover an interval

   Two subtrees are needed because a single subtree may not be able to
   efficiently cover an interval.  Figure 10 shows the smallest subtree
   that contains [7, 9) in a 9-element tree.  The smallest single
   subtree that contains the interval is [0, 9) but this is the entire
   tree.  Using two subtrees, the interval can be described by [7, 8)
   and [8, 9).

                   +===================+
                   |      [0, 9)       |
                   +===================+
                      /             |
          +----------------+        |
          |     [0, 8)     |        |
          +----------------+        |
           /              \         |
      +--------+      +--------+    |
      | [0, 4) |      | [4, 8) |    |
      +--------+      +--------+    |
       /      \        /      \     |
   +-----+ +-----+ +-----+ +-----+  |
   |[0,2)| |[2,4)| |[4,6)| |[6,8)|  |
   +-----+ +-----+ +-----+ +-----+  |
     / \     / \     / \     / \    |
   +-+ +-+ +-+ +-+ +-+ +-+ +-+ +=+ +=+
   |0| |1| |2| |3| |4| |4| |6| |7| |8|
   +-+ +-+ +-+ +-+ +-+ +-+ +-+ +=+ +=+

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      Figure 10: An example showing an inefficient choice of a single
                                  subtree

5.  Issuance Logs

   This section defines the structure of an _issuance log_.

   An issuance log describes an append-only sequence of _entries_
   (Section 5.3), identified consecutively by an index value, starting
   from zero.  Each entry is an assertion that the CA has certified.
   The entries in the issuance log are represented as a Merkle Tree,
   described in Section 2.1 of [RFC9162].

   Unlike [RFC6962] and [RFC9162], an issuance log does not have a
   public submission interface.  The log only contains entries which the
   log operator, i.e. the CA, chose to add.  As entries are added, the
   Merkle Tree is updated to be computed over the new sequence.

   A snapshot of the log is known as a _checkpoint_. A checkpoint is
   identified by its _tree size_, that is the number of elements
   comitted to the log at the time.  Its contents can be described by
   the Merkle Tree Hash (Section 2.1.1 of [RFC9162]) of entries zero
   through tree_size - 1.

   Cosigners (Section 5.4) sign assertions about the state of the
   issuance log.  A Merkle Tree CA operates a combination of an issuance
   log and one or more CA cosigners (Section 5.5) that authenticate the
   log state and certifies the contents.  External cosigners may also be
   deployed to assert correct log operation or provide other services to
   relying parties (Section 7.3).

5.1.  Log Parameters

   An issuance log has the following parameters:

   *  A log ID, which uniquely identifies the log.  See Section 5.2.

   *  A collision-resistant cryptographic hash function.  SHA-256 [SHS]
      is RECOMMENDED.

   *  A minimum index, which is the index of the first log entry which
      is available.  See Section 5.6.1.  This value changes over the
      lifetime of the log.

   Throughout this document, the hash algorithm in use is referred to as
   HASH, and the size of its output in bytes is referred to as
   HASH_SIZE.

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5.2.  Log IDs

   Each issuance log is identified by a _log ID_, which is a trust
   anchor ID [I-D.ietf-tls-trust-anchor-ids].

   An issuance log's log ID determines an X.509 distinguished name
   (Section 4.1.2.4 of [RFC5280]).  The distinguished name has a single
   relative distinguished name, which has a single attribute.  The
   attribute has type id-rdna-trustAnchorID, defined below:

   id-rdna-trustAnchorID OBJECT IDENTIFIER ::= {
       iso(1) identified-organization(3) dod(6) internet(1) security(5)
       mechanisms(5) pkix(7) rdna(25) TBD}

   The attribute's value is a RELATIVE-OID containing the trust anchor
   ID's ASN.1 representation.  For example, the distinguished name for a
   log named 32473.1 would be represented in syntax of [RFC4514] as:

   1.3.6.1.5.5.7.25.TBD=#0d0481fd5901

   For initial experimentation, early implementations of this design
   will:

   1.  Use UTF8String to represent the attribute's value rather than
       RELATIVE-OID.  The UTF8String contains trust anchor ID's ASCII
       representation, e.g. 324731.1.

   2.  Use the OID 1.3.6.1.4.1.44363.47.1 instead of id-rdna-
       trustAnchorID.

   For example, the distinguished name for a log named 32473.1 would be
   represented in syntax of [RFC4514] as:

   1.3.6.1.4.1.44363.47.1=#0c0733323437332e31

5.3.  Log Entries

   Each entry in the log is a MerkleTreeCertEntry, defined with the TLS
   presentation syntax below.  A MerkleTreeCertEntry describes
   certificate information that the CA has validated and certified.

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   struct {} Empty;

   enum {
       null_entry(0), tbs_cert_entry(1), (2^16-1)
   } MerkleTreeCertEntryType;

   struct {
       MerkleTreeCertEntryType type;
       select (type) {
          case null_entry: Empty;
          case tbs_cert_entry: opaque tbs_cert_entry_data[N];
          /* May be extended with future types. */
       }
   } MerkleTreeCertEntry;

   When type is tbs_cert_entry, N is the number of bytes needed to
   consume the rest of the input.  A MerkleTreeCertEntry is expected to
   be decoded in contexts where the total length of the entry is known.

   tbs_cert_entry_data contains the DER [X.690] encoding of a
   TBSCertificateLogEntry, defined below:

   TBSCertificateLogEntry  ::=  SEQUENCE  {
         version             [0]  EXPLICIT Version DEFAULT v1,
         issuer                   Name,
         validity                 Validity,
         subject                  Name,
         subjectPublicKeyInfoHash OCTET STRING,
         issuerUniqueID      [1]  IMPLICIT UniqueIdentifier OPTIONAL,
         subjectUniqueID     [2]  IMPLICIT UniqueIdentifier OPTIONAL,
         extensions          [3]  EXPLICIT Extensions OPTIONAL }

   The version, issuer, validity, subject, issuerUniqueID,
   subjectUniqueID, and extensions fields have the corresponding
   semantics as in Section 4.1.2 of [RFC5280], with the exception of
   subjectPublicKeyInfoHash. subjectPublicKeyInfoHash contains the hash
   of subject's public key as a SubjectPublicKeyInfo (Section 4.1.2.7 of
   [RFC5280]).  The hash uses the log's hash function (Section 5.1) and
   is computed over the SubjectPublicKeyInfo's DER [X.690] encoding.
   The issuer field MUST be the issuance log's log ID as an X.509
   distinguished name, as described in Section 5.2.

   When type is null_entry, the entry does not represent any
   information.  The entry at index zero of every issuance log MUST be
   of type null_entry.  Other entries MUST NOT use null_entry.
   null_entry exists to avoid zero serial numbers in the certificate
   format (Section 6.1).

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   MerkleTreeCertEntry is an extensible structure.  Future documents may
   define new values for MerkleTreeCertEntryType, with corresponding
   semantics.  See Section 5.5 and Section 12.5 for additional
   discussion.

5.4.  Cosigners

   This section defines a log _cosigner_. A cosigner follows some
   append-only view of the log and signs subtrees (Section 4) consistent
   with that view.  The signatures generated by a cosigner are known as
   _cosignatures_. All subtrees signed by a cosigner MUST be consistent
   with each other.  The cosigner may be external to the log, in which
   case it might ensure consistency by checking consistency proofs.  The
   cosigner may be operated together with the log, in which case it can
   trust its log state.

   A cosignature MAY implicitly make additional statements about a
   subtree, determined by the cosigner's role.  This document defines
   one concrete cosigner role, a CA cosigner (Section 5.5), to
   authenticate the log and certify entries.  Other documents and
   specific deployments may define other cosigner roles, to perform
   different functions in a PKI.  For example, [TLOG-WITNESS] defines a
   cosigner that only checks the log is append-only, and [TLOG-MIRROR]
   defines a cosigner that mirrors a log.

   Each cosigner has a public key and a _cosigner ID_, which uniquely
   identifies the cosigner.  The cosigner ID is a trust anchor ID
   [I-D.ietf-tls-trust-anchor-ids].  By identifying the cosigner, the
   cosigner ID specifies both the public key and the additional
   statements made by the cosigner's signatures.  If a single operator
   performs multiple cosigner roles in an ecosystem, each role MUST use
   a distinct cosigner ID and SHOULD use a distinct key.

   A single cosigner, with a single cosigner ID and public key, MAY
   generate cosignatures for multiple logs.  In this case, signed
   subtrees only need to be consistent with others for the same log.

5.4.1.  Signature Format

   A cosigner computes a cosignature for a subtree in some log by
   signing a MTCSubtreeSignatureInput, defined below using the TLS
   presentation language (Section 3 of [RFC8446]):

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   opaque HashValue[HASH_SIZE];

   /* From Section 4.1 of draft-ietf-tls-trust-anchor-ids */
   opaque TrustAnchorID<1..2^8-1>;

   struct {
       TrustAnchorID log_id;
       uint64 start;
       uint64 end;
       HashValue hash;
   } MTCSubtree;

   struct {
       uint8 label[16] = "mtc-subtree/v1\n\0";
       TrustAnchorID cosigner_id;
       MTCSubtree subtree;
   } MTCSubtreeSignatureInput;

   log_id MUST be the issuance log's ID (Section 5.2), in its binary
   representation (Section 3 of [I-D.ietf-tls-trust-anchor-ids]). start
   and end MUST define a valid subtree of the log, and hash MUST be the
   subtree's hash value in the cosigner's view of the log.  The label is
   a fixed prefix for domain separation.  Its value MUST be the string
   mtc-subtree/v1, followed by a newline (U+000A), followed by a zero
   byte (U+0000). cosigner_id MUST be the cosigner ID, in its binary
   representation.

   The resulting signature is known as a _subtree signature_. When start
   is zero, the resulting signature describes the checkpoint with tree
   size end and is also known as a _checkpoint signature_.

   For each supported log, a cosigner retains its checkpoint signature
   with the largest end.  This is known as the cosigner's _current_
   checkpoint.  If the cosigner's current checkpoint has tree size
   tree_size, it MUST NOT generate a signature for a subtree [start,
   end) if start > 0 and end > tree_size.  That is, a cosigner can only
   sign a non-checkpoint subtree if it is contained in its current
   checkpoint.  In a correctly-operated cosigner, every signature made
   by the cosigner can be proven consistent with its current checkpoint
   with a subtree consistency proof (Section 4.4).  As a consequence, a
   cosigner that signs a subtree is held responsible for all the entries
   in the tree of size matching the subtree end, even if the
   corresponding checkpoint is erroneously unavailable.

   Before signing a subtree, the cosigner MUST ensure that hash is
   consistent with its log state.  Different cosigner roles may obtain
   this assurance differently.  For example, a cosigner may compute the
   hash from its saved log state (e.g. if it is the log operator or

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   maintains a copy of the log) or by verifying a subtree consistency
   proof (Section 4.4) from its current checkpoint.  When a cosigner
   signs a subtree, it is held responsible _both_ for the subtree being
   consistent with its other signatures, _and_ for the cosigner-specific
   additional statements.

   Cosigners SHOULD publish their current checkpoint, along with the
   checkpoint signature.

   [[TODO: CT and tlog put timestamps in checkpoint signatures.  Do we
   want them here?  In CT and tlog, the timestamps are monotonically
   increasing as the log progresses, but we also sign subtrees.  We can
   separate subtree and checkpoint signatures, with timestamps only in
   the latter, but it's unclear if there is any benefit to this.]]

5.4.2.  Signature Algorithms

   The cosigner's public key specifies both the key material and the
   signature algorithm to use with the key material.  In order to change
   key or signature parameters, a cosigner operator MUST deploy a new
   cosigner, with a new cosigner ID.  Signature algorithms MUST fully
   specify the algorithm parameters, such as hash functions used.  This
   document defines the following signature algorithms:

   *  ECDSA with P-256 and SHA-256 [FIPS186-5]

   *  ECDSA with P-384 and SHA-384 [FIPS186-5]

   *  Ed25519 [RFC8032]

   *  ML-DSA-44 [FIPS204]

   *  ML-DSA-65 [FIPS204]

   *  ML-DSA-87 [FIPS204]

   Other documents or deployments MAY define other signature schemes and
   formats.  Log clients that accept cosignatures from some cosigner are
   assumed to be configured with all parameters necessary to verify that
   cosigner's signatures, including the signature algorithm and version
   of the signature format.

5.5.  Certification Authority Cosigners

   A _CA cosigner_ is a cosigner (Section 5.4) that certifies the
   contents of a log.

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   When a CA cosigner signs a subtree, it makes the additional statement
   that it has certified each entry in the subtree.  For example, a
   domain-validating CA states that it has performed domain validation
   for each entry, at some time consistent with the entry's validity
   dates.  CAs are held responsible for every entry in every subtree
   they sign.  Proving an entry is included (Section 4.3) in a CA-signed
   subtree is sufficient to prove the CA certified it.

   What it means to certify an entry depends on the entry type:

   *  To certify an entry of type null_entry is a no-op.  A CA MAY
      freely certify null_entry without being held responsible for any
      validation.

   *  To certify an entry of type tbs_cert_entry is to certify the
      TBSCertificateLogEntry, as defined in Section 5.3.

   Entries are extensible.  Future documents MAY define type values and
   what it means to certify them.  A CA MUST NOT sign a subtree if it
   contains an entry with type that it does not recognize.  Doing so
   would certify that the CA has validated the information in some not-
   yet-defined entry format.  Section 12.5 further discusses security
   implications of new formats.

   A CA operator MAY operate multiple CA cosigners that all certify the
   same log in parallel.  This may be useful when, e.g., rotating CA
   keys.  In this case, each CA instance MUST have a distinct name.  The
   CA operator's ACME server can return all CA cosignatures together in
   a single certificate, with the application protocol selecting the
   cosignatures to use.  Section 8 describes how this is done in TLS
   [RFC8446].

   If the CA operator additionally operates a traditional X.509 CA, that
   CA key MUST be distinct from any Merkle Tree CA cosigner keys.

5.6.  Publishing Logs

   _[[NOTE: This section is written to avoid depending on a specific
   serving protocol.  The current expectation is that a Web PKI
   deployment would derive from [TLOG-TILES], to match the direction of
   Certificate Transparency and pick up improvements made there._

   _For now, we avoid a normative reference on [TLOG-TILES] and also
   capture the fact that the certificate construction is independent of
   the choice of protocol.  Similar to how the CT ecosystem is migrating
   to a tiled interface, were someone to improve on [TLOG-TILES], a PKI
   could migrate to that new protocol without impacting certificate
   verification._

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   _That said, this is purely a starting point for describing the
   design.  We expect the scope of this document, and other related
   documents to adapt as the work evolves across the IETF, C2SP,
   Certificate Transparency, and other communities.]]_

   Issuance logs are intended to be publicly accessible in some form, to
   allow monitors to detect misissued certificates.

   The access method does not affect certificate interoperability, so
   this document does not prescribe a specific protocol.  An individual
   issuance log MAY be published in any form, provided other parties in
   the PKI are able to consume it.  Relying parties SHOULD define log
   serving requirements, including the allowed protocols and expected
   availability, as part of their policies on which CAs to support.  See
   also Section 10.3.

   For example, a log ecosystem could use [TLOG-TILES] to serve logs.
   [TLOG-TILES] improves on [RFC6962] and [RFC9162] by exposing the log
   as a collection of cacheable, immutable "tiles".  This works well
   with a variety of common HTTP [RFC9110] serving architectures.  It
   also allows log clients to request arbitrary tree nodes, so log
   clients can fetch the structures described in Section 4.

5.6.1.  Log Pruning

   Over time, an issuance log's entries will expire and likely be
   replaced with certificate renewals.  As this happens, the total size
   of the log grows, even if the unexpired subset remains fixed.  To
   mitigate this, issuance logs MAY be _pruned_, as described in this
   section.

   Pruning makes some prefix of the log unavailable, without changing
   the tree structure.  It may be used to reduce the serving cost of
   long-lived logs, where any entries have long expired.  Section 10.3
   discusses policies on when pruning may be permitted.  This section
   discusses how it is done and the impact on log structure.

   An issuance log is pruned by updating its _minimum index_ parameter
   (Section 5.1).  The minimum index is the index of the first log entry
   that the log publishes.  (See Section 5.6.)  It MUST be less than or
   equal to the tree size of the log's current checkpoint, and also
   satisfy any availability policies set by relying parties who trust
   the CA.

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   An entry is said to be _available_ if its index is greater than or
   equal to the minimum index.  A checkpoint is said to be available if
   its tree size is greater than the minimum index.  A subtree [start,
   end) is said to be available if end is greater than the minimum
   index.

   Log protocols MUST serve enough information to allow a log client to
   efficiently obtain the following:

   *  Signatures over the latest checkpoint by the CA's cosigners
      (Section 5.5)

   *  Any individual available log entry (Section 5.3)

   *  The hash value of any available checkpoint

   *  An inclusion proof (Section 2.1.3 of [RFC9162]) for any available
      entry to any containing checkpoint

   *  A consistency proof (Section 2.1.4 of [RFC9162]) between any two
      available checkpoints

   *  The hash value of any available subtree (Section 4)

   *  A subtree inclusion proof (Section 4.3) for any available entry in
      any containing subtree

   *  A subtree consistency proof (Section 4.4) between any available
      subtree to any containing checkpoint

   Meeting these requirements requires a log to retain some information
   about pruned entries.  Given a node [start, end) in the Merkle Tree,
   if end is less than or equal to the minimum index, the node's
   children MAY be discarded in favor of the node's hash.

   Figure 11 shows an example pruned tree with 13 elements, where the
   minimum index is 7.  It shows the original tree, followed by the
   pruned tree.  The pruned tree depicts the nodes that MUST be
   available or computable.  Note that entry 6 MAY be discarded, only
   the hash of entry 6 must be available.

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                   +-----------------------------+
                   |            [0, 13)          |
                   +-----------------------------+
                      /                       \
          +----------------+             +----------------+
          |     [0, 8)     |             |     [8, 13)    |
          +----------------+             +----------------+
           /              \                 /          |
      +--------+      +--------+      +---------+      |
      | [0, 4) |      | [4, 8) |      | [8, 12) |      |
      +--------+      +--------+      +---------+      |
       /      \        /      \         /      \       |
   +-----+ +-----+ +-----+ +-----+ +------+ +-------+  |
   |[0,2)| |[2,4)| |[4,6)| |[6,8)| |[8,10)| |[10,12)|  |
   +-----+ +-----+ +-----+ +-----+ +------+ +-------+  |
     / \     / \     / \     / \     / \      / \      |
   +=+ +=+ +=+ +=+ +=+ +=+ +=+ +=+ +=+ +=+ +==+ +==+ +==+
   |0| |1| |2| |3| |4| |5| |6| |7| |8| |9| |10| |11| |12|
   +=+ +=+ +=+ +=+ +=+ +=+ +=+ +=+ +=+ +=+ +==+ +==+ +==+

                   +-----------------------------+
                   |            [0, 13)          |
                   +-----------------------------+
                      /                       \
          +----------------+             +----------------+
          |     [0, 8)     |             |     [8, 13)    |
          +----------------+             +----------------+
           /              \                 /          |
      +--------+      +--------+      +---------+      |
      | [0, 4) |      | [4, 8) |      | [8, 12) |      |
      +--------+      +--------+      +---------+      |
                       /      \         /      \       |
                   +-----+ +-----+ +------+ +-------+  |
                   |[4,6)| |[6,8)| |[8,10)| |[10,12)|  |
                   +-----+ +-----+ +------+ +-------+  |
                             / \     / \      / \      |
                           +-+ +=+ +=+ +=+ +==+ +==+ +==+
                           |6| |7| |8| |9| |10| |11| |12|
                           +-+ +=+ +=+ +=+ +==+ +==+ +==+

        Figure 11: An example showing the minimum nodes that must be
                          available after pruning

   Logs MAY retain additional nodes, or expect log clients to compute
   required nodes from other nodes.  For example, in Figure 11, the
   log's serving protocol MAY instead serve [0, 2) and [2, 4), with the
   log client computing [0, 4) from those values.

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6.  Certificates

   This section defines how to construct Merkle Tree Certificates, which
   are X.509 Certificates [RFC5280] that assert the information in an
   issuance log entry.  A Merkle Tree Certificate is constructed from
   the following:

   *  A TBSCertificateLogEntry (Section 5.3) contained in the issuance
      log (Section 5)

   *  A subject public key whose hash matches the TBSCertificateLogEntry

   *  A subtree (Section 4) that contains the log entry

   *  Zero or more signatures (Section 5.4) over the subtree, which
      together satisfy relying party requirements (Section 7.3)

   For any given TBSCertificateLogEntry, there are multiple possible
   certificates that may prove the entry is certified by the CA and
   publicly logged, varying by choice of subtree and signatures.
   Section 6.1 defines how the certificate is constructed based on those
   choices.  Section 6.2 and Section 6.3 define two profiles of Merkle
   Tree Certificates, full certificates and signatureless certificates,
   and how to select the subtree and signatures for them.

6.1.  Certificate Format

   The information is encoded in an X.509 Certificate [RFC5280] as
   follows:

   The TBSCertificate's version, issuer, validity, subject,
   issuerUniqueID, subjectUniqueID, and extensions MUST match the
   corresponding fields of the TBSCertificateLogEntry.  Per Section 5.3,
   this means issuer MUST be the issuance log's log ID as an X.509
   distinguished name, as described in Section 5.2.

   The TBSCertificate's serialNumber MUST contain the zero-based index
   of the TBSCertificateLogEntry in the log.  Section 4.1.2.2 of
   [RFC5280] forbids zero as a serial number, but Section 5.3 defines a
   null_entry type for use in entry zero, so the index will be positive.
   This encoding is intended to avoid implementation errors by having
   the serial numbers and indices off by one.

   The TBSCertificate's subjectPublicKeyInfo contains the specified
   public key.  Its hash MUST match the TBSCertificateLogEntry's
   subjectPublicKeyInfoHash.

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   The TBSCertificate's signature and the Certificate's
   signatureAlgorithm MUST contain an AlgorithmIdentifier whose
   algorithm is id-alg-mtcProof, defined below, and whose parameters is
   omitted.

   id-alg-mtcProof OBJECT IDENTIFIER ::= {
       iso(1) identified-organization(3) dod(6) internet(1) security(5)
       mechanisms(5) pkix(7) algorithms(6) TBD}

   For initial experimentation, early implementations of this design
   will use the OID 1.3.6.1.4.1.44363.47.0 instead of id-alg-mtcProof.

   The signatureValue contains an MTCProof structure, defined below
   using the TLS presentation language (Section 3 of [RFC8446]):

   opaque HashValue[HASH_SIZE];

   struct {
       TrustAnchorID cosigner_id;
       opaque signature<0..2^16-1>;
   } MTCSignature;

   struct {
       uint64 start;
       uint64 end;
       HashValue inclusion_proof<0..2^16-1>;
       MTCSignature signatures<0..2^16-1>;
   } MTCProof;

   start and end MUST contain the corresponding parameters of the chosen
   subtree. inclusion_proof MUST contain a subtree inclusion proof
   (Section 4.3) for the log entry and the subtree. signatures contains
   the chosen subtree signatures.  In each signature, cosigner_id
   contains the cosigner ID (Section 5.4) in its binary representation
   (Section 3 of [I-D.ietf-tls-trust-anchor-ids]), and signature
   contains the signature value as described in Section 5.4.1.

   The MTCProof is encoded into the signatureValue with no additional
   ASN.1 wrapping.  The most significant bit of the first octet of the
   signature value SHALL become the first bit of the bit string, and so
   on through the least significant bit of the last octet of the
   signature value, which SHALL become the last bit of the bit string.

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6.2.  Full Certificates

   A _full certificate_ is a Merkle Tree certificate which contains
   sufficient signatures to allow a relying party to trust the choice of
   subtree, without any predistributed information beyond the
   cosigner(s) parameters.  Full certificates can be issued without
   significant processing delay.

   When issuing a certificate, the CA first adds the
   TBSCertificateLogEntry to its issuance log.  It then schedules a job
   to construct a checkpoint and collect cosignatures.  The job proceeds
   as follows:

   1.  The CA signs the checkpoint with its key(s) (Section 5.5).

   2.  Using the procedure in Section 4.5, the CA determines the two
       subtrees that cover the entries added between this checkpoint and
       the most recent checkpoint.

   3.  The CA signs each subtree with its key(s) (Section 5.4).

   4.  The CA requests sufficient checkpoint cosignatures (Section 5.4)
       from external cosigners to meet relying party requirements
       (Section 7.3).

   5.  The CA requests subtree cosignatures (Appendix C.2) from the
       cosigners above.

   6.  For each certificate in the interval, the CA constructs
       certificates (Section 6.1) using the covering subtree.

   Steps 4 and 5 are analogous to requesting SCTs from CT logs in
   Certificate Transparency, except that a single run of this job
   collects signatures for many certificates at once.  The CA MAY
   request signatures from a redundant set of cosigners and select the
   ones that complete first.

   This document does not prescribe the specific cosigner roles, or a
   particular protocol for requesting cosignatures.  Protocols for
   cosigners MAY vary depending on the needs for that cosigner.  A
   consistency-only cosigner, such as [TLOG-WITNESS], might only require
   a checkpoint signature and consistency proof, while a mirroring
   cosigner, such as [TLOG-MIRROR] might require the full log contents.

   A cosigner MAY expose a private interface for the CA, to reduce
   denial-of-service risk, or a cosigner MAY expose a public interface
   for other parties to request additional cosignatures.  The latter may
   be useful if a relying party requires a cosigner that the CA does not

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   communicate with.  In this case, an authenticating party MAY request
   cosignatures and add them to the certificate.  However, it is
   RECOMMENDED that the CA collect cosignatures for the authenticating
   party.  This simplifies deployment, as relying party policies change
   over time.

   This document does not place any requirements on how frequently this
   job runs.  More frequent runs results in lower issuance delay, but
   higher signing overhead.  It is RECOMMENDED that CAs run at most one
   instance of this job at a time, starting the next instance after the
   previous one completes.  A single run collects signatures for all
   entries since the most recent checkpoint, so there is little benefit
   to overlapping them.  Less frequent runs may also aid relying parties
   that wish to directly audit signatures, as described in Section 5.2
   of [AuditingRevisited], though this document does not define such a
   system.

6.3.  Signatureless Certificates

   A _signatureless certificate_ is a Merkle Tree certificate which
   contains no signatures and instead assumes the relying party had
   predistributed information about which subtrees were trusted.
   Signatureless certificates are an optional size optimization.  They
   require a processing delay to construct, and only work in a
   sufficiently up-to-date relying party.  Authenticating parties thus
   SHOULD deploy a corresponding full certificate alongside any
   signatureless certificate, and use some application-protocol-specific
   mechanism to select between the two.  Section 8 discusses such a
   mechanism for TLS [RFC8446].

6.3.1.  Landmarks

   A signatureless certificate is constructed based on a _landmark
   sequence_, which is a sequence of _landmarks_. Landmarks are agreed-
   upon tree sizes across the ecosystem for optimizing certificates.
   Landmarks SHOULD be allocated by the CA, but they can also be
   allocated by some other coordinating party.  It is possible, but NOT
   RECOMMENDED, for multiple landmark sequences to exist per CA.
   Landmarks are allocated to balance minimizing the delay in obtaining
   a signatureless certificate with minimizing the size of the relying
   party's predistributed state.

   A landmark sequence has the following fixed parameters:

   *  base_id: An OID arc for trust anchor IDs of individual landmarks

   *  max_landmarks: A positive integer, describing the maximum number
      of landmarks that may contain unexpired certificates at any time

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   *  landmark_url: Some URL to fetch the current list of landmarks

   Landmarks are numbered consecutively from zero.  Each landmark has a
   trust anchor ID, determined by appending the landmark number to
   base_id.  For example, the trust anchor ID for landmark 42 of a
   sequence with base_id of 32473.1 would be 32473.1.42.

   Each landmark specifies a tree size.  The first landmark, numbered
   zero, is always a tree size of zero.  The sequence of tree sizes MUST
   be append-only and monotonically increasing.

   Landmarks determine _landmark subtrees_: for each landmark, other
   than number zero, let tree_size be the landmark's tree size and
   prev_tree_size be that of the previous landmark.  As described in
   Section 4.5, select the one or two subtrees that cover
   [prev_tree_size, tree_size).  Each of those subtrees is a landmark
   subtree.  Landmark zero has no landmark subtrees.

   The most recent max_landmarks landmarks are said to be _active_.
   Landmarks MUST be allocated such that, at any given time, only active
   landmarks contain unexpired certificates.  The active landmark
   subtrees are those determined by the active landmarks.  There are at
   most 2 * max_landmarks active landmark subtrees at any time.  Every
   unexpired entry will be contained in one or more landmark subtree, or
   between the last landmark subtree and the latest checkpoint.  Active
   landmark subtrees are predistributed to the relying party as trusted
   subtrees, as described in Section 7.4.

   It is RECOMMENDED that landmarks be allocated following the procedure
   described in Section 6.3.2.  If landmarks are allocated incorrectly
   (e.g. past landmarks change, or max_landmarks is inaccurate), there
   are no security consequences, but some older certificates may fail to
   validate.

   Relying parties will locally retain up to 2 * max_landmarks hashes
   (Section 7.4) per CA, so max_landmarks should be set to balance the
   delay between landmarks and the amount of state the relying party
   must maintain.  Using the recommended procedure above, a CA with a
   maximum certificate lifetime of 7 days, allocating a landmark every
   hour, will have a max_landmarks of 168.  The client state is then 336
   hashes, or 10,752 bytes with SHA-256.

   landmark_url MUST serve a resource with Content-Type: text/plain;
   charset=utf-8 and the following lines.  Each line MUST be terminated
   by a newline character (U+000A):

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   *  Two space-separated non-negative decimal integers: <last_landmark>
      <num_active_landmarks>.  This line MUST satisfy the following,
      otherwise it is invalid:

      -  num_active_landmarks <= max_landmarks

      -  num_active_landmarks <= last_landmark

   *  num_active_landmarks + 1 lines each containing a single non-
      negative decimal integer, containing a tree size.  Numbered from
      zero to num_active_landmarks, line i contains the tree size for
      landmark last_landmark - i.  The integers MUST be monotonically
      decreasing and lower or equal to the log's latest tree size.

6.3.2.  Allocating Landmarks

   It is RECOMMENDED that landmarks be allocated using the following
   procedure:

   1.  Select some time_between_landmarks duration.  Define a series of
       consecutive, non-overlapping time intervals, each of duration
       time_between_landmarks.

   2.  At most once per time interval, append the latest checkpoint tree
       size to the landmark sequence if it is greater than the last
       landmark's tree size.

   To ensure that only active landmarks contain unexpired certificates,
   set max_landmarks to ceil(max_cert_lifetime / time_between_landmarks)
   + 1, where max_cert_lifetime is the CA's maximum certificate
   lifetime.

6.3.3.  Constructing Signatureless Certificates

   Given a TBSCertificateLogEntry in the issuance log and a landmark
   sequence, a signatureless certificate is constructed as follows:

   1.  Wait for the first landmark to be allocated that contains the
       entry.

   2.  Determine the landmark's subtrees and select the one that
       contains the entry.

   3.  Construct a certificate (Section 6.1) using the selected subtree
       and no signatures.

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   Before sending this certificate, the authenticating party SHOULD
   obtain some application-protocol-specific signal that implies the
   relying party has been configured with the corresponding landmark.
   (Section 7.4 defines how relying parties are configured.)  The trust
   anchor ID of the landmark may be used as an efficient identifier in
   the application protocol.  Section 8 discusses how to do this in TLS
   [RFC8446].

6.4.  Size Estimates

   The inclusion proofs in full and signatureless certificates scale
   logarithmically with the size of the subtree.  These sizes can be
   estimated with the CA's issuance rate.  The byte counts below assume
   the issuance log's hash function is SHA-256.

   Some organizations have published statistics which can be used to
   estimate this rate for the Web PKI.  As of June 9th, 2025:

   *  [LetsEncrypt] reported around 558,000,000 active certificates for
      a single CA

   *  [MerkleTown] reported around 2,100,000,000 unexpired certificates
      in CT logs, across all CAs

   *  [MerkleTown] reported an issuance rate of around 444,000
      certificates per hour, across all CAs

   The current issuance rate across the Web PKI may not necessarily be
   representative of the Web PKI after a transition to short-lived
   certificates.  Assuming a certificate lifetime of 7 days, and that
   subscribers will update their certificates 75% of the way through
   their lifetime (see Section 10.4), every certificate will be reissued
   every 126 hours.  This gives issuance rate estimates of around
   4,400,000 certificates per hour and 17,000,000 certificates per hour,
   for the first two values above.  Note the larger estimate is across
   all CAs, while subtrees would only span one CA.

   Using the per-CA short lifetime estimate, if the CA mints a
   checkpoint every 2 seconds, full certificate subtrees will span
   around 2,500 certificates, leading to 12 hashes in the inclusion
   proof, or 384 bytes.  Full certificates additionally must carry a
   sufficient set of signatures to meet relying party requirements.

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   If a new landmark is allocated every hour, signatureless certificate
   subtrees will span around 4,400,000 certificates, leading to 23
   hashes in the inclusion proof, giving an inclusion proof size of 736
   bytes, with no signatures.  This is significantly smaller than a
   single ML-DSA-44 signature, 2,420 bytes, and almost ten times smaller
   than the three ML-DSA-44 signatures necessary to include post-quantum
   SCTs.

   The proof sizes grow logarithmically, so 32 hashes, or 1024 bytes, is
   sufficient for subtrees of up to 2^32 (4,294,967,296) certificates.

7.  Relying Parties

   This section discusses how relying parties verify Merkle Tree
   Certificates.

7.1.  Trust Anchors

   In order to accept certificates from a Merkle Tree CA, a relying
   party MUST be configured with:

   *  The log ID (Section 5.2)

   *  A set of supported cosigners, as pairs of cosigner ID and public
      key

   *  A policy on which combinations of cosigners to accept in a
      certificate (Section 7.3)

   *  An optional list of trusted subtrees, with their hashes, that are
      known to be consistent with the relying party's cosigner
      requirements (Section 7.4)

   *  A list of revoked ranges of indices (Section 7.5)

   [[TODO: Define some representation for this.  In a trust anchor,
   there's a lot of room for flexibility in what the client stores.  In
   principle, we could even encode some of this information in an X.509
   intermediate certificate, if an application wishes to use this with a
   delegation model with intermediates, though the security story
   becomes more complex.  Decide how/whether to do that.]]

7.2.  Verifying Certificate Signatures

   When verifying the signature on an X.509 certificate (Step (a)(1) of
   Section 6.1.3 of [RFC5280]) whose issuer is a Merkle Tree CA, the
   relying party performs the following procedure:

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   1.  Check that the TBSCertificate's signature field is id-alg-
       mtcProof with omitted parameters.  If either check fails, abort
       this process and fail verification.

   2.  Decode the signatureValue as an MTCProof, as described in
       Section 6.1.

   3.  Let index be the certificate's serial number.  If index is
       contained in one of the relying party's revoked ranges
       (Section 7.5), abort this process and fail verification.

   4.  Construct a TBSCertificateLogEntry as follows:

       1.  Copy the version, issuer, validity, subject, issuerUniqueID,
           subjectUniqueID, and extensions fields from the
           TBSCertificate.

       2.  Set subjectPublicKeyInfoHash to the hash of the DER encoding
           of subjectPublicKeyInfo.

   5.  Construct a MerkleTreeCertEntry of type tbs_cert_entry with
       contents the TBSCertificateLogEntry.  Let entry_hash be the hash
       of the entry, MTH({entry}) = HASH(0x00 || entry), as defined in
       Section 2.1.1 of [RFC9162].

   6.  Let expected_subtree_hash be the result of evaluating the
       MTCProof's inclusion_proof for entry index, with hash entry_hash,
       of the subtree described by the MTCProof's start and end,
       following the procedure in Section 4.3.2.  If evaluation fails,
       abort this process and fail verification.

   7.  If [start, end) matches a trusted subtree (Section 7.4), check
       that expected_subtree_hash is equal to the trusted subtree's
       hash.  Return success if it matches and failure if it does not.

   8.  Otherwise, check that the MTCProof's signatures contain a
       sufficient set of valid signatures from cosigners to satisfy the
       relying party's cosigner requirements (Section 7.3).
       Unrecognized cosigners MUST be ignored.  Signatures are verified
       as described in Section 5.4.1.  The hash field of the MTCSubtree
       is set to expected_subtree_hash.

   This procedure only replaces the signature verification portion of
   X.509 path validation.  The relying party MUST continue to perform
   other checks, such as checking expiry.

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7.3.  Trusted Cosigners

   A relying party's cosigner policy determines the sets of cosigners
   that must sign a view of the issuance log before it is trusted.

   This document does not prescribe a particular policy, but gives
   general guidance.  Relying parties MAY implement policies other than
   those described below, and MAY incorporate cosigners acting in roles
   not described in this document.

   In picking trusted cosigners, the relying party SHOULD ensure the
   following security properties:

   Authenticity:  The relying party only accepts entries certified by
      the CA

   Transparency:  The relying party only accepts entries that are
      publicly accessible, so that monitors, particularly the subject of
      the certificate, can notice any unauthorized certificates

   Relying parties SHOULD ensure authenticity by requiring a signature
   from the most recent CA cosigner key.  If the CA is transitioning
   from an old to new key, the relying party SHOULD accept both until
   certificates that predate the new key expire.  This is analogous to
   the signature in a traditional X.509 certificate.

   While a CA signature is sufficient to prove a subtree came from the
   CA, this is not enough to ensure the certificate is visible to
   monitors.  A misbehaving CA might not operate the log correctly,
   either presenting inconsistent versions of the log to relying parties
   and monitors, or refuse to publish some entries.

   To mitigate this, relying parties SHOULD ensure transparency by
   requiring a quorum of signatures from additional cosigners.  At
   minimum, these cosigners SHOULD enforce a consistent view of the log.
   For example, [TLOG-WITNESS] describes a lightweight "witness"
   cosigner role that checks this with consistency proofs.  This is not
   sufficient to ensure durable logging.  Section 7.5 discusses
   mitigations for this.  Alternatively, a relying party MAY require
   cosigners that serve a copy of the log, in addition to enforcing a
   consistent view.  For example, [TLOG-MIRROR] describes a "mirror"
   cosigner role.

   Relying parties MAY accept the same set of additional cosigners
   across issuance logs.

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   Cosigner roles are extensible without changes to certificate
   verification itself.  Future specifications and individual
   deployments MAY define other cosigner roles to incorporate into
   relying party policies.

   Section 10.2 discusses additional deployment considerations in
   cosigner selection.

7.4.  Trusted Subtrees

   As an optional optimization, a relying party MAY incorporate a
   periodically updated, predistributed list of active landmark
   subtrees, determined as described in Section 6.3.1.  The relying
   party configures these as trusted subtrees, allowing it to accept
   signatureless certificates (Section 6.3) constructed against those
   subtrees.

   Before configuring the subtrees as trusted, the relying party MUST
   obtain assurance that each subtree is consistent with checkpoints
   observed by a sufficient set of cosigners (see Section 5.4) to meet
   its cosigner requirements.  It is not necessary that the cosigners
   have generated signatures over the specific subtrees, only that they
   are consistent.

   This criteria can be checked given:

   *  Some _reference checkpoint_ that contains the latest landmark

   *  For each cosigner, either:

      -  A cosignature on the reference checkpoint

      -  A cosigned checkpoint containing the referenced checkpoint and
         a valid Merkle consistency proof (Section 2.1.4 of [RFC9162])
         between the two

   *  For each subtree, a valid subtree consistency proof (Section 4.4)
      between the subtree and the reference checkpoint

   [[TODO: The subtree consistency proofs have many nodes in common.  It
   is possible to define a single "bulk consistency proof" that verifies
   all the hashes at once, but it's a lot more complex.]]

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   This document does not prescribe how relying parties obtain this
   information.  A relying party MAY, for example, use an application-
   specific update service, such as the services described in [CHROMIUM]
   and [FIREFOX].  If the relying party considers the service
   sufficiently trusted (e.g. if the service provides the trust anchor
   list or certificate validation software), it MAY trust the update
   service to perform these checks.

   The relying party SHOULD incorporate its trusted subtree
   configuration in application-protocol-specific certificate selection
   mechanisms, to allow an authenticating party to select a
   signatureless certificate.  The trust anchor IDs of the landmarks may
   be used as efficient identifiers in the application protocol.
   Section 8 discusses how to do this in TLS [RFC8446].

7.5.  Revocation by Index

   For each supported Merkle Tree CA, the relying party maintains a list
   of revoked ranges of indices.  This allows a relying party to
   efficiently revoke entries of an issuance log, even if the contents
   are not necessarily known.  This may be used to mitigate the security
   consequences of misbehavior by a CA, or other parties in the
   ecosystem.

   When a relying party is first configured to trust a CA, it SHOULD be
   configured to revoke all entries from zero up to but not including
   the first available unexpired certificate at the time.  This
   revocation SHOULD be periodically updated as entries expire and logs
   are pruned (Section 5.6.1).  In particular, when CAs prune entries,
   relying parties SHOULD be updated to revoke all newly unavailable
   entries.  This gives assurance that, even if some unavailable entry
   had not yet expired, the relying party will not trust it.  It also
   allows monitors to start monitoring a log without processing expired
   entries.

   A misbehaving CA might correctly construct a globally consistent log,
   but refuse to make some entries or intermediate nodes available.
   Consistency proofs between checkpoints and subtrees would pass, but
   monitors cannot observe the entries themselves.  Relying parties
   whose cosigner policies (Section 7.3) do not require durable logging
   (e.g. via [TLOG-MIRROR]) are particularly vulnerable to this.  In
   this case, the indices of the missing entries will still be known, so
   relying parties can use this mechanism to revoke the unknown entries,
   possibly as an initial, targeted mitigation before a complete CA
   removal.

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   When a CA is found to be untrustworthy, relying parties SHOULD remove
   trust in that CA.  To minimize the compatibility impact of this
   mitigation, index-based revocation can be used to only distrust
   entries after some index, while leaving existing entries accepted.
   This is analogous to the [SCTNotAfter] mechanism used in some PKIs.

8.  Use in TLS

   TLS implementations that authenticate with or accept Merkle Tree
   certificates SHOULD support trust anchor IDs
   ([I-D.ietf-tls-trust-anchor-ids]) for certificate selection.

   A full certificate has a trust anchor ID of the corresponding log ID
   (Section 5.2).  The authenticating party can obtain this information
   either by parsing the certificate's issuer field or via out-of-band
   information as described in Section 3.2 of
   [I-D.ietf-tls-trust-anchor-ids].

   [[TODO: Ideally we would negotiate cosigners.
   https://github.com/tlswg/tls-trust-anchor-ids/issues/54 has a sketch
   of how one might do this, though other designs are possible.
   Negotiating cosigners allows the ecosystem to manage cosigners
   efficiently, without needing to collect every possible cosignature
   and send them all at once.  This is wasteful, particularly with post-
   quantum algorithms.]]

   A full certificate MAY be used without signals about what the relying
   party trusts.  As with other choices of default certificates, an
   authenticating party that does so assumes that the relying party
   trusts the issuing CA, e.g. because the CA is relatively ubiquitous
   among the relying parties that it supports.

   A signatureless certificate has a trust anchor ID of the
   corresponding landmark, as described in Section 6.3.1.  This can be
   configured in the authenticating party via out-of-band information,
   as described in Section 3.2 of [I-D.ietf-tls-trust-anchor-ids].  A
   relying party that has been configured with trusted subtrees
   (Section 7.4) derived from a set of landmarks SHOULD be configured to
   support those landmarks' trust anchor IDs.  TLS certificate selection
   will then correctly determine whether the signatureless certificate
   is compatible with the relying party.

   [[TODO: We can do slightly better.  If the relying party supports
   landmark 42, it can be assumed to also support landmark 41, 40, 39,
   etc. https://github.com/tlswg/tls-trust-anchor-ids/issues/62
   discusses how to fit this into the trust anchor IDs framework.  This
   allows the client to summarize its state with one ID per CA.]]

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   Authenticating parties SHOULD preferentially use signatureless
   certificates over full certificates, when both are supported by the
   relying party.  A signatureless certificate asserts the same
   information as its full counterpart, but is expected to be smaller.
   A signatureless certificate SHOULD NOT be used without a signal that
   the relying party trusts the corresponding landmark subtree.  Even if
   the relying party is assumed to trust the issuing CA, the relying
   party may not have sufficiently up-to-date trusted subtrees
   predistributed.

9.  ACME Extensions

   This section describes how to issue Merkle Tree certificates using
   ACME [RFC8555].

   When downloading the certificate (Section 7.4.2 of [RFC8555]), ACME
   clients supporting Merkle Tree certificates SHOULD send "application/
   pem-certificate-chain-with-properties" in their Accept header
   (Section 12.5.1 of [RFC9110]).  ACME servers issuing Merkle Tree
   certificates SHOULD then respond with that content type and include
   trust anchor ID information as described in Section 6 of
   [I-D.ietf-tls-trust-anchor-ids].  Section 8 decribes the trust anchor
   ID assignments for full and signatureless certificates.

   When processing an order for a Merkle Tree certificate, the ACME
   server moves the order to the "valid" state once the corresponding
   entry is sequenced in the issuance log.  The order's certificate URL
   then serves the full certificate, constructed as described in
   Section 6.2.

   The full certificate response SHOULD additionally carry a alternate
   URL for the signatureless certificate, as described Section 7.4.2 of
   [RFC8555].  Before the signatureless certificate is available, the
   alternate URL SHOULD return a HTTP 503 (Service Unavailable)
   response, with a Retry-After header (Section 10.2.3 of [RFC9110])
   estimating when the certificate will become available.  Once the next
   landmark is allocated, the ACME server constructs a signatureless
   certificate, as described in Section 6.3 and serves it from the
   alternate URL.

   ACME clients supporting Merkle Tree certificates SHOULD support
   fetching alternate chains.  If an alternate chain returns an HTTP 503
   with a Retry-After header, as described above, the client SHOULD
   retry the request at the specified time.

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10.  Deployment Considerations

10.1.  Operational Costs

10.1.1.  Certification Authority Costs

   While Merkle Tree certificates expects CAs to operate logs, the costs
   of these logs are expected to be much lower than a CT log from
   [RFC6962] or [RFC9162]:

   Section 5.6 does not constrain the API to the one defined in
   [RFC6962] or [RFC9162].  If the PKI uses a tile-based protocol, such
   as [TLOG-TILES], the issuance log benefits from the improved caching
   properties of such designs.

   Unlike a CT log, an issuance log does not have public submission
   APIs.  Log entries are only added by the CA directly.  The costs are
   thus expected to scale with the CA's own operations.

   A CA only needs to produce a digital signature for every checkpoint,
   rather than for every certificate.  The lower signature rate
   requirements could allow more secure and/or economical key storage
   choices.

   Individual entries are kept small and do not scale with public key or
   signature sizes.  This mitigates growth from post-quantum algorithms.
   Public keys in entries are replaced with fixed-sized hashes.  There
   are no signatures in entries themselves, and only signatures on the
   very latest checkpoint are retained.  Every new checkpoint completely
   subsumes the old checkpoint, so there is no need to retain older
   signatures.  Likewise, a subtree is only signed if contained in
   another signed checkpoint.

   Log pruning (Section 5.6.1) allows a long-lived log to serve only the
   more recent entries, scaling with the size of the retention window,
   rather than the log's total lifetime.

   Mirrors of the log can also reduce CA bandwidth costs, because
   monitors can fetch data from mirrors instead of CAs directly.  In
   PKIs that deploy mirrors as part of cosigner policies, relying
   parties could set few availability requirements on CAs, as described
   in Section 10.3.

10.1.2.  Cosigner Costs

   The costs of cosigners vary by cosigner role.  A consistency-checking
   cosigner, such as [TLOG-WITNESS], requires very little state and can
   be run with low cost.

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   A mirroring cosigner, such as [TLOG-MIRROR], performs comparable
   roles as CT logs, but several of the cost-saving properties in
   Section 10.1.1 also apply: improved protocols, smaller entries, less
   frequent signatures, and log pruning.  While a mirror does need to
   accommodate another party's (the CA's) growth rate, it grows only
   from new issuances from that one CA.  If one CA's issuance rate
   exceeds the mirror's capacity, that does not impact the mirror's
   copies of other CAs.  Mirrors also do not need to defend against a
   client uploading a large number of existing certificates all at once.
   Submissions are also naturally batched and serialized.

10.1.3.  Monitor Costs

   In a CT-based PKI, every log carries a potentially distinct subset of
   active certificates, so monitors must check the contents of every CT
   log.  At the same time, certificates are commonly synchronized
   between CT logs.  As a result, a monitor will typically download each
   certificate multiple times, once for every log.  In Merkle Tree
   Certificates, each entry appears in exactly one log.  A relying party
   might require a log to be covered by a quorum of mirrors, but each
   mirror is cryptographically verified to serve the same contents.
   Once a monitor has obtained some entry from one mirror, it does not
   need to download it from the others.

   In addition to downloading each entry only once, the entries
   themselves are smaller, as discussed in Section 10.1.1.

10.2.  Choosing Cosigners

   In selecting trusted cosigners and cosigner requirements
   (Section 7.3), relying parties navigate a number of trade-offs:

   A consistency-checking cosigner, such as [TLOG-WITNESS], is very
   cheap to run, but does not guarantee durable logging, while a
   mirroring cosigner is more expensive and may take longer to cosign
   structures.  Requiring a mirror signature provides stronger
   guarantees to the relying party, which in turn can reduce the
   requirements on CAs (see Section 10.3), however it may cause
   certificate issuance to take longer.  That said, mirrors are
   comparable to CT logs, if not cheaper (see Section 10.1), so they may
   be appropriate in PKIs where running CT logs is already viable.

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   Relying parties that require larger quorums of trusted cosigners can
   reduce the trust placed in any individual cosigner.  However, these
   larger quorums result in larger, more expensive full certificates.
   The cost of this will depend on how frequently the signatureless
   optimization occurs in a given PKI.  Conversely, relying parties that
   require smaller quorums have smaller full certificates, but place
   more trust in their cosigners.

   Relying party policies also impact monitor operation.  If a relying
   party accepts any one of three cosigners, monitors SHOULD check the
   checkpoints of all three.  Otherwise, a malicious CA may send
   different split views to different cosigners.  More generally,
   monitors SHOULD check the checkpoints in the union of all cosigners
   trusted by all supported relying parties.  This is an efficient check
   because, if the CA is operating correctly, all cosigners will observe
   the same tree.  Thus the monitor only needs to check consistency
   proofs between the checkpoints, and check the log contents themselves
   once.  Monitors MAY also rely on other parties in the transparency
   ecosystem to perform this check.

10.3.  Log Availability

   CAs and mirrors are expected to serve their log contents over HTTP.
   It is possible for the contents to be unavailable, either due to
   temporary service outage or because the log has been pruned
   (Section 5.6.1).  If some resources are unavailable, they may not be
   visible to monitors.

   As in CT, PKIs which deploy Merkle Tree certificates SHOULD establish
   availability policies, adhered to by trusted CAs and mirrors, and
   enforced by relying party vendors as a condition of trust.  Exact
   availability policies for these services are out of scope for this
   document, but this section provides some general guidance.

   Availability policies SHOULD specify how long an entry must be made
   available, before a CA or mirror is permitted to prune the entry.  It
   is RECOMMENDED to define this using a _retention period_, which is
   some time after the entry has expired.  In such a policy, an entry
   could only be pruned if it, and all preceding entries, have already
   expired for the retention period.  Policies MAY opt to set different
   retention periods between CAs and mirrors.  Permitting limited log
   retention is analogous to the CT practice of temporal sharding
   [CHROME-CT], except that a pruned issuance log remains compatible
   with older, unupdated relying parties.

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   Such policies impact monitors.  If the retention period is, e.g. 6
   months, this means that monitors are expected to check entries of
   interest within 6 months.  It also means that a new monitor may only
   be aware of a 6 month history of entries issued for a particular
   domain.

   If historical data is not available to verify the retention period,
   such as information in another mirror or a trusted summary of
   expiration dates of entries, it may not be possible to confirm
   correct behavior.  This is mitigated by the revocation process
   described in Section 7.5: if a CA were to prune a forward-dated entry
   and, in the 6 months when the entry was available, no monitor noticed
   the unusual expiry, an updated relying party would not accept it
   anyway.

   The log pruning process simply makes some resources unavailable, so
   availability policies SHOULD constrain log pruning in the same way as
   general resource availability.  That is, if it would be a policy
   violation for the log to fail to serve a resource, it should also be
   a policy violation for the log to prune such that the resource is
   removed, and vice versa.

   PKIs that require mirror cosignatures (Section 7.3) can impose
   minimal to no availability requirements on CAs, all without
   compromising transparency goals.  If a CA never makes some entry
   available, mirrors will be unable to update.  This will prevent
   relying parties from accepting the undisclosed entries.  However, a
   CA which is persistently unavailable may not offer sufficient benefit
   to be used by authenticating parties or trusted by relying parties.

   However, if a mirror's interface becomes unavailable, monitors may be
   unable to check for unauthorized issuance, if the entries are not
   available in another mirror.  This does compromise transparency
   goals.  As such, availability policies SHOULD set availability
   expectations on mirrors.  This can also be mitigated by using
   multiple mirrors, either directly enforced in cosigner requirements,
   or by keeping mirrors up-to-date with each other.

   In PKIs that do not require mirroring cosigners, the CA's serving
   endpoint is more crucial for monitors.  Such PKIs thus SHOULD set
   availability requirements on CAs.

   In each of these cases, availability failures can be mitigated by
   revoking the unavailable entries by index, as described in
   Section 7.5, likely as a first step in a broader distrust.

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10.4.  Certificate Renewal

   When an authenticating party requests a certificate, the
   signatureless certificate will not be available until the next
   landmark is ready.  From there, the signatureless certificate will
   not be available until relying parties receive new trusted subtrees.

   To maximize coverage of the signatureless certificate optimization,
   authenticating parties performing routine renewal SHOULD request a
   new Merkle Tree certificate some time before the previous Merkle Tree
   certificate expires.  Renewing around 75% into the previous
   certificate's lifetime is RECOMMENDED.  Authenticating parties
   additionally SHOULD retain both the new and old certificates in the
   certificate set until the old certificate expires.  As the new
   subtrees are delivered to relying parties, certificate negotiation
   will transition relying parties to the new certificate, while
   retaining the old certificate for relying parties that are not yet
   updated.

   The above also applies if the authenticating party is performing a
   routine key rotation alongside the routine renewal.  In this case,
   certificate negotiation would pick the key as part of the certificate
   selection.  This slightly increases the lifetime of the old key but
   maintains the size optimization continuously.

   If the service is rotating keys in response to a key compromise, this
   option is not appropriate.  Instead, the service SHOULD immediately
   discard the old key and request a full certificate and the revocation
   of the previous certificate.  This will interrupt the size
   optimization until the new signatureless certificate is available and
   relying parties are updated.

10.5.  Multiple CA Keys

   The separation between issuance logs and CA cosigners gives CAs
   additional flexibility in managing keys.  A CA operator wishing to
   rotate keys, e.g. to guard against compromise of older key material,
   or upgrade to newer algorithms, could retain the same issuance log
   and sign its checkpoints and subtrees with both keys in parallel,
   until relying parties are all updated.  Older relying parties would
   verify the older signatures, while newer relying parties would verify
   the newer signatures.  A cosignature negotiation mechanism in the
   application protocol (see Section 8) would avoid using extra
   bandwidth for the two signatures.

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11.  Privacy Considerations

   The Privacy Considerations described in Section 9 of
   [I-D.ietf-tls-trust-anchor-ids] apply to its use with Merkle Tree
   Certificates.

   In particular, relying parties that share an update process for
   trusted subtrees (Section 7.4) will fetch the same stream of updates.
   However, updates may reach different users at different times,
   resulting in some variation across users.  This variation may
   contribute to a fingerprinting attack [RFC6973].  If the Merkle Tree
   CA trust anchors are sent unconditionally in trust_anchors, this
   variation will be passively observable.  If they are sent
   conditionally, e.g. with the DNS mechanism, the trust anchor list
   will require active probing.

12.  Security Considerations

12.1.  Authenticity

   A key security requirement of any PKI scheme is that relying parties
   only accept assertions that were certified by a trusted certification
   authority.  Merkle Tree certificates achieve this by ensuring the
   relying party only accepts authentic subtree hashes:

   *  In full certificates, the relying party's cosigner requirements
      (Section 7.3) are expected to include some signature by the CA's
      cosigner.  The CA's cosigner (Section 5.5) is defined to certify
      the contents of every checkpoint and subtree that it signs.

   *  In signatureless certificates, the cosigner requirements are
      checked ahead of time, when the trusted subtrees are
      predistributed (Section 7.4).

   Given such a subtree hash, computed over entries that the CA
   certified, it then must be computationally infeasible to construct an
   entry not on this list, and some inclusion proof, such that inclusion
   proof verification succeeds.  This requires using a collision-
   resistant hash in the Merkle Tree construction.

   Log entries contain public key hashes, so it must additionally be
   computationally infeasible to compute a public key whose hash matches
   the entry, other than the intended public key.  This also requires a
   collision-resistant hash.

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12.2.  Transparency

   The transparency mechanisms in this document do not prevent a CA from
   issuing an unauthorized certificate.  Rather, they provide comparable
   security properties as Certificate Transparency [RFC9162] in ensuring
   that all certificates are either rejected by relying parties, or
   visible to monitors and, in particular, the subject of the
   certificate.

   Compared to Certificate Transparency, some of the responsibilities of
   a log have moved to the CA.  All signatures generated by the CA in
   this system are assertions about some view of the CA's issuance log.
   However, a CA does not need to function correctly to ensure
   transparency properties.  Relying parties are expected to require a
   quorum of additional cosigners, which together enforce properties of
   the log (Section 7.3) and prevent or detect CA misbehavior:

   A CA might violate the append-only property of its log and present
   different views to different parties.  However, each individual
   cosigner will only follow a single append-only view of the log
   history.  Provided the cosigners are correctly operated, relying
   parties and monitors will observe consistent views between each
   other.  Views that were not cosigned at all may not be detected, but
   they also will not be accepted by relying parties.

   If the CA sends one view to some cosigners and another view to other
   cosigners, it is possible that multiple views will be accepted by
   relying parties.  However, in that case monitors will observe that
   cosigners do not match each other.  Relying parties can then react by
   revoking the inconsistent indices (Section 7.5), and likely removing
   the CA.  If the cosigners are mirrors, the underlying entries in both
   views will also be visible.

   A CA might correctly construct its log, but refuse to serve some
   unauthorized entry, e.g. by feigning an outage or pruning the log
   outside the retention policy (Section 10.3).  If the relying party
   requires cosignatures from trusted mirrors, the entry will either be
   visible to monitors in the mirrors, or have never reached a mirror.
   In the latter case, the entry will not have been cosigned, so the
   relying party would not accept it.  If the relying party accepts log
   views without a trusted mirror, the unauthorized entry may not be
   available.  However, the existence of _some_ entry at that index will
   be visible, so monitors will know the CA is failing to present an
   entry.  Relying parties can then react by revoking the undisclosed
   entries by index (Section 7.5), and likely removing the CA.

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12.3.  Public Key Hashes

   Unlike Certificate Transparency, the mechanisms in this document do
   not provide the subject public keys, only the hashed values.  This is
   intended to reduce log serving costs, particularly with large post-
   quantum keys.  As a result, monitors look for unrecognized hashes
   instead of unrecognized keys.  Any unrecognized hash, even if the
   preimage is unknown, indicates an unauthorized certificate.

   This optimization complicates studies of weak public keys, e.g.
   [SharedFactors].  Such studies will have to retrieve the public keys
   separately, such as by connecting to the TLS servers, or fetching
   from the CA if it retains the unhashed key.  This document does not
   define a mechanism for doing this, or require that CAs or mirrors
   retain unhashed keys.  The transparency mechanisms in this protocol
   are primarily intended to allow monitors to observe certificate
   issuance.

12.4.  Non-Repudiation

   When a monitor finds an unauthorized certificate issuance in a log or
   mirror, it must be possible to prove the CA indeed certified the
   information in the entry.  However, only the latest checkpoint
   signature is retained by the transparency ecosystem, so it may not be
   possible to reconstruct the exact certificate seen by relying
   parties.

   However, per Section 5.5, any checkpoint signature is a binding
   assertion by the CA that it has certified every entry in the
   checkpoint.  Thus, given _any_ signed checkpoint that contains the
   unauthorized entry, a Merkle inclusion proof (Section 2.1.3 of
   [RFC9162]) is sufficient to prove the CA issued the entry.  This is
   analogous to how, in Section 3.2.1 of [RFC9162], CAs are held
   accountable for signed CT precertificates.

   The transparency ecosystem does not retain unhashed public keys, so
   it also may not be possible to construct a complete certificate from
   the checkpoint signature and inclusion proof.  However, if the log
   entry's subjectPublicKeyInfoHash does not correspond to an authorized
   key for the subject of the certificate, the entry is still
   unauthorized.  A Merkle Tree CA is held responsible for all log
   entries it certifies, whether or not the preimage of the hash is
   known.

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12.5.  New Log Entry Types

   MerkleTreeCertEntry (Section 5.3) is extensible and permits protocol
   extensions to define new formats for the CA to certify.  This means
   older CAs, cosigners, relying parties, and monitors might interact
   with new entries:

   Section 5.3 and Section 5.5 forbid a CA from logging or signing
   entries that it does not recognize.  A CA cannot faithfully claim to
   certify information if it does not understand it.  This is analogous
   to how a correctly-operated X.509 can never sign an unrecognized
   X.509 extension.

   External cosigners may or may not interact with the unrecognized
   entries.  [TLOG-MIRROR] and [TLOG-WITNESS] describe cosigners whose
   roles do not interpret the contents of log entries.  New entry types
   MAY be added without updating them.  If a cosigner role does
   interpret a log entry, it MUST define how it interacts with unknown
   ones.

   If a relying party trusts an issuance log, but the issuance log
   contains an unrecognized entry, the entry will not cause it to accept
   an unexpected certificate.  In Section 7.2, the relying party
   constructs the MerkleTreeCertEntry that it expects.  The unrecognized
   entry will have a different type value, so the proof will never
   succeed, assuming the underlying hash function remains collision-
   resistant.

   If a monitor observes an entry with unknown type, it may not be able
   to determine if it is of interest.  For example, it may be unable to
   tell whether it covers some relevant DNS name.  Until the monitor is
   updated to reflect the current state of the PKI, the monitor may be
   unable to detect all misissued certificates.

   This situation is analogous to the addition of a new X.509 extension.
   When relying parties add support for log entry types or new X.509
   extensions, they SHOULD coordinate with monitors to ensure the
   transparency ecosystem is able to monitor the new formats.

12.6.  Certificate Malleability

   An ASN.1 structure like X.509’s Certificate is an abstract data type
   that is independent of its serialization.  There are multiple
   encoding rules for ASN.1.  Commonly, protocols use DER [X.690], such
   as Section 4.4.2 of [RFC8446].  This aligns with Section 4.1.1.3 of
   [RFC5280], which says X.509 signatures are computed over the DER-
   encoded TBSCertificate.  After signature verification, applications
   can assume the DER-encoded TBSCertificate is not malleable.

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   While the signature verification process in Section 7.2 first
   transforms the TBSCertificate into a TBSCertificateLogEntry, it
   preserves this non-malleability.  There is a unique valid DER
   encoding for every abstract TBSCertificate structure, so malleability
   of the DER-encoded TBSCertificate reduces to malleability of the
   TBSCertificate value:

   *  The version, issuer, validity, subject, issuerUniqueID,
      subjectUniqueID, and extensions fields are copied from the
      TBSCertificate to the TBSCertificateLogEntry unmodified, so they
      are directly authenticated by the inclusion proof.

   *  serialNumber is omitted from TBSCertificateLogEntry, but its value
      determines the inclusion proof index, which authenticates it.

   *  The redundant signature field in TBSCertificate is omitted from
      TBSCertificateLogEntry, but Section 7.2 checks for an exact value,
      so no other values are possible.

   *  subjectPublicKeyInfo is hashed as subjectPublicKeyInfoHash in
      TBSCertificateLogEntry.  Provided the underlying hash function is
      collision-resistant, no other values are possible for a given log
      entry.

   Some non-conforming X.509 implementations use a BER [X.690] parser
   instead of DER and then, instead of computing the canonical DER
   encoding per Section 4.1.1.3 of [RFC5280], verify the signature over
   the particular BER encoding received.  Those implementations may
   still assume the received BER-encoded TBSCertificate is not
   malleable.  However, as BER encoding is not unique, this breaks the
   above assumptions in Merkle Tree Certificates.  To preserve non-
   malleability, these non-conforming implementations MUST do the
   following when verifying a Merkle Tree Certificate:

   *  When parsing the outermost TBSCertificate SEQUENCE tag and length,
      reparse with a conforming DER parser.

   *  When copying the version, issuer, validity, subject,
      issuerUniqueID, subjectUniqueID, and extensions fields, either
      reparse with a conforming DER parser, or copy over the observed
      BER encodings.

   *  Reparse the serialNumber field with a conforming DER parser.

   *  Reparse the signature field with a conforming DER parser.
      Equivalently, check for an exact match for the expected, DER-
      encoded value.

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   *  When hashing subjectPublicKeyInfo, either reparse with a
      conforming DER parser, or hash the observed BER encoding.

   Section 5.3 requires that the TBSCertificateLogEntry in a
   MerkleTreeCertEntry be DER-encoded, so applying a stricter parser
   will be compatible with conforming CAs.  While these existing non-
   conforming implementations may be unable to switch to a DER parser
   due to compatibility concerns, Merkle Tree Certificates is new, so
   there is no existing deployment of malformed BER-encoded
   TBSCertificateLogEntry structures.

   The above only ensures the TBSCertificate portion is non-malleable.
   In Merkle Tree Certificates, similar to ECDSA X.509 signature, the
   signature value is malleable.  Multiple MTCProof structures may prove
   a single TBSCertificate structure.  Additionally, in all X.509-based
   protocols, a BER-based parser for the outer, unsigned Certificate
   structure will admit malleability in those portions of the encoding.
   Applications that derive a unique identifier from the Certificate
   MUST instead use the TBSCertificate, or some portion of it, for
   Merkle Tree Certificates.

13.  IANA Considerations

13.1.  Module Identifier

   IANA is requested to add the following entry in the "SMI Security for
   PKIX Module Identifier" registry [RFC7299]:

                +=========+=================+============+
                | Decimal | Description     | References |
                +=========+=================+============+
                | TBD     | id-mod-mtc-2025 | [this-RFC] |
                +---------+-----------------+------------+

                                 Table 1

13.2.  Algorithm

   IANA is requested to add the following entry to the "SMI Security for
   PKIX Algorithms" registry [RFC7299]:

                +=========+=================+============+
                | Decimal | Description     | References |
                +=========+=================+============+
                | TBD     | id-alg-mtcProof | [this-RFC] |
                +---------+-----------------+------------+

                                 Table 2

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13.3.  Relative Distinguished Name Attribute

   IANA is requested to add the following entry to the "SMI Security for
   PKIX Relative Distinguished Name Attribute" registry
   [I-D.ietf-lamps-x509-alg-none]:

             +=========+=======================+============+
             | Decimal | Description           | References |
             +=========+=======================+============+
             | TBD     | id-rdna-trustAnchorID | [this-RFC] |
             +---------+-----------------------+------------+

                                 Table 3

14.  References

14.1.  Normative References

   [FIPS186-5]
              "Digital Signature Standard (DSS)", National Institute of
              Standards and Technology (U.S.),
              DOI 10.6028/nist.fips.186-5, February 2023,
              <https://doi.org/10.6028/nist.fips.186-5>.

   [FIPS204]  "Module-lattice-based digital signature standard",
              National Institute of Standards and Technology (U.S.),
              DOI 10.6028/nist.fips.204, August 2024,
              <https://doi.org/10.6028/nist.fips.204>.

   [I-D.ietf-tls-trust-anchor-ids]
              Beck, B., Benjamin, D., O'Brien, D., and K. Nekritz, "TLS
              Trust Anchor Identifiers", Work in Progress, Internet-
              Draft, draft-ietf-tls-trust-anchor-ids-02, 15 September
              2025, <https://datatracker.ietf.org/doc/html/draft-ietf-
              tls-trust-anchor-ids-02>.

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <https://www.rfc-editor.org/rfc/rfc2119>.

   [RFC3629]  Yergeau, F., "UTF-8, a transformation format of ISO
              10646", STD 63, RFC 3629, DOI 10.17487/RFC3629, November
              2003, <https://www.rfc-editor.org/rfc/rfc3629>.

   [RFC4648]  Josefsson, S., "The Base16, Base32, and Base64 Data
              Encodings", RFC 4648, DOI 10.17487/RFC4648, October 2006,
              <https://www.rfc-editor.org/rfc/rfc4648>.

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   [RFC5280]  Cooper, D., Santesson, S., Farrell, S., Boeyen, S.,
              Housley, R., and W. Polk, "Internet X.509 Public Key
              Infrastructure Certificate and Certificate Revocation List
              (CRL) Profile", RFC 5280, DOI 10.17487/RFC5280, May 2008,
              <https://www.rfc-editor.org/rfc/rfc5280>.

   [RFC5912]  Hoffman, P. and J. Schaad, "New ASN.1 Modules for the
              Public Key Infrastructure Using X.509 (PKIX)", RFC 5912,
              DOI 10.17487/RFC5912, June 2010,
              <https://www.rfc-editor.org/rfc/rfc5912>.

   [RFC8032]  Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
              Signature Algorithm (EdDSA)", RFC 8032,
              DOI 10.17487/RFC8032, January 2017,
              <https://www.rfc-editor.org/rfc/rfc8032>.

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <https://www.rfc-editor.org/rfc/rfc8174>.

   [RFC8446]  Rescorla, E., "The Transport Layer Security (TLS) Protocol
              Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,
              <https://www.rfc-editor.org/rfc/rfc8446>.

   [RFC8555]  Barnes, R., Hoffman-Andrews, J., McCarney, D., and J.
              Kasten, "Automatic Certificate Management Environment
              (ACME)", RFC 8555, DOI 10.17487/RFC8555, March 2019,
              <https://www.rfc-editor.org/rfc/rfc8555>.

   [RFC9110]  Fielding, R., Ed., Nottingham, M., Ed., and J. Reschke,
              Ed., "HTTP Semantics", STD 97, RFC 9110,
              DOI 10.17487/RFC9110, June 2022,
              <https://www.rfc-editor.org/rfc/rfc9110>.

   [RFC9162]  Laurie, B., Messeri, E., and R. Stradling, "Certificate
              Transparency Version 2.0", RFC 9162, DOI 10.17487/RFC9162,
              December 2021, <https://www.rfc-editor.org/rfc/rfc9162>.

   [SHS]      "Secure hash standard", National Institute of Standards
              and Technology (U.S.), DOI 10.6028/nist.fips.180-4, 2015,
              <https://doi.org/10.6028/nist.fips.180-4>.

   [X.690]    ITU-T, "Information technology - ASN.1 encoding Rules:
              Specification of Basic Encoding Rules (BER), Canonical
              Encoding Rules (CER) and Distinguished Encoding Rules
              (DER)", ISO/IEC 8824-1:2021 , February 2021.

14.2.  Informative References

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   [APPLE-CT] Apple, "Apple's Certificate Transparency policy", 5 March
              2021, <https://support.apple.com/en-us/HT205280>.

   [AuditingRevisited]
              Heimberger, L., Patton, C., and B. Westerbaan, "Private
              SCT Auditing, Revisited", 25 April 2025,
              <https://eprint.iacr.org/2025/556.pdf>.

   [CABF-153] CA/Browser Forum, "Ballot 153 – Short-Lived Certificates",
              11 November 2015, <https://cabforum.org/2015/11/11/ballot-
              153-short-lived-certificates/>.

   [CABF-SC081]
              CA/Browser Forum, "Ballot SC081v3: Introduce Schedule of
              Reducing Validity and Data Reuse Periods", 11 April 2025,
              <https://cabforum.org/2025/04/11/ballot-sc081v3-introduce-
              schedule-of-reducing-validity-and-data-reuse-periods/>.

   [CHROME-CT]
              Google Chrome, "Chrome Certificate Transparency Policy",
              17 March 2022,
              <https://googlechrome.github.io/CertificateTransparency/
              ct_policy.html>.

   [CHROMIUM] Chromium, "Component Updater", 3 March 2022,
              <https://chromium.googlesource.com/chromium/src/+/main/
              components/component_updater/README.md>.

   [FIREFOX]  Mozilla, "Firefox Remote Settings", 20 August 2022,
              <https://wiki.mozilla.org/Firefox/RemoteSettings>.

   [I-D.ietf-lamps-x509-alg-none]
              Benjamin, D., "Unsigned X.509 Certificates", Work in
              Progress, Internet-Draft, draft-ietf-lamps-x509-alg-none-
              10, 5 September 2025,
              <https://datatracker.ietf.org/doc/html/draft-ietf-lamps-
              x509-alg-none-10>.

   [LetsEncrypt]
              Let's Encrypt, "Let's Encrypt Stats", 7 March 2023,
              <https://letsencrypt.org/stats/>.

   [MerkleTown]
              Cloudflare, Inc., "Merkle Town", 7 March 2023,
              <https://ct.cloudflare.com/>.

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   [RFC4514]  Zeilenga, K., Ed., "Lightweight Directory Access Protocol
              (LDAP): String Representation of Distinguished Names",
              RFC 4514, DOI 10.17487/RFC4514, June 2006,
              <https://www.rfc-editor.org/rfc/rfc4514>.

   [RFC6962]  Laurie, B., Langley, A., and E. Kasper, "Certificate
              Transparency", RFC 6962, DOI 10.17487/RFC6962, June 2013,
              <https://www.rfc-editor.org/rfc/rfc6962>.

   [RFC6973]  Cooper, A., Tschofenig, H., Aboba, B., Peterson, J.,
              Morris, J., Hansen, M., and R. Smith, "Privacy
              Considerations for Internet Protocols", RFC 6973,
              DOI 10.17487/RFC6973, July 2013,
              <https://www.rfc-editor.org/rfc/rfc6973>.

   [RFC7299]  Housley, R., "Object Identifier Registry for the PKIX
              Working Group", RFC 7299, DOI 10.17487/RFC7299, July 2014,
              <https://www.rfc-editor.org/rfc/rfc7299>.

   [SCTNotAfter]
              Adrian, D., "How to distrust a CA without any certificate
              errors", March 2025,
              <https://dadrian.io/blog/posts/sct-not-after/>.

   [SharedFactors]
              Våge, H. F. and University of Bergen, "Finding shared RSA
              factors in the Certificate Transparency logs", 13 May
              2022, <https://bora.uib.no/bora-
              xmlui/bitstream/handle/11250/3001128/
              Masters_thesis__for_University_of_Bergen.pdf>.

   [SIGNED-NOTE]
              C2SP, "Note", April 2025, <https://c2sp.org/signed-note>.

   [STH-Discipline]
              Barnes, R., "STH Discipline & Security Considerations", 3
              March 2017, <https://mailarchive.ietf.org/arch/msg/trans/
              Zm4NqyRc7LDsOtV56EchBIT9r4c/>.

   [TLOG-CHECKPOINT]
              C2SP, "Transparency Log Checkpoints", March 2024,
              <https://c2sp.org/tlog-checkpoint>.

   [TLOG-MIRROR]
              C2SP, "Transparency Log Mirrors", July 2025,
              <https://c2sp.org/tlog-mirror>.

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   [TLOG-TILES]
              C2SP, "Tiled Transparency Logs", June 2025,
              <https://c2sp.org/tlog-tiles>.

   [TLOG-WITNESS]
              C2SP, "Transparency Log Witness Protocol", June 2025,
              <https://c2sp.org/tlog-witness>.

Appendix A.  ASN.1 Module

   MerkleTreeCertificates
     { iso(1) identified-organization(3) dod(6) internet(1)
       security(5) mechanisms(5) pkix(7) id-mod(0)
       id-mod-mtc-2025(TBD) }

   DEFINITIONS IMPLICIT TAGS ::=
   BEGIN

   IMPORTS
     SIGNATURE-ALGORITHM
     FROM AlgorithmInformation-2009  -- in [RFC5912]
       { iso(1) identified-organization(3) dod(6) internet(1)
         security(5) mechanisms(5) pkix(7) id-mod(0)
         id-mod-algorithmInformation-02(58) }
     ATTRIBUTE
     FROM PKIX-CommonTypes-2009 -- in [RFC5912]
       { iso(1) identified-organization(3) dod(6) internet(1)
         security(5) mechanisms(5) pkix(7) id-mod(0)
         id-mod-pkixCommon-02(57) } ;
     TrustAnchorID
     FROM TrustAnchorIDs-2025 -- in [I-D.ietf-tls-trust-ancohor-ids]
       { iso(1) identified-organization(3) dod(6) internet(1)
         security(5) mechanisms(5) pkix(7) id-mod(0)
         id-mod-trustAnchorIDs-2025(TBD) }

   TBSCertificateLogEntry  ::=  SEQUENCE  {
         version             [0]  EXPLICIT Version DEFAULT v1,
         issuer                   Name,
         validity                 Validity,
         subject                  Name,
         subjectPublicKeyInfoHash OCTET STRING,
         issuerUniqueID      [1]  IMPLICIT UniqueIdentifier OPTIONAL,
         subjectUniqueID     [2]  IMPLICIT UniqueIdentifier OPTIONAL,
         extensions          [3]  EXPLICIT Extensions OPTIONAL }

   id-alg-mtcProof OBJECT IDENTIFIER ::= {
       iso(1) identified-organization(3) dod(6) internet(1) security(5)
       mechanisms(5) pkix(7) algorithms(6) TBD}

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   sa-mtcProof SIGNATURE-ALGORITHM ::= {
      IDENTIFIER id-alg-mtcProof
      PARAMS ARE absent
   }

   id-rdna-trustAnchorID OBJECT IDENTIFIER ::= {
       iso(1) identified-organization(3) dod(6) internet(1) security(5)
       mechanisms(5) pkix(7) rdna(25) TBD}

   at-trustAnchorID ATTRIBUTE ::= {
      TYPE TrustAnchorID
      IDENTIFIED BY id-rdna-trustAnchorID
   }

   END

Appendix B.  Merkle Tree Structure

   This non-normative section describes how the Merkle Tree structure
   relates to the binary representations of indices.  It is included to
   help implementors understand the procedures described in Section 4.

B.1.  Binary Representations

   Within a Merkle Tree whose size is a power of two, the binary
   representation of an leaf's index gives the path to that leaf.  The
   leaf is a left child if the least-significant bit is unset and a
   right child if it is set.  The next bit indicates the direction of
   the parent node, and so on.  Figure 12 demonstrates this in a Merkle
   Tree of size 8:

          +----------------+
          |     [0, 8)     |        level 3
          +----------------+
           /              \
      +--------+      +--------+
      | [0, 4) |      | [4, 8) |    level 2
      +--------+      +--------+
       /      \        /      \
   +-----+ +-----+ +-----+ +-----+
   |[0,2)| |[2,4)| |[4,6)| |[6,8)|  level 1
   +-----+ +-----+ +-----+ +-----+
     / \     / \     / \     / \
   +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+
   |0| |1| |2| |3| |4| |5| |6| |7|  level 0
   +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+

                Figure 12: An example Merkle Tree of size 8

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   The binary representation of 4 is 0b100.  It is the left (0) child of
   [4, 6), which is the left (0) child of [4, 8), which is the right (1)
   child of [0, 8).

   Each level in the tree corresponds to a bit position and can be
   correspondingly numbered, with 0 indicating the least-significant bit
   and the leaf level, and so on.  In this numbering, a node's level can
   be determined as follows: if the node is a root of subtree [start,
   end), the node's level is BIT_WIDTH(end - start - 1).

   Comparing two indices determines the relationship between two paths.
   The highest differing bit gives the level at which paths from root to
   leaf diverge.  For example, the bit representations of 4 and 6 are
   0b100 and 0b110, respectively.  The highest differing bit is bit 1.
   Bits 2 and up are the same between the two indices.  This indicates
   that the paths from the root to leaves 4 and 6 diverge when going to
   level 2 to level 1.

   This can be generalized to arbitrary-sized Merkle Trees.  Figure 13
   depicts a Merkle Tree of size 6:

          +--------------+
          |     [0, 6)   |   level 3
          +--------------+
           /          |
      +--------+      |
      | [0, 4) |      *      level 2
      +--------+      |
       /      \       |
   +-----+ +-----+ +-----+
   |[0,2)| |[2,4)| |[4,6)|   level 1
   +-----+ +-----+ +-----+
     / \     / \     / \
   +-+ +-+ +-+ +-+ +-+ +-+
   |0| |1| |2| |3| |4| |5|   level 0
   +-+ +-+ +-+ +-+ +-+ +-+

                Figure 13: An example Merkle Tree of size 6

   When the size of a Merkle Tree is not a power of two, some levels on
   the rightmost edge of the tree are skipped.  The rightmost edge is
   the path to the last element.  The skipped levels can be seen in its
   binary representation.  Here, the last element is 5, which has binary
   representation 0b101.  When a bit is set, the corresponding node is a
   right child.  When it is unset, the corresponding node is skipped.

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   In a tree of the next power of two size, the skipped nodes in this
   path are where there _would_ have been a right child, had there been
   enough elements to construct one.  Without a right child, the hash
   operation is skipped and a skipped node has the same value as its
   singular child.  Figure 14 depicts this for a tree of size 6.

          +----------------+
          |     [0, 6)     |        level 3
          +----------------+
           /              \
      +--------+      +--------+
      | [0, 4) |      | [4, 6) |    level 2
      +--------+      +--------+
       /      \        /      \
   +-----+ +-----+ +-----+ +-----+
   |[0,2)| |[2,4)| |[4,6)| |     |  level 1
   +-----+ +-----+ +-----+ +-----+
     / \     / \     / \     / \
   +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+
   |0| |1| |2| |3| |4| |5| | | | |  level 0
   +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+

      Figure 14: An example Merkle Tree of size 6, viewed as a subset
                            of a tree of size 8

   Zero bits also indicate skipped nodes in paths that have not yet
   diverged from the rightmost edge (i.e. the path to the last element),
   when viewed from root to leaf.  In the example, the binary
   representation of 4 is 0b100.  While bit 0 and bit 1 are both unset,
   they manifest in the tree differently.  Bit 0 indicates that 4 is a
   right child.  However, at bit 1, 0b100 has not yet diverged from the
   last element, 0b101.  That instead indicates a skipped node, not a
   left child.

B.2.  Inclusion Proof Evaluation

   The procedure in Section 4.3.2 builds up a subtree hash in r by
   staring from entry_hash and iteratively hashing elements of
   inclusion_proof on the left or right.  That means this procedure,
   when successful, must return _some_ hash that contains entry_hash.

   Treating [start, end) as a Merkle Tree of size end - start, the
   procedure hashes by based on the path to index.  Within this smaller
   Merkle Tree, it has index fn = index - start (first number), and the
   last element has index sn = end - start - 1 (second number).

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   Step 4 iterates through inclusion_proof and the paths to fn and sn in
   parallel.  As the procedure right-shifts fn and sn and looks at the
   least-significant bit, it moves up the two paths, towards the root.
   When sn is zero, the procedure has reached the top of the tree.  The
   procedure checks that the two iterations complete together.

   Iterating from level 0 up, fn and sn will initially be different.
   While they are different, step 4.2 hashes on the left or right based
   on the binary representation, as discussed in Appendix B.1.

   Once fn = sn, the remainder of the path is on the right edge.  At
   that point, the condition in step 4.2 is always true.  It only
   incorporates proof entries on the left, once per set bit.  Unset bits
   are skipped.

   Inclusion proofs can also be evaluated by considering these two
   stages separately.  The first stage consumes l1 = BIT_WIDTH(fn XOR
   sn) proof entries.  The second stage consumes l2 = POPCOUNT(fn >> l1)
   proof entries.  A valid inclusion proof must then have l1 + l2
   entries.  The first l1 entries are hashed based on fn's least
   significant bits, and the remaining l2 entries are hashed on the
   left.

B.3.  Consistency Proof Structure

   A subtree consistency proof for [start, end) and the tree of n
   elements is similar to an inclusion proof for element end - 1.  If
   one starts from end - 1's hash, incorporating the whole inclusion
   proof should reconstruct root_hash and incorporating a subset of the
   inclusion proof should reconstruct node_hash.  Thus end - 1's hash
   and this inclusion proof can prove consistency.  A subtree
   consistency proof in this document applies two optimizations over
   this construction:

   1.  Instead of starting at level 0 with end - 1, the proof can start
       at a higher level.  Any ancestor of end - 1 shared by both the
       subtree and the overall tree is a valid starting node to
       reconstruct node_hash and root_hash.  Use the highest level with
       a commmon ancestor.  This truncates the inclusion proof portion
       of the consistency proof.

   2.  If this starting node is the entire subtree, omit its hash from
       the consistency proof.  The verifier is assumed to already know
       node_hash.

   A Merkle consistency proof, defined in Section 2.1.4 of [RFC9162],
   applies these same optimizations.

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   Figure 15 depicts a subtree consistency proof between the subtree [0,
   6) and the Merkle Tree of size 8.  The consistency proof begins at
   level 1, or node [4, 6).  The inclusion proof portion is similarly
   truncated to start at level 1: [6, 8) and [0, 4).  If the consistency
   proof began at level 0, the starting node would be leaf 5, and the
   consistency proof would additionally include leaf 4.

          +----------------+
          |     [0, 6)     |         level 3
          +----------------+
           /           |
      +========+  +--------+
      | [0, 4) |  | [4, 6) |         level 2
      +========+  +--------+
       /      \        |
   +-----+ +-----+ +~~~~~+
   |[0,2)| |[2,4)| |[4,6)|           level 1
   +-----+ +-----+ +~~~~~+
     / \     / \     / \
   +-+ +-+ +-+ +-+ +-+ +-+
   |0| |1| |2| |3| |4| |5|           level 0
   +-+ +-+ +-+ +-+ +-+ +-+

          +----------------+
          |     [0, 8)     |         level 3
          +----------------+
           /              \
      +========+      +--------+
      | [0, 4) |      | [4, 8) |     level 2
      +========+      +--------+
       /      \        /      \
   +-----+ +-----+ +~~~~~+ +=====+
   |[0,2)| |[2,4)| |[4,6)| |[6,8)|   level 1
   +-----+ +-----+ +~~~~~+ +=====+
     / \     / \     / \     / \
   +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+
   |0| |1| |2| |3| |4| |5| |6| |7|   level 0
   +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+

       Figure 15: A subtree consistency proof that starts at level 1
                             instead of level 0

   Note that the truncated inclusion proof may include nodes from lower
   levels, if the corresponding level was skipped on the right edge.
   Figure 16 depicts a subtree consistency proof between the subtree [0,
   6) and the Merkle Tree of size 7.  As above, the starting node is [4,
   6) at level 1.  The inclusion proof portion includes leaf 6 at level

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   0.  This is because leaf 6 is taking the place of its skipped parent
   at level 1.  (A skipped node can be thought of as a duplicate of its
   singular child.)

          +----------------+
          |     [0, 6)     |       level 3
          +----------------+
           /           |
      +========+  +--------+
      | [0, 4) |  | [4, 6) |       level 2
      +========+  +--------+
       /      \        |
   +-----+ +-----+ +~~~~~+
   |[0,2)| |[2,4)| |[4,6)|         level 1
   +-----+ +-----+ +~~~~~+
     / \     / \     / \
   +-+ +-+ +-+ +-+ +-+ +-+
   |0| |1| |2| |3| |4| |5|         level 0
   +-+ +-+ +-+ +-+ +-+ +-+

          +----------------+
          |     [0, 7)     |       level 3
          +----------------+
           /              \
      +========+      +--------+
      | [0, 4) |      | [4, 7) |   level 2
      +========+      +--------+
       /      \        /    |
   +-----+ +-----+ +~~~~~+ +=+
   |[0,2)| |[2,4)| |[4,6)| |6|     level 1
   +-----+ +-----+ +~~~~~+ +=+
     / \     / \     / \    |
   +-+ +-+ +-+ +-+ +-+ +-+ +-+
   |0| |1| |2| |3| |4| |5| |6|     level 0
   +-+ +-+ +-+ +-+ +-+ +-+ +-+

     Figure 16: The interaction between inclusion proof truncation and
                               skipped levels

B.4.  Consistency Proof Verification

   The procedure in Section 4.4.3 is structured similarly to inclusion
   proof evaluation (Appendix B.2).  It iteratively builds two hashes,
   fr and sr, which are expected to equal node_hash and root_hash,
   respectively.  Everything hashed into fr is also hashed into sr, so
   success demonstrates that root_hash contains node_hash.

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   Step 2 initializes fn (first number), sn (second number), and tn
   (third number) to follow, respectively, the paths to start, end - 1
   (the last element of the subtree), and n - 1 (the last element of the
   tree).

   Steps 3 and 4 then skip to the starting node, described in
   Appendix B.3.  The starting node may be:

   *  The entire subtree [start, end) if [start, end) is directly
      contained in the tree.  This will occur if end is n (step 3), or
      if [start, end) is full (exiting step 4 because fn is sn).

   *  Otherwise, the highest full subtree along the right edge of
      [start, end).  This corresponds to the process exiting step 4
      because LSB(sn) is not set.

   Steps 5 and 6 initialize the hashes fr and sr:

   *  In the first case above, fn will equal sn after truncation.  Step
      5 will then initialize the hashes to node_hash because consistency
      proof does not need to include the starting node.

   *  In the second case above, fn is less than sn.  Step 6 will then
      initialize the hashes to the first value in the consistency proof.

   Step 7 incorporates the remainder of the consistency proof into fr
   and sr:

   *  All hashes are incorporated into sr, with hashing on the left or
      right determined the same as in inclusion proof evaluation.

   *  A subset of the hashes are incorporated into fr.  It skips any
      hash on the right because those contain elements greater than end
      - 1.  It also stops incorporating when fn and sn have converged.

   This reconstructs the hashes of the subtree and full tree, which are
   then compared to expected values in step 8.

   In the case when fn is sn in step 5, the condition in step 7.2.1 is
   always false, and fr is always equal to node_hash in step 8.  In this
   case, steps 6 through 8 are equivalent to verifying an inclusion
   proof for the truncated subtree [fn, sn + 1) and truncated tree tn +
   1.

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Appendix C.  Extensions to Tiled Transparency Logs (To Be Removed)

   [[TODO: This section is expected to be removed.  It is sketched here
   purely for illustrative purposes, until the features are defined
   somewhere else, e.g. in the upstream tlog documents.]]

C.1.  Subtree Signed Note Format

   A subtree, with signatures, can be represented as a signed note
   [SIGNED-NOTE].  Trust anchor IDs can be converted into log origins
   and cosigner names by concatenating the ASCII string oid/1.3.6.1.4.1.
   and the ASCII representation of the trust anchor ID.  For example,
   the checkpoint origin for a log named 32473.1 would be
   oid/1.3.6.1.4.1.32473.1.

   The note body is a sequence of the following lines, each terminated
   by a newline character (U+000A):

   *  The log origin

   *  Two space-separated, non-negative decimal integers, <start> <end>

   *  The subtree hash, as single hash encoded in base64

   Each note signature has a key name of the cosigner name.  The
   signature's key ID is computed using the reserved signature type in
   [SIGNED-NOTE], and a fixed string, as follows:

   key ID = SHA-256(key name || 0x0A || 0xFF || "mtc-subtree/v1")[:4]

   A subtree whose start is zero can also be represented as a checkpoint
   [TLOG-CHECKPOINT].  A corresponding subtree signature can be
   represented as a note signature using a key ID computed as follows:

   key ID = SHA-256(key name || 0x0A || 0xFF || "mtc-checkpoint/v1")[:4]

   The only difference between the two forms is the implicit
   transformation from the signed note text to the MTCSubtree structure.

C.2.  Requesting Subtree Signatures

   This section defines the sign-subtree cosigner HTTP endpoint for
   clients to obtain subtree signatures from non-CA cosigners, such as
   mirrors and witnesses.  It may be used by the CA when assembling a
   certificate, or by an authenticating party to add a cosignature to a
   certificate that the CA did not themselves obtain.

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   The cosigner MAY expose this endpoint publicly to general
   authenticating parties, or privately to the CA.  The latter is
   sufficient if the CA is known to automatically request cosignatures
   from this cosigner when constructing certificates.  If private,
   authenticating the CA is out of scope for this document.

   Clients call this endpoint as POST <prefix>/sign-subtree, where
   prefix is some URL prefix.  For a mirror or witness, the URL prefix
   is the submission prefix.  The client's request body MUST be a
   sequence of:

   *  The requested subtree as a signed note (Appendix C.1), with zero
      or more signatures.  The endpoint MAY require signatures from the
      CA as a DoS mitigation, as described below.

   *  A blank line

   *  A checkpoint, signed by the requested cosigner.  The checkpoint's
      tree size must be at least end.

   *  A blank line

   *  Zero or more subtree consistency proof (Section 4.4) lines.  Each
      line MUST encode a single hash in base64 [RFC4648].  The client
      MUST NOT send more than 63 consistency proof lines.

   Each line MUST terminate in a newline character (U+000A).

   The cosigner performs the following steps:

   1.  Check that the checkpoint contains signatures from itself

   2.  Check that the subtree consistency proof proves consistency
       between the subtree hash and the checkpoint

   3.  If all checks pass, cosign the subtree, as described in
       Section 5.4

   On success, the response body MUST be a sequence of one or more note
   signature lines [SIGNED-NOTE], each starting with an em dash
   character (U+2014) and ending with a newline character (U+000A).  The
   signatures MUST be cosignatures from the cosigner key(s) on the
   subtree.

   Instead of statelessly validating checkpoints by signature, the
   cosigner MAY statefully check the requested checkpoint against
   internal witness or mirror state.  In this case, if the cosigner
   needs a newer checkpoint, it responds with a "409 Conflict" with its

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   latest signed checkpoint.  In this case, the subtree cosigning SHOULD
   remember and accept the last few signed checkpoints, to minimize
   conflicts.

   If operating statefully, the subtree cosigner process only needs read
   access to the mirror or witness state and can freely operate on stale
   state without violating any invariants.

   Mirrors MAY choose to check subtree hashes by querying their log
   state, instead of evaluating proofs.

   Publicly-exposed subtree cosigning endpoints MAY mitigate DoS in a
   variety of techniques:

   *  Only cosigning recent subtrees, as old subtrees do not need to be
      co-signed

   *  Caching subtree signatures

   *  Requiring a CA signature on the subtree; CAs are only expected to
      sign two subtrees (Section 4.5) for each checkpoint

   *  Rate-limiting requests

Acknowledgements

   This document stands on the shoulders of giants and builds upon
   decades of work in TLS authentication, X.509, and Certificate
   Transparency.  The authors would like to thank all those who have
   contributed over the history of these protocols.

   The authors additionally thank Bob Beck, Ryan Dickson, Aaron Gable,
   Nick Harper, Dennis Jackson, Matt Mueller, Chris Patton, Ryan Sleevi,
   and Emily Stark for many valuable discussions and insights which led
   to this document.  We wish to thank Mia Celeste in particular, whose
   implementation of an earlier draft revealed several pitfalls.

   The idea to mint tree heads infrequently was originally described by
   Richard Barnes in [STH-Discipline].  The size optimization in Merkle
   Tree Certificates is an application of this idea to the certificate
   itself.

Change log

      *RFC Editor's Note:* Please remove this section prior to
      publication of a final version of this document.

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Since draft-davidben-tls-merkle-tree-certs-00

   *  Simplify hashing by removing the internal padding to align with
      block size. #72

   *  Avoid the temptation of floating points. #66

   *  Require lifetime to be a multiple of batch_duration. #65

   *  Rename window to validity window. #21

   *  Split Assertion into Assertion and AbridgedAssertion.  The latter
      is used in the Merkle Tree and HTTP interface.  It replaces
      subject_info by a hash, to save space by not serving large post-
      quantum public keys.  The original Assertion is used everywhere
      else, including BikeshedCertificate. #6

   *  Add proper context to every node in the Merkle Tree. #32

   *  Clarify we use a single CertificateEntry. #11

   *  Clarify we use POSIX time. #1

   *  Elaborate on CA public key and signature format. #27

   *  Miscellaneous changes.

Since draft-davidben-tls-merkle-tree-certs-01

   *  Minor editorial changes

Since draft-davidben-tls-merkle-tree-certs-02

   *  Replace the negotiation mechanism with TLS Trust Anchor
      Identifiers.

Since draft-davidben-tls-merkle-tree-certs-03

   *  Switch terminology from "subscriber" to "authenticating party".

   *  Use <1..2^24-1> encoding for all certificate types in the
      CertificateEntry TLS message

   *  Clarify discussion and roles in transparency ecosystem

   *  Update references

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Since draft-davidben-tls-merkle-tree-certs-04

   Substantially reworked the design.  The old design was essentially
   the landmark checkpoint and CA-built logs ideas, but targeting only
   the optimized and slow issuance path, and with a more bespoke tree
   structure:

   In both draft-04 and draft-05, a CA looks like today’s CAs except
   that they run some software to publish what they issue and sign tree
   heads to certify certificates in bulk.

   In draft-04, the CA software publishes certificates in a bunch of
   independent Merkle Trees.  This is very easy to do as a collection of
   highly cacheable, immutable static files because each tree is
   constructed independently, and never appended to after being built.
   In draft-05, the certificates are published in a single Merkle Tree.
   The [TLOG-TILES] interface allows such trees to also use highly
   cacheable, immutable static files.

   In draft-04, there only are hourly tree heads.  Clients are
   provisioned with tree heads ahead of time so we can make small,
   inclusion-proof-only certificates.  In draft-05, the ecosystem must
   coordinate on defining "landmark" checkpoints.  Clients are
   provisioned with subtrees describing landmark checkpoints ahead of
   time so we can make small, inclusion-proof-only certificates.

   In draft-04, each tree head is independent.  In draft-05, each
   landmark checkpoint contains all the previous checkpoints.

   In draft-04, the independent tree heads were easily prunable.  In
   draft-05, we define how to prune a Merkle Tree.

   In draft-04, there is no fast issuance mode.  In draft-05, frequent,
   non-landmark checkpoints can be combined with inclusion proofs and
   witness signatures for fast issuance.  This is essentially an STH and
   inclusion proof in CT.

Since draft-davidben-tls-merkle-tree-certs-05

   *  Add some discussion on malleability

   *  Discuss the monitoring impacts of the responsibility shift from CA
      with log quorum to CA+log with mirror quorum

   *  Sketch out a more concrete initial ACME extension

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Since draft-davidben-tls-merkle-tree-certs-06

   *  Fix mistyped reference

   *  Removed now unnecessary placeholder text

   *  First draft at IANA registration and ASN.1 module

   *  Added a prose version of the procedure to select subtrees

   *  Rename 'landmarks checkpoint' to 'landmarks'

   *  Clarify and fix an off-by-one error in recommended landmark
      allocation scheme

   *  Add some diagrams to the Overview section

Since draft-davidben-tls-merkle-tree-certs-07

   *  Clarify landmark zero

   *  Clarify signature verification process

   *  Improve subtree consistency proof verification algorithm

   *  Add an appendix that explains the Merkle Tree proof procedures

Authors' Addresses

   David Benjamin
   Google LLC
   Email: davidben@google.com

   Devon O'Brien
   Email: devon.obrien@gmail.com

   Bas Westerbaan
   Cloudflare
   Email: bas@cloudflare.com

   Luke Valenta
   Cloudflare
   Email: lvalenta@cloudflare.com

Benjamin, et al.          Expires 22 April 2026                [Page 76]
Internet-Draft          Merkle Tree Certificates            October 2025

   Filippo Valsorda
   Geomys
   Email: ietf@filippo.io

Benjamin, et al.          Expires 22 April 2026                [Page 77]