Merkle Mountain Range for Immediately Verifiable and Replicable Commitments

Internet-Draft	MMRIVER	November 2024
Bryce	Expires 30 May 2025	[Page]

Abstract

This specification describes the COSE encoding of proofs for post-order traversal binary Merkle trees, also known as history trees and Merkle mountain ranges. Proving and verifying are defined in terms of the cryptographic asynchronous accumulator described by ReyzinYakoubov. The technical advantages of post-order traversal binary Merkle trees are discussed in CrosbyWallachStorage and PostOrderTlog.¶

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.¶

A complete MMR(n) defines an mmr with n nodes where no equal height sibling trees exist.¶
i shall be the index of any node, including leaf nodes, in the MMR¶
g shall be the zero based height of a node in the tree.¶
H(x) shall be the SHA-256 digest of any value x¶
|| shall mean concatenation of raw byte representations of the referenced values.¶

In this specification, all numbers are unsigned 64 bit integers. The maximum height of a single tree is 64 (which will have g=63 for its peak).¶

Table 1: Verifiable Data Structure Algorithms
Name	Value	Description	Reference
MMRIVER_SHA256	TBD_1 (requested assignment 3)	Linearly addressed, position committing, MMR implementations, such as the MMRIVER ledger	This document

4. Inclusion Proof

The CBOR representation of an inclusion proof is¶

inclusion-proof = bstr .cbor [

  ; zero based index of a tree node
  index: uint

  ; path proving the node's inclusion
  inclusion-path: [ + bstr ]
]

Note that the inclusion path for the index leads to a single permanent node in the tree. This node will initially be a peak in the accumulator, as the tree grows it will eventually be "buried" by a new peak.¶

4.1. inclusion_proof_path

inclusion_proof_path(i, c) is used to produce the verification paths for inclusion proofs and consistency proofs.¶

Given:¶

c the index of the last node in any tree which contains i.¶
i the index of the mmr node whose verification path is required.¶

And the methods:¶

index_height (Section 9.1) which obtains the zero based height g of any node.¶

And the constraints:¶

i <= c¶

We define inclusion_proof_path as¶

  def inclusion_proof_path(i, c):

    path = []

    g = index_height(i)

    while True:

      # The sibling of i is at i +/- 2^(g+1)
      siblingoffset = (2 << g)

      # If the index after i is higher, it is the left parent,
      # and i is the right sibling
      if index_height(i+1) > g:

        # The witness to the right sibling is offset behind i
        isibling = i - siblingoffset + 1

        # The parent of a right sibling is stored immediately
        # after
        i += 1
      else:

        # The witness to a left sibling is offset ahead of i
        isibling = i + siblingoffset - 1

        # The parent of a left sibling is stored immediately after
        # its right sibling
        i += siblingoffset

      # When the computed sibling exceeds the range of MMR(C+1),
      # we have completed the path
      if isibling > c:
          return path

      path.append(isibling)

      # Set g to the height of the next item in the path.
      g += 1

5. Receipt of Inclusion

The cbor representation of an inclusion proof is:¶

protected-header-map = {
  &(alg: 1) => int
  &(vds: 395) => 3
  * cose-label => cose-value
}

alg (label: 1): REQUIRED. Signature algorithm identifier. Value type: int.¶
vds (label: 395): REQUIRED. verifiable data structure algorithm identifier. Value type: int.¶

The unprotected header for an inclusion proof signature is:¶

inclusion-proofs = [ + inclusion-proof ]

verifiable-proofs = {
  &(inclusion-proof: -1) => inclusion-proofs
}

unprotected-header-map = {
  &(vdp: 396) => verifiable-proofs
  * cose-label => cose-value
}

The payload of an MMRIVER inclusion proof signature is the tree peak committing to the nodes inclusion, or the node itself where the proof path is empty. The algorithm included_root (Section 5.2) obtains this value.¶

The payload MUST be detached. Detaching the payload forces verifiers to recompute the root from the inclusion proof, this protects against implementation errors where the signature is verified but the payload merkle root does not match the inclusion proof.¶

5.1. Verifying the Receipt of inclusion

The inclusion proof and signature are verified in order. First the verifiers applies the inclusion proof to a possible entry (set member) bytes. The result is the merkle root implied by the inclusion proof path for the candidate value. The COSE Sign1 payload MUST be set to this value. Second the verifier checks the signature of the COSE Sign1. If the resulting signature verifies, the Receipt has proved inclusion of the entry in the verifiable data structure. If the resulting signature does not verify, the signature may have been tampered with.¶

It is recommended that implementations return a single boolean result for Receipt verification operations, to reduce the chance of accepting a valid signature over an invalid inclusion proof.¶

As the proof must be processed prior to signature verification the implementation SHOULD check the lengths of the proof paths are appropriate for the provided tree sizes.¶

5.2. included_root

The algorithm included_root calculates the accumulator peak for the provided proof and node value.¶

Given:¶

i is the index the nodeHash is to be shown at¶
nodehash the value whose inclusion is to be shown¶
proof is the path of sibling values committing i.¶

And the methods:¶

index_height (Section 9.1) which obtains the zero based height g of any node.¶
hash_pospair64 (Section 8.3.1) which applies H to the new node position and its children.¶

We define included_root as¶

  def included_root(i, nodehash, proof):

    root = nodehash

    g = index_height(i)

    for sibling in proof:

      # If the index after i is higher, it is the left parent,
      # and i is the right sibling

      if index_height(i + 1) > g:

        # The parent of a right sibling is stored immediately after

        i = i + 1

        # Set `root` to `H(i+1 || sibling || root)`
        root = hash_pospair64(i + 1, sibling, root)
      else:

        # The parent of a left sibling is stored immediately after
        # its right sibling.

        i = i + (2 << g)

        # Set `root` to `H(i+1 || root || sibling)`
        root = hash_pospair64(i + 1, root, sibling)

      # Set g to the height of the next item in the path.
      g = g + 1

    # If the path length was zero, the original nodehash is returned
    return root

6. Consistency Proof

A consistency proof shows that the accumulator, defined in ReyzinYakoubov, for tree-size-1 is a prefix of the accumulator for tree-size-2.¶

The signature is over the complete accumulator for tree-size-2 obtained using the proof and the, supplied, possibly empty, list of right-peaks which complete the accumulator for tree-size-2.¶

The receipt of consistency is defined so that a chain of cumulative consistency proofs can be verified together.¶

The cbor representation of a consistency proof is:¶

consistency-path = [ * bstr ]

consistency-proof =  bstr .cbor [

  ; previous tree size
  tree-size-1: uint

  ; latest tree size
  tree-size-2: uint

  ; the inclusion path from each accumulator peak in
  ; tree-size-1 to its new peak in tree-size-2.
  consistency-paths: [ + consistency-path ]

  ; the additional peaks that
  ; complete the accumulator for tree-size-2,
  ; when appended to those produced by the consistency paths
  right-peaks: [ *bstr ]
]

6.1. consistency_proof_path

Produces the verification paths for inclusion of the peaks of tree-size-1 under the peaks of tree-size-2.¶

right-peaks are obtained by invoking peaks(tree-size-2 - 1), and discarding length(proofs) from the left.¶

Given:¶

ifrom is the last index of tree-size-1¶
ito is the last index of tree-size-2¶

And the methods:¶

inclusion_proof_path (Section 4.1)¶
Section 9.2 ¶

And the constraints:¶

ifrom <= ito¶

We define consistency_proof_paths as¶

  def consistency_proof_paths(ifrom, ito):

    proof = []

    for i in peaks(ifrom):
      proof.append(inclusion_proof_path(i, ito))

    return proof

7. Receipt of Consistency

The cbor representation of an inclusion proof for MMRIVER is:¶

protected-header-map = {
  &(alg: 1) => int
  &(vds: 395) => 3
  * cose-label => cose-value
}

alg (label: 1): REQUIRED. Signature algorithm identifier. Value type: int.¶
vds (label: 395): REQUIRED. verifiable data structure algorithm identifier. Value type: int.¶

The unprotected header for an inclusion proof signature is:¶

consistency-proofs = [ + consistency-proof ]

verifiable-proofs = {
  &(consistency-proof: -2) => consistency-proof
}

unprotected-header-map = {
  &(vdp: 396) => verifiable-proofs
  * cose-label => cose-value
}

The payload MUST be detached. Detaching the payload forces verifiers to recompute the roots from the consistency proofs. This protects against implementation errors where the signature is verified but the payload is not genuinely produced by the included proof.¶

7.1. Verifying the Receipt of consistency

Verification accommodates verifying the result of a cumulative series of consistency proofs.¶

Perform the following for each consistency-proof in the list, verifying the signature with the output of the last.¶

Initialize current proof as the first consistency-proof.¶
Initialize accumulatorfrom to the peaks of tree-size-1 in the current proof.¶
Initialize ifrom to tree-size-1 - 1 from the current proof.¶
Initialize proofs to the consistency-paths from the current proof.¶
Apply the algorithm consistent_roots (Section 7.1.1)¶
Apply the peaks algorithm to obtain the accumulator for tree-size-2¶
From the peaks for tres-size-2, discard from the left the number of roots returned by consistent_roots.¶
Create the consistent accumulator by appending the remaining peaks to the consistent roots.¶
If there are no remaining proofs, use the consistent accumulator as the detached payload and verify the signature of the COSE Sign1.¶

It is recommended that implementations return a single boolean result for Receipt verification operations, to reduce the chance of accepting a valid signature over an invalid consistency proof.¶

As the proof must be processed prior to signature verification the implementation SHOULD check the lengths of the proof paths are appropriate for the provided tree sizes.¶

7.1.1. consistent_roots

consistent_roots returns the descending height ordered list of elements from the accumulator for the consistent future state.¶

Implementations MUST require that the number of peaks returned by Section 9.2(ifrom) equals the number of entries in accumulatorfrom.¶

Given:¶

ifrom the last index in the complete MMR from which consistency was proven.¶
accumulatorfrom the node values corresponding to the peaks of the accumulator for tree-size-1¶
proofs the inclusion proofs for each node in accumulatorfrom for tree-size-2¶

And the methods:¶

included_root (Section 5.2)¶
Section 9.2 ¶

We define consistent_roots as¶

  def consistent_roots(ifrom, accumulatorfrom, proofs):

    frompeaks = peaks(ifrom)

    # if length(frompeaks) != length(proofs) -> ERROR

    roots = []
    for i in range(len(accumulatorfrom)):
      root = included_root(
          frompeaks[i], accumulatorfrom[i], proofs[i])

      if roots and roots[-1] == root:
          continue
      roots.append(root)

    return roots

8. Appending a leaf

An algorithm for appending to a tree maintained in post order layout is provided. Implementation defined methods for interacting with storage are specified.¶

8.1. add_leaf_hash

When a new node is appended, if its height matches the height of its immediate predecessor, then the two equal height siblings MUST be merged. Merging is defined as the append of a new node which takes the adjacent peaks as its left and right children. This process MUST proceed until there are no more completable sub trees.¶

add_leaf_hash(f) adds the leaf hash value f to the tree.¶

Given:¶

f the leaf value resulting from H(x) for the caller defined leaf value x¶
db an interface supporting the Section 8.2.2 and Section 8.2.1 implementation defined storage methods.¶

And the methods:¶

index_height (Section 9.1)¶
Section 8.3.1 ¶

We define add_leaf_hash as¶

  def add_leaf_hash(db, f: bytes):

    # Set g to 0, the height of the leaf item f
    g = 0

    # Set i to the result of invoking Append(f)
    i = db.append(f)

    # If index_height(i) is greater than g (#looptarget)
    while index_height(i) > g:

      # Set ileft to the index of the left child of i,
      # which is i - 2^(g+1)

      ileft = i - (2 << g)

      # Set iright to the index of the the right child of i,
      # which is i - 1

      iright = i - 1

      # Set v to H(i + 1 || Get(ileft) || Get(iright))
      # Set i to the result of invoking Append(v)

      i = db.append(
        hash_pospair64(i+1, db.get(ileft), db.get(iright)))

      # Set g to the height of the new i, which is g + 1
      g += 1

    return i

8.2. Implementation defined storage methods

The following methods are assumed to be available to the implementation. Very minimal requirements are specified.¶

Informally, the storage must be array like and have no gaps.¶

8.2.1. Get

Reads the value from the tree at the supplied index.¶

The read MUST be consistent with any other calls to Append or Get within the same algorithm invocation.¶

Get MAY fail for transient reasons.¶

8.2.2. Append

Appends new node to storage and returns the index that will be occupied by the node provided to the next call to append.¶

The implementation MUST guarantee that the results of Append are immediately available to Get calls in the same invocation of the algorithm.¶

Append MUST return the node i identifying the node location which comes next.¶

The implementation MAY defer commitment to underlying persistent storage.¶

Append MAY fail for transient reasons.¶

8.3. Node values

Interior nodes in the MUST prefix the value provided to H(x) with pos.¶

The value v for any interior node MUST be H(pos || Get(LEFT_CHILD) || Get(RIGHT_CHILD))¶

The algorithm for leaf addition is provided the result of H(x) directly.¶

8.3.1. hash_pospair64

Returns H(pos || a || b), which is the value for the node identified by index pos - 1¶

Editors note: How this draft accommodates hash alg agility is tbd.¶

Given:¶

pos the size of the MMR whose last node index is pos - 1¶
a the first value to include in the hash after pos¶
b the second value to include in the hash after pos¶

And the constraints:¶

pos < 2^64¶
a and b MUST be hashes produced by the appropriate hash alg.¶

We define hash_pospair64 as¶

  def hash_pospair64(pos, a, b):

    # Note: Hash algorithm agility is tbd, this example uses SHA-256
    h = hashlib.sha256()

    # Take the big endian representation of pos
    h.update(pos.to_bytes(8, byteorder="big", signed=False))
    h.update(a)
    h.update(b)
    return h.digest()

9. Essential supporting algorithms

9.1. index_height

index_height(i) returns the zero based height g of the node index i¶

Given:¶

i the index of any mmr node.¶

We define index_height as¶

  def index_height(i) -> int:
    pos = i + 1
    while not all_ones(pos):
      pos = pos - most_sig_bit(pos) + 1

    return bit_length(pos) - 1

9.2. peaks

peaks(i) returns the peak indices for MMR(i+1), which is also its accumulator.¶

Assumes MMR(i+1) is complete, implementations can check for this condition by testing the height of i+1¶

Given:¶

i the index of any mmr node.¶

We define peaks¶

  def peaks(i):
    peak = 0
    peaks = []
    s = i+1
    while s != 0:
      # find the highest peak size in the current MMR(s)
      highest_size = (1 << log2floor(s+1)) - 1
      peak = peak + highest_size
      peaks.append(peak-1)
      s -= highest_size

    return peaks

Appendix B. Assumed bit primitives

B.1. log2floor

Returns the floor of log base 2 x¶

  def log2floor(x):
    return x.bit_length() - 1

B.2. most_sig_bit

Returns the mask for the the most significant bit in pos¶

  def most_sig_bit(pos) -> int:
    return 1 << (pos.bit_length() - 1)

The following primitives are assumed for working with bits as they commonly have library or hardware support.¶

B.3. bit_length

The minimum number of bits to represent pos. b011 would be 2, b010 would be 2, and b001 would be 1.¶

  def bit_length(pos):
    return pos.bit_length()

B.4. all_ones

Tests if all bits, from the most significant that is set, are 1, b0111 would be true, b0101 would be false.¶

  def all_ones(pos) -> bool:
    msb = most_sig_bit(pos)
    mask = (1 << (msb + 1)) - 1
    return pos == mask

B.5. ones_count

Count of set bits. For example ones_count(b101) is 2¶

B.6. trailing_zeros

  (v & -v).bit_length() - 1

Appendix C. Implementation Status

Note to RFC Editor: Please remove this section as well as references to BCP205 before AUTH48.¶

This section records the status of known implementations of the protocol defined by this specification at the time of posting of this Internet-Draft, and is based on a proposal described in BCP205. The description of implementations in this section is intended to assist the IETF in its decision processes in progressing drafts to RFCs. Please note that the listing of any individual implementation here does not imply endorsement by the IETF. Furthermore, no effort has been spent to verify the information presented here that was supplied by IETF contributors. This is not intended as, and must not be construed to be, a catalog of available implementations or their features. Readers are advised to note that other implementations may exist.¶

According to BCP205, "this will allow reviewers and working groups to assign due consideration to documents that have the benefit of running code, which may serve as evidence of valuable experimentation and feedback that have made the implemented protocols more mature. It is up to the individual working groups to use this information as they see fit".¶

C.1. Implementers

C.1.1. DataTrails

An open-source implementation was initiated and is maintained by Data Trails Inc. - DataTrails.¶

Uses SHA-256 as the hash alg¶

C.1.1.1. Implementation Name

An application demonstrating the concepts is available at https://app.datatrails.ai/.¶

C.1.1.2. Implementation URL

An open-source implementation is available at:¶

https://github.com/datatrails/go-datatrails-merklelog¶

C.1.1.3. Maturity

Used in production. SEMVER unstable (no backwards compat declared yet)¶

C.1.2. Robin Bryce (1)

C.1.2.1. Implementation URL

A minimal reference implementation of this draft. Used to generate the test vectors in this draft, is available at:¶

https://github.com/robinbryce/draft-bryce-cose-merkle-mountain-range-proofs/blob/main/algorithms.py¶

C.1.2.2. Maturity

Reference only¶

C.1.3. Robin Bryce (2)

C.1.3.1. Implementation URL

A minimal tiled log implementation¶

https://github.com/robinbryce/mmriver-tiles-ts¶

C.1.3.2. Maturity

Prototype¶

C.1.4. Mimblewimble

Is specifically committing to positions as we describe, but is committing zero based indices, and uses BLAKE2B as the HASH-ALG. Accounting for those differences, their commitment trees would be compatible with this draft.¶

C.1.4.1. Implementation URL

An implementation is available here:¶

https://github.com/mimblewimble/grin/blob/master/doc/mmr.md (Grin is a rust implementation of the mimblewimble protocol)¶
https://github.com/BeamMW/beam/blob/master/core/merkle.cpp (Beam is a C++ implementation of the mimblewimble protocol)¶

C.1.5. Herodotus

C.1.5.1. Implementation URL

https://github.com/HerodotusDev/rust-accumulators¶

Production, supports keccak, posiedon & pedersen hash algs¶

Editors note: test vectors, based on a SHA256 instantiation, are currently provided in a separate document. These should inlined if the draft is accepted: https://github.com/robinbryce/draft-bryce-cose-merkle-mountain-range-proofs/blob/main/test-vectors.md¶