A Merkle tree that pairs an odd node with itself also collides on duplicated tails of 2 or more leaves and on several copies of the last leaf: test every tail length, split like RFC 6962, and hash the empty list as SHA-256() · id PC-247
· checked_on 2026-11-12
· revision of PC-227
· license CC-BY-4.0
1) Do not pair an odd node with itself: split like RFC 6962 section 2.1 (largest power of two smaller than n, 0x00 prefix for leaves, 0x01 for inner nodes), or promote the odd node unchanged (the same tree), and/or commit the leaf count where the verifier checks it. 2) Hash the empty list as SHA-256() of the empty string (RFC 6962 MTH({})), not recursion into an empty slice. 3) Check: for every N you accept, build the root for the list and for the list plus a copy of its last t leaves for t = 1..N, and plus t copies of the last leaf; no root may equal the original. A check of only one extra leaf is too narrow. 4) Treat the 0x00/0x01 prefixes as a separate protection against a leaf passing for an inner node; they do not remove the duplication.
#!/usr/bin/env python3
"""merkle_odd.py 2.0 — find duplicated-tail collisions in a Merkle root; commit the count or do not self-pair.
usage: python3 -I merkle_odd.py --selftest
A tree that pairs an odd last node with itself (hash(x||x)) gives one root for
several lists: [a,b,c] and [a,b,c,c], but also [a..f] and [a..f,e,f] (a duplicated
tail of 2 leaves) and [a..e] with 1, 2 or 3 extra copies of e. Checking only
"N odd, plus one copy of the last leaf" misses these. tail_collisions() appends a
copy of the last t leaves (t = 1..N) and t copies of the last leaf, and reports
every variant whose root equals the original. exhaustive() compares all lists up
to a length over a small alphabet. RFC 6962 (split at the largest power of two
below n, 0x00/0x01 prefixes, empty list = SHA-256() of the empty string),
promoting the odd node, and committing the leaf count give no collision in these
checks. This selftest uses fixture leaves (no network).
Exit 0: pass. Exit 1: unexpected result. Exit 2: usage.
"""
from __future__ import annotations
import hashlib
import itertools
import sys
def sha(b: bytes) -> bytes:
return hashlib.sha256(b).digest()
def leaf(x: bytes) -> bytes:
return sha(b"\x00" + x)
def node(left: bytes, right: bytes) -> bytes:
return sha(b"\x01" + left + right)
def naive(leaves):
"""The bug: an odd last node is paired with itself."""
if not leaves:
raise ValueError("naive tree: empty list has no root")
nodes = [leaf(x) for x in leaves]
while len(nodes) > 1:
if len(nodes) % 2 == 1:
nodes.append(nodes[-1])
nodes = [node(nodes[i], nodes[i + 1]) for i in range(0, len(nodes), 2)]
return nodes[0]
def promote(leaves):
"""An odd last node moves up a level unchanged."""
if not leaves:
return sha(b"")
nodes = [leaf(x) for x in leaves]
while len(nodes) > 1:
nxt = [node(nodes[i], nodes[i + 1]) for i in range(0, len(nodes) - 1, 2)]
if len(nodes) % 2 == 1:
nxt.append(nodes[-1])
nodes = nxt
return nodes[0]
def rfc6962(leaves):
"""RFC 6962 section 2.1 Merkle Tree Hash, including the empty list."""
n = len(leaves)
if n == 0:
return sha(b"")
if n == 1:
return leaf(leaves[0])
k = 1
while k * 2 < n:
k *= 2
return node(rfc6962(leaves[:k]), rfc6962(leaves[k:]))
def bare(leaves):
"""RFC 6962 split without the 0x00/0x01 prefixes: a leaf can then pass for an inner node."""
n = len(leaves)
if n == 1:
return sha(leaves[0])
k = 1
while k * 2 < n:
k *= 2
return sha(bare(leaves[:k]) + bare(leaves[k:]))
def with_count(root_fn):
"""Commit the leaf count (8 bytes, big-endian) next to the root."""
def rooted(leaves):
return sha(b"\x02" + len(leaves).to_bytes(8, "big") + root_fn(leaves))
return rooted
def tail_variants(leaves, max_copies=None):
"""Lists that a self-pairing tree may confuse with `leaves`."""
n = len(leaves)
out = []
for t in range(1, n + 1):
out.append(("tail", t, list(leaves) + list(leaves[-t:])))
for m in range(1, (max_copies or n) + 1):
if n:
out.append(("copies", m, list(leaves) + [leaves[-1]] * m))
return out
def tail_collisions(root_fn, leaves, max_copies=None):
base = root_fn(leaves)
return [(kind, t) for kind, t, v in tail_variants(leaves, max_copies) if root_fn(v) == base]
def exhaustive(root_fn, alphabet=(b"a", b"b"), max_len=8, min_len=1):
"""All lists of length min_len..max_len over alphabet; pairs of distinct lists with one root."""
seen = {}
hits = []
for n in range(min_len, max_len + 1):
for items in itertools.product(alphabet, repeat=n):
r = root_fn(list(items))
if r in seen:
hits.append((seen[r], items))
else:
seen[r] = items
return hits
L = [bytes([c]) for c in b"abcdefghijklmnop"]
def selftest() -> None:
a_f = L[:6]
a_e = L[:5]
# the known case and the narrow check
assert naive(L[:3]) == naive(L[:3] + [L[2]]), "abc vs abcc"
assert naive(a_f) != naive(a_f + [L[5]]), "old check passes for n=6"
assert naive(a_f) == naive(a_f + [L[4], L[5]]), "tail of 2 leaves"
for m in (1, 2, 3):
assert naive(a_e) == naive(a_e + [L[4]] * m), f"n=5 with {m} copies"
assert ("tail", 2) in tail_collisions(naive, a_f)
assert {("copies", 1), ("copies", 2), ("copies", 3)} <= set(tail_collisions(naive, a_e)), "n=5 copies 1-3"
assert ("copies", 1) not in tail_collisions(naive, a_f), "n=6: one extra copy does not collide"
assert tail_collisions(naive, L[:12]) and not any(
k == "copies" and t == 1 for k, t in tail_collisions(naive, L[:12])), "n=12: copy-1 check misses"
# the fixes: no tail collision for n = 1..16
fixes = {"rfc6962": rfc6962, "promote": promote,
"count+naive": with_count(naive), "count+rfc6962": with_count(rfc6962)}
naive_hits = 0
for n in range(1, 17):
naive_hits += len(tail_collisions(naive, L[:n]))
for name, fn in fixes.items():
assert not tail_collisions(fn, L[:n]), f"{name} n={n}"
assert naive_hits > 0
# exhaustive: all lists of 1..8 leaves over {a,b}
assert exhaustive(naive), "naive has collisions"
for name, fn in fixes.items():
assert not exhaustive(fn), f"{name} exhaustive"
# empty list: RFC 6962 MTH({}) = SHA-256()
assert rfc6962([]) == hashlib.sha256(b"").digest()
assert promote([]) == rfc6962([])
assert with_count(rfc6962)([]) != rfc6962([])
assert not exhaustive(rfc6962, max_len=4, min_len=0)
try:
naive([])
except ValueError:
pass
else:
raise AssertionError("naive([]) must raise ValueError")
# prefixes are a separate protection: without them a two-leaf list equals a one-leaf list of two hashes
assert bare(L[:2]) == bare([sha(L[0]) + sha(L[1])]), "bare tree: leaf passes for inner node"
assert rfc6962(L[:2]) != rfc6962([leaf(L[0]) + leaf(L[1])]), "prefixed tree separates them"
assert not tail_collisions(bare, L[:6]), "prefix removal alone does not bring the tail collision back"
# RFC 6962 section 2.1 prefixes, written out: 0x00 for a leaf, 0x01 for an inner node
assert leaf(b"x") == hashlib.sha256(b"\x00x").digest(), "leaf prefix"
assert node(b"l", b"r") == hashlib.sha256(b"\x01lr").digest(), "node prefix"
# promotion is the same tree as RFC 6962
for n in range(0, 17):
assert promote(L[:n]) == rfc6962(L[:n]), f"promote==rfc n={n}"
def main(argv=None) -> int:
argv = sys.argv[1:] if argv is None else argv
if argv == ["--selftest"]:
try:
selftest()
except (AssertionError, RecursionError) as e:
print("unexpected result", e)
return 1
print("pass")
return 0
print(__doc__.strip(), file=sys.stderr)
return 2
if __name__ == "__main__":
sys.exit(main())
Merkle rootCVE-2012-2459RecursionErrorA manifest or batch is attested by a Merkle root; a verifier rebuilds the tree and accepts a list that repeats the end of the original list, because both lists hash to the same root. The team added the one-extra-leaf check and considers the tree safe.
With hash(x||x) on an odd node, [a,b,c] and [a,b,c,c] share a root, but so do [a..f] and [a..f,e,f] (a duplicated tail of 2 leaves, although [a..f] and [a..f,f] differ) and [a..e] with 1, 2 or 3 extra copies of e. A check of N plus one copy of the last leaf passes for N = 6 and N = 12 and still lets collisions through. A fixed tree that raises RecursionError on an empty list also fails the verifier instead of returning the RFC 6962 value.
On every level with an odd number of nodes the last node is hashed with itself, so a padding copy at any level looks like a real copy. The duplicated part can be the last leaf, the last pair or a longer tail. The RFC 6962 split uses the shape of the tree (largest power of two below n) instead, and defines the hash of an empty list as SHA-256 of the empty string.
Save the snippet as merkle_odd.py and run `python3 -I merkle_odd.py --selftest` (fixture leaves, no network). Pass: last line is `pass` and the exit code is 0. It asserts that the naive tree collides for [a,b,c] and [a,b,c,c], does not collide for [a..f] with one extra f but does for [a..f,e,f], collides for five leaves with 1 to 3 extra copies, and that RFC 6962, promotion and a committed leaf count show no tail collision for lists of 1 to 16 leaves and no collision among all lists up to 8 leaves over two symbols; that the empty list hashes to SHA-256 of the empty string and the naive tree raises ValueError on it; and the two prefixes. Without arguments the script prints usage and exits 2. Checked 2026-11-12 with Python 3.13.5: selftest pass, usage exit 2, 13 one-line broken variants each exit 1 (mutaties.py), among them the check of only one tail length, no copies, no count, a wrong empty-list value, a split at k <= n and missing prefixes. Fixture leaves only: no real manifest, signature or audit path was checked.
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