BIP: 374 source
  Layer: Applications
  Title: Discrete Log Equality Proofs
  Author: Andrew Toth <andrewstoth@gmail.com>
          Ruben Somsen <rsomsen@gmail.com>
          Sebastian Falbesoner <sebastian.falbesoner@gmail.com>
  Comments-URI: https://github.com/bitcoin/bips/wiki/Comments:BIP-0374
  Status: Draft
  Type: Standards Track
  License: BSD-2-Clause
  Created: 2024-12-26
  Post-History: https://gist.github.com/andrewtoth/df97c3260cc8d12f09d3855ee61322ea
                https://groups.google.com/g/bitcoindev/c/MezoKV5md7s

This document proposes a standard for 64-byte zero-knowledge discrete logarithm equality proofs (DLEQ proofs) over an elliptic curve. For given elliptic curve points A, B, C, G, and a scalar a known only to the prover where A = a⋅G and C = a⋅B, the prover proves knowledge of a without revealing anything about a. This can, for instance, be useful in ECDH: if A and B are ECDH public keys, and C is their ECDH shared secret computed as C = a⋅B, the proof establishes that the same secret key a is used for generating both A and C without revealing a.

This document is licensed under the 2-clause BSD license.

BIP352 requires senders to compute output scripts using ECDH shared secrets from the same secret keys used to sign the inputs. Generating an incorrect signature will produce an invalid transaction that will be rejected by consensus. An incorrectly generated output script can still be consensus-valid, meaning funds may be lost if it gets broadcast. By producing a DLEQ proof for the generated ECDH shared secrets, the signing entity can prove to other entities that the output scripts have been generated correctly without revealing the private keys.

All conventions and notations are used as defined in BIP327.

The basic proof generation uses a random scalar k, the secret a, and the point being proven C = a⋅B.

• Let R₁ = k⋅G.
• Let R₂ = k⋅B.
• Let e = hash(R₁ || R₂).
• Let s = (k + e⋅a).

Verifying the proof involves recreating R₁ and R₂ with only e and s as follows:

• Let R₁ = s⋅G - e⋅A.
• Let R₂ = s⋅B - e⋅C.

• s⋅G - e⋅A = (k + e⋅a)⋅G - e⋅A = k⋅G + e⋅(a⋅G) - e⋅A = k⋅G + e⋅A - e⋅A = k⋅G.
• s⋅B - e⋅C = (k + e⋅a)⋅B - e⋅C = k⋅B + e⋅(a⋅B) - e⋅C = k⋅B + e⋅C - e⋅C = k⋅B.

The following generates a proof that the result of a⋅B and the result of a⋅G are both generated from the same scalar a without having to reveal a.

Input:

• The secret key a: a 256-bit unsigned integer
• The public key B: a point on the curve
• Auxiliary random data r: a 32-byte array^[1]
• The generator point G: a point on the curve^[2]
• An optional message m: a 32-byte array^[3]

• Fail if a = 0 or a ≥ n.
• Fail if is_infinite(B).
• Let A = a⋅G.
• Let C = a⋅B.
• Let t be the byte-wise xor of bytes(32, a) and hash_BIP0374/aux(r).
• Let rand = hash_{BIP0374/nonce}(t || cbytes(A) || cbytes(C)).
• Let k = int(rand) mod n.
• Fail if k = 0.
• Let R₁ = k⋅G.
• Let R₂ = k⋅B.
• Let m' = m if m is provided, otherwise an empty byte array.
• Let e = int(hash_{BIP0374/challenge}(cbytes(A) || cbytes(B) || cbytes(C) || cbytes(G) || cbytes(R₁) || cbytes(R₂) || m')).
• Let s = (k + e⋅a) mod n.
• Let proof = bytes(32, e) || bytes(32, s).
• If VerifyProof(A, B, C, proof) (see below) returns failure, abort.
• Return the proof proof.

The following verifies the proof generated in the previous section. If the following algorithm succeeds, the points A and C were both generated from the same scalar. The former from multiplying by G, and the latter from multiplying by B.

Input:

• The public key of the secret key used in the proof generation A: a point on the curve
• The public key used in the proof generation B: a point on the curve
• The result of multiplying the secret and public keys used in the proof generation C: a point on the curve
• A proof proof: a 64-byte array
• The generator point used in the proof generation G: a point on the curve^[2]
• An optional message m: a 32-byte array^[3]

• Fail if any of is_infinite(A), is_infinite(B), is_infinite(C), is_infinite(G)
• Let e = int(proof[0:32]).
• Let s = int(proof[32:64]); fail if s ≥ n.
• Let R₁ = s⋅G - e⋅A.
• Fail if is_infinite(R₁).
• Let R₂ = s⋅B - e⋅C.
• Fail if is_infinite(R₂).
• Let m' = m if m is provided, otherwise an empty byte array.
• Fail if e ≠ int(hash_{BIP0374/challenge}(cbytes(A) || cbytes(B) || cbytes(C) || cbytes(G) || cbytes(R₁) || cbytes(R₂) || m')).
• Return success iff no failure occurred before reaching this point.

This proposal is compatible with all older clients.

A reference python implementation is included here. Test vectors can be generated by running ./bip-0374/gen_test_vectors.py which will produce a CSV file of random test vectors for both generating and verifying proofs. These can be run against the reference implementation with ./bip-0374/run_test_vectors.py.

1. ^ Why include auxiliary random data? The auxiliary random data should be set to fresh randomness for each proof. The same rationale and recommendations from BIP340 should be applied.
2. a b Why include the generator point G as an input? While all other BIPs have used the generator point from secp256k1, passing it as an input here lets this algorithm be used for other curves.
3. a b Why include a message as an input? This could be useful for protocols that want to authorize on a compound statement, not just knowledge of a scalar. This allows the protocol to combine knowledge of the scalar and the statement.

Thanks to josibake, Tim Ruffing, benma, stratospher, waxwing, Yuval Kogman and all others who participated in discussions on this topic.