This BIP specifies a standard way to generate and verify DLEQ proofs. This is motivated by sending to silent payments in PSBTs. However, there are also other uses where DLEQs could be useful, so it would be good to have this BIP for others to reference.

This is inspired by https://github.com/discreetlogcontracts/dlcspecs/blob/master/ECDSA-adaptor.md#proof-of-discrete-logarithm-equality, but is a little more specific.
There is an implementation of that already at https://github.com/BlockstreamResearch/secp256k1-zkp/blob/master/src/modules/ecdsa_adaptor/dleq_impl.h, which this BIP attempts to be compatible with.

Pull request here https://github.com/bitcoin/bips/pull/1689

<pre>
  BIP: ?
  Title: Discrete Log Equality Proofs over secp256k1
  Author: Andrew Toth <andre...@gmail.com>
          Ruben Somsen <rso...@gmail.com>
  Comments-URI: TBD
  Status: Draft
  Type: Standards Track
  License: BSD-2-Clause
  Created: 2024-06-29
  Post-History: TBD
</pre>

== Introduction ==

=== Abstract ===

This document proposes a standard for 64-byte zero-knowledge ''discrete logarithm equality proofs'' (DLEQ proofs) over the elliptic curve ''secp256k1''. For given elliptic curve points ''A'', ''B'', and ''C'', the prover proves knowledge of a scalar ''a'' such that ''A = a⋅G'' and ''C = a⋅B'' 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''.

=== Copyright ===

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

=== Motivation ===

[https://github.com/bitcoin/bips/blob/master/bip-0352.mediawiki#specification 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.

== Specification ==

All conventions and notations are used as defined in [https://github.com/bitcoin/bips/blob/master/bip-0327.mediawiki#user-content-Notation BIP327].

=== DLEQ Proof Generation ===

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

The algorithm ''Prove(a, B, r)'' is defined as:
* Fail if ''a = 0'' or ''a &ge; 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_BIP?/aux(r)''.
* Let ''rand = hash_DLEQ(t || cbytes(A) || cytes(C))''.
* Let ''k = int(rand) mod n''.
* Fail if ''k = 0''.
* Let ''R₁ = k⋅G''.
* Let ''R₂ = k⋅B''.
* Let ''e = int(hash_DLEQ(cbytes(A) || cbytes(B) || cbytes(C) || cbytes(R₁) || cbytes(R₂)))''.
* Let ''proof = bytes(32, e) || bytes(32, (k + ea) mod n)''.
* If ''VerifyProof(A, B, C, proof)'' (see below) returns failure, abort.
* Return the proof ''proof''.

=== DLEQ Proof Verification ===

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 algorithm ''VerifyProof(A, B, C, proof)'' is defined as:
* Let ''e = int(proof[0:32])''.
* Let ''s = int(proof[32:64])''; fail if ''s &ge; n''.
* Let ''R₁ = s⋅G - e⋅A''.
* Fail if ''is_infinite(R₁)''.
* Fail if ''not has_even_y(R₁)''.
* Let ''R₂ = s⋅B - e⋅C''.
* Fail if ''is_infinite(R₂)''.
* Fail if ''not has_even_y(R₂)''.
* Fail if ''e ≠ int(hash_BIP?/DLEQ(cbytes(A) || cbytes(B) || cbytes(C) || cbytes(R₁) || cbytes(R₂)))''.
* Return success iff no failure occurred before reaching this point.

== Test Vectors and Reference Code ==

TBD

== Changelog ==

TBD

== Footnotes ==

<references />

== Acknowledgements ==

TBD