serai/crypto/dleq

Discrete Log Equality

Implementation of discrete log equality both within a group and across groups, the latter being extremely experimental, for curves implementing the ff/group APIs. This library has not undergone auditing and the cross-group DLEq proof has no formal proofs available.

Cross-Group DLEq

The present cross-group DLEq is based off MRL-0010, which isn't computationally correct as while it proves both keys have the same discrete-log value for the G'/H' component, yet doesn't prove a lack of a G/H component. Accordingly, it was augmented with a pair of Schnorr Proof of Knowledges, proving a known G'/H' component, guaranteeing a lack of a G/H component (assuming an unknown relation between G/H and G'/H').

The challenges for the ring signatures were also merged, removing one-element from each bit's proof with only a slight reduction to challenge security (as instead of being uniform over each scalar field, they're uniform over the mutual bit capacity of each scalar field). This reduction is identical to the one applied to the proved-for scalar, and accordingly should not reduce overall security. It does create a lack of domain separation, yet that shouldn't be an issue.

The following variants are available:

• ClassicLinear. This is only for reference purposes, being the above described proof, with no further optimizations.

• ConciseLinear. This proves for 2 bits at a time, not increasing the signature size for both bits yet decreasing the amount of commitments/challenges in total.

• EfficientLinear. This provides ring signatures in the form ((R_G, R_H), s), instead of (e, s), and accordingly enables a batch verification of their final step. It is the most performant, and also the largest, option.

• CompromiseLinear. This provides signatures in the form ((R_G, R_H), s) AND proves for 2-bits at a time. While this increases the amount of steps in verifying the ring signatures, which aren't batch verified, and decreases the amount of items batched (an operation which grows in efficiency with quantity), it strikes a balance between speed and size.

The following numbers are from benchmarks performed with Secp256k1/Ed25519 on a Intel i7-118567:

Algorithm	Size	Performance
ClassicLinear	56829 bytes (+27%)	157ms (0%)
ConciseLinear	44607 bytes (Reference)	156ms (Reference)
EfficientLinear	65145 bytes (+46%)	122ms (-22%)
CompromiseLinear	48765 bytes (+9%)	137ms (-12%)

CompromiseLinear is the best choce by only being marginally sub-optimal regarding size, yet still achieving most of the desired performance improvements. That said, neither the original postulation (which had flaws) nor any construction here has been proven nor audited. Accordingly, they are solely experimental, and none are recommended.

All proofs are suffixed Linear in the hope a logarithmic proof makes itself available, which would likely immediately become the most efficient option.