gnark's Groth16 commitment extension unsound for more than one commitment

QFinch assessment

The summary is that the proof of knowledge associated to a commitment is crucial to bind the commitment to the actual circuit variables that were supposed to be committed. However, the same σ is used for all proofs of knowledge for the commitments, which allows mixing between them, making it possible to fix the value of all but one commitment before choosing the circuit variable assignments.

Potential impact

It is a soundness issue - in case of multiple commitments used inside the circuit the prover is able to choose all but the last commitment. As we use the commitments for optimized non-native multiplication, lookup checks etc. as random challenges, then it could impact the soundness of the whole circuit. However, using multiple commitments has been discouraged due to the additional cost to the verifier and it has not been supported in the recursive in-circuit Groth16 verifier and Solidity verifier. So we expect the impact of the issue be very small - only for the users who have implemented the native Groth16 verifier or are using it with multiple commitments. We do not have information of such users.