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Zero-knowledge proofs of satisfiability of linear equations over a group are often used as a building block of more complex protocols. In particular, in an asymmetric bilinear group we often have two commitments in different sides of the pairing, and we want to prove that they open to the same value. This problem was tackled by González, Hevia and Ràfols (ASIACRYPT 2015), who presented an aggregated proof, in the QA-NIZK setting, consisting of only four group elements. In this work, we present a more efficient proof, which is based on the same assumptions and consists of three group elements. We argue that our construction is optimal in terms of proof size.

QA-NIZK Arguments of Same Opening for Bilateral Commitments