Paper 2024/1488

Compact Proofs of Partial Knowledge for Overlapping CNF Formulae

Abstract

At CRYPTO '94, Cramer, Damgaard, and Schoenmakers introduced a general technique for constructing honest-verifier zero-knowledge proofs of partial knowledge (PPK), where a prover Alice wants to prove to a verifier Bob she knows $$ witnesses for $$ claims out of $$ claims without revealing the indices of those $$ claims. Their solution starts from a base honest-verifier zero-knowledge proof of knowledge $$ and requires to run in parallel $$ execution of the base protocol, giving a complexity of $$, where $$ is the communication complexity of the base protocol. However, modern practical scenarios require communication-efficient zero-knowledge proofs tailored to handle partial knowledge in specific application-dependent formats. In this paper we propose a technique to compose a large class of $$-protocols for atomic statements into $$-protocols for PPK over formulae in conjunctive normal form (CNF) that overlap, in the sense that there is a common subset of literals among all clauses of the formula. In such formulae, the statement is expressed as a conjunction of $$ clauses, each of which consists of a disjunction of $$ literals (i.e., each literal is an atomic statement) and $$ literals are shared among clauses. The prover, for a threshold parameter $$, proves knowledge of at least $$ witnesses for $$ distinct literals in each clause. At the core of our protocol, there is a new technique to compose $$-protocols for regular CNF relations (i.e., when $$) that exploits the overlap among clauses and that we then generalize to formulae where $$ providing improvements over state-of-the-art constructions.