International Association
for Cryptologic Research

At CRYPTO 1994, Cramer, Damg{\aa}rd and Schoenmakers proposed a general method to construct proofs of knowledge (PoKs), especially for $k$-out-of-$n$ partial knowledge, of which relations can be expressed in disjunctive normal form (DNF). Since then, proofs of $k$-out-of-$n$ partial knowledge have attracted much attention and some efficient constructions have been proposed. However, many practical scenarios require efficient PoK protocols for partial knowledge in other forms. In this paper, we mainly focus on PoK protocols for $k$-conjunctive normal form ($k$-CNF) relations, which have $n$ statements and can be expressed as follows: (i) $k$ statements constitute a clause via ``OR'' operations, and (ii) the relation consists of multiple clauses via ``AND'' operations. We propose an alternative Sigma protocol (called DAG-$\Sigmaup$ protocol) for $k$-CNF relations, by turning these relations into directed acyclic graphs (DAGs). Our DAG-$\Sigmaup$ protocol achieves less communication cost and smaller computational overhead compared with Cramer et al.'s general method.