Short Non-interactive Zero-Knowledge Proofs

We show that probabilistically checkable proofs can be used to shorten non-interactive zero-knowledge proofs. We obtain publicly verifiable non-interactive zero-knowledge proofs for circuit satisfiability with adaptive and unconditional soundness where the size grows quasi-linearly in the number of gates. The zero-knowledge property relies on the existence of trapdoor permutations, or it can be based on a specific number theoretic assumption related to factoring to get better efficiency. As an example of the latter, we suggest a non-interactive zero-knowledge proof for circuit satisfiability based on the Naccache-Stern cryptosystem consisting of a quasi-linear number of bits. This yields the shortest known non-interactive zero-knowledge proof for circuit satisfiability.