Compact zero-knowledge arguments for Blum integers

Abstract

We present a communication-efficient zero-knowledge proof of knowledge for the factorization of Blum integers, a special class of integers of the form n = p q , where p and q are distinct prime numbers satisfying p ≡ q ≡ 3 mod 4 and p ≃ q ≃ n . Existing protocols for proving such statements often incur significant communication costs, especially when demonstrating that p and q are of nearly equal size. We leverage the MPC-in-the-head paradigm, a cryptographic technique that transforms secure multi-party computation protocols into efficient zero-knowledge proof systems. In our protocol, the prover uses additive sharing of p and q over the integers. This approach simplifies proving the size relationship p ≃ q ≃ n and the congruence p ≡ q ≡ 3 mod 4 without requiring costly range proofs. To verify the primality of p and q , we employ the Boneh-Franklin biprimality test. Our protocol achieves a significant reduction in communication complexity. For a 2048-bit integer n and 128-bit security, we construct a proof as small as 12.3 KB, with prover and verifier computational costs comparable to existing protocols that require over 131 KB.