Compact zero-knowledge arguments for Blum integers
Résumé
We present a communication-efficient zero-knowledge argument of knowledge for the factorization of Blum integers, a special class of integers of the form n = pq, 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 an argument as small as 6.3 KB, with prover and verifier computational costs comparable to existing protocols that require over 131 KB.
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