The Statistical Zero-knowledge Proof for Blum Integer Based on Discrete Logarithm

Blum integers (BL), which has extensively been used in the domain of cryptography, are integers with form p1q2 , where p and q are different primes both ≡ 3 mod 4 and k1 and k2 are odd integers. These integers can be divided two types: 1) M = pq, 2) M = p1q2 , where at least one of k1 and k2 is greater than 1. In [3], Bruce Schneier has already proposed an open problem: it is unknown whether there exists a truly practical zero-knowledge proof for M(= pq) ∈ BL. In this paper, we construct two statistical zeroknowledge proofs based on discrete logarithm, which satisfies the two following properties: 1) the prover can convince the verifier M ∈ BL ; 2) the prover can convince the verifier M = pq or M = p1q2 , where at least one of k1 and k2 is more than 1. In addition, we propose a statistical zero-knowledge proof in which the prover proves that a committed integer a is not equal to 0.