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**2003/232**

**Title:**

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

**Authors:**
Chunming Tang, Zhuojun Liu, Jinwang Liu

**Abstract:**

Blum integers (BL), which has extensively been used in the domain

of cryptography, are integers with form p^{k_1}q^{k_2}, where

p and q are different primes both \equiv
3\hspace{4pt}mod\hspace{4pt}4 and k_1 and k_2 are odd

integers. These integers can be divided two types: 1) M=pq, 2)

M=p^{k_1}q^{k_2}, where at least one of k_1 and k_2 is

greater than 1.\par

In \cite{dbk3}, Bruce Schneier has already proposed an open

problem: {\it it is unknown whether there exists a truly practical

zero-knowledge proof for M(=pq)\in BL}. In this paper, we

construct two statistical zero-knowledge proofs based on discrete

logarithm, which satisfies the two following properties: 1) the

prover can convince the verifier M\in BL ; 2) the prover can

convince the verifier M=pq or M=p^{k_1}q^{k_2}, where at least

one of k_1 and k_2 is more than 1.\par

In addition, we propose a statistical zero-knowledge proof in

which the prover proves that a committed integer a is not equal

to 0.\par

**ePrint:**
https://eprint.iacr.org/2003/232

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