【摘要】 Blum整数(BL)已经被广泛应用在密码学领域中,它是形式为pk1qk2的整数,其中p和q是模4余3的不同素数,而k1和k2是奇数,这种整数通常被分为两种,即:I∶={M|M=pq}和II∶={M|M=pk1qk2} ,其中k1和k2至少有一个大于1的奇数.Bruce Schneier中提出了一个开问题:不知道是否存在一个证明整数M∈BL且M∈I的实用零知识证明系统.该文基于离散对数构造了两个具有如下基质的零知识证明系统:1)证明者能确信验者M∈BL;2)证明者能确信验者M∈I或M∈II.另外,也构造了证明一个秘密整数a不等于零的零知识证明系统.更多还原

【Abstract】 Blum integers （BL）, which have extensively been used in the domain of cryptography, are integers with form pk₁qk₂, where p and q are different primes both ≡3 mod 4 and k₁ and k₂ are odd integers. These integers can be divided into two types:I∶={M|M=pq}, and II∶={M|M=pk₁qk₂}, where at least one of k₁ and k₂ is greater than 1. 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 zero-knowledge proof systems 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∈I or M∈II. In addition, we also propose a statistical zero-knowledge proof in which the prover proves that a committed integer a is not equal to 0.更多还原