On The Distinctness Of Modular Reductions Of Maximal Length Sequences Over Z/(pq)

The study of architectures to produce nonlinear sequences with desirable good properties is thought to be the foremost research subject on the design of stream ciphers. Researches on sequences over rings have shown that a class of nonlinear sequences can be deduced from injectivity of compressing maps on primitive sequences over rings. Firstly, this dissertation studies the distinctness of modular reductions of primitive sequences over Z/(pq), where p and q are two different odd primes with p < q. Let a and b be two primitive sequences generated by a primitive polynomial f(x) of degree n over Z/(pq). The main results proved here are:1. For the case of n = 1, if gcd(p–1, q–1) = 2 and (p–1) / ordp(2) is congruent to (q–1) / ordq(2) modulo 2, then a≡b mod 2 if and only if a = b.2. For the case of n > 1, if there exist a nonnegative integer S and a primitive elementξin Z/(pq) such that x S -ξ≡0 mod (f(x), pq), and either (q - 1) is not divisible by (p - 1) or 2(p - 1) divides (q - 1), then a≡b mod 2 if and only if a = b.3. For the case of n > 1, if there exist a nonnegative integer S and a primitive elementξin Z/(pq) such that x S -ξ≡0 mod (f(x), pq), then a≡b mod M if and only if a = b, where M > 2 and gcd(M, pq) = 1. Secondly, we discuss the partial period distribution of the sequence deduced from the primitive sequence over Z/(pq) modulo 2 and the result is:4. Let Tp = pn -1, Tq = qn -1, T = lcm(Tp, Tq). For s∈{0, 1}, denote by PL(s) the proportion of number of s occurring in a segment with L length of a mod 2. Then we have (?)Where ln is natual logarithm and (?)Furthermore, the method in conclusion 4 can be reused to get a similar result on the partial period distribution of the sequence deduced from the primitive sequence over Z/(pq) modulo M, where gcd(M, pq) = 1.