On the distinctness of modular reductions of primitive sequences over Z/(232−1)

Abstract

This paper studies the distinctness of modular reductions of primitive sequences over $${\mathbf{Z}/(2^{32}-1)}$$ . Let f(x) be a primitive polynomial of degree n over $${\mathbf{Z}/(2^{32}-1)}$$ and H a positive integer with a prime factor coprime with 232?1. Under the assumption that every element in $${\mathbf{Z}/(2^{32}-1)}$$ occurs in a primitive sequence of order n over $${\mathbf{Z}/(2^{32}-1)}$$ , it is proved that for two primitive sequences $${\underline{a}=(a(t))_{t\geq 0}}$$ and $${\underline{b}=(b(t))_{t\geq 0}}$$ generated by f(x) over $${\mathbf{Z}/(2^{32}-1), \underline{a}=\underline{b}}$$ if and only if $${a\left( t\right) \equiv b\left( t\right) \bmod{H}}$$ for all t ? 0. Furthermore, the assumption is known to be valid for n between 7 and 100, 000, the range of which is sufficient for practical applications.