Distribution Properties of Binary Sequences Derived from Primitive Sequences Modulo Square-free Odd Integers

Abstract

Recently, a class of nonlinear sequences, modular reductions of primitive sequences over integer residue rings, was proposed and has attracted much attention. In particular, modulo 2 reductions of primitive sequences over $\mathbf {Z}/(2^{31}-1)$ were used in the ZUC algorithm. In this paper, we study the distribution properties of modulo 2 reductions of primitive sequences over $\mathbf {Z}/(M)$, where M is a square-free odd integer. Let $\underline{a}$ be a primitive sequence of order n over $\mathbf {Z}/(M)$ with period T and $\left[ \underline{a}\right] _{\text {mod}\, 2}$ the modulo 2 reduction of $\underline{a}$. With the estimate of exponential sums over $\mathbf {Z}/(M)$, the proportion $f_{s}$ of occurrences of s within a segment of $\left[ \underline{a}\right] _{\text {mod}\, 2}$ of length $\mu T$ is estimated, where $s\in \left\{ 0,1\right\}$ and $0<\mu \le 1$. Based on this estimate, it is further shown that for given M and $\mu$, $f_{s}$ tends to $\frac{M+1-2s}{2M}$ as $n\rightarrow \infty$. This result implies that there exists a small imbalance between 0 and 1 in $\left[ \underline{a}\right] _{\text {mod}\, 2}$, which should be taken into full consideration in the design of stream ciphers based on $\left[ \underline{a}\right] _{\text {mod}\, 2}$.

This work was supported by NSF of China (Nos. 61872383, 61402524, 61872359 and 61602510). The work of Qun-Xiong Zheng was also supported by Young Elite Scientists Sponsorship Program by CAST (2016QNRC001) and by National Postdoctoral Program for Innovative Talents (BX201600188) and by China Postdoctoral Science Foundation funded project (2017M611035).