Balance and pattern distribution of sequences derived from pseudorandom subsets of $\mathbb{Z}_q$

TL;DR

This paper analyzes the balance and pattern distribution of sequences derived from pseudorandom subsets of integers modulo q, providing conditions under which these sequences exhibit desirable pseudorandom properties.

Contribution

The paper introduces a unified framework for analyzing sequences from pseudorandom subsets, extending previous results on quadratic residues and primitive roots.

Findings

Sequence (s_n) is balanced when T is smaller than q.

Sequence (t_n) is balanced when T is about (1 - 2^{-1/(m-1)})q.

Sequence (u_n) is balanced when T is approximately q/2.

Abstract

Let $q$ be a positive integer and $\mathcal{S}=\left\{x_0,x_1,\ldots,x_{T-1}\right\}\subseteq\mathbb{Z}_q=\{0,1,\ldots,q-1\}$ with $$0\leq x_0<x_1<\ldots< x_{T-1}\leq q-1.$$ We derive from $\mathcal{S}$ three (finite) sequences. 1. For an integer $M\geq 2$ let $(s_n)$ be the $M$-ary sequence defined by \begin{eqnarray*} s_n\equiv x_{n+1}-x_n \bmod M, \qquad n=0,1,\ldots, T-2. \end{eqnarray*} 2. For an integer $m\geq 2$ let $(t_n)$ be the binary sequence defined by \begin{eqnarray*} t_n=\left\{\begin{array}{ll} 1, & \hbox{if } 1\leq x_{n+1}-x_n\leq m-1, \\ 0, & \hbox{otherwise}, \end{array}\right. \qquad n=0,1,\ldots, T-2. \end{eqnarray*} 3. Let $(u_n)$ be the characteristic sequence of $\mathcal{S}$, \begin{eqnarray*} u_n=\left\{\begin{array}{ll} 1, & \hbox{if } n\in \mathcal{S}, \\ 0, & \hbox{otherwise}, \end{array}\right. \qquad n=0,1,\ldots, q-1. \end{eqnarray*} We study the balance and pattern distribution of the sequences $(s_n)$, $(t_n)$ and $(u_n)$. For sets $\mathcal{S}$ with desirable pseudorandom properties, more precisely, sets with low correlation measures, we show the following: 1. The sequence $(s_n)$ is (asymptotically) balanced and has uniform pattern distribution if $T$ is of smaller order of magnitude than $q$. 2. The sequence $(t_n)$ is balanced and has uniform pattern distribution if $T$ is approximately $\left(1-\frac{1}{2^{1/(m-1)}}\right)q$. 3. The sequence $(u_n)$ is balanced and has uniform pattern distribution if $T$ is approximately $\frac{q}{2}$. These results are motivated by earlier results for the sets of quadratic residues and primitive roots modulo a prime. We unify these results and derive many further (asymptotically) balanced sequences with uniform pattern distribution from pseudorandom subsets.