Binary sequences derived from differences of consecutive quadratic residues

TL;DR

This paper investigates binary sequences derived from quadratic residues modulo a prime, analyzing their linear complexity, balance, and pattern distribution, with implications for cryptographic sequence design.

Contribution

It introduces and studies two new binary sequences from quadratic residues, providing conditions for maximal linear complexity and analyzing their statistical properties.

Findings

Sequence $(d_n)$ can attain maximal linear complexity under certain conditions.

Sequence $(d_n)$ is unbalanced, with about 1/3 zeros and 2/3 ones for large primes.

Sequence $(t_n)$ is balanced and exhibits uniform pattern distribution for large primes.

Abstract

For a prime $p\ge 5$ let $q_0,q_1,\ldots,q_{(p-3)/2}$ be the quadratic residues modulo $p$ in increasing order. We study two $(p-3)/2$-periodic binary sequences $(d_n)$ and $(t_n)$ defined by $d_n=q_n+q_{n+1}\bmod 2$ and $t_n=1$ if $q_{n+1}=q_n+1$ and $t_n=0$ otherwise, $n=0,1,\ldots,(p-5)/2$. For both sequences we find some sufficient conditions for attaining the maximal linear complexity $(p-3)/2$. Studying the linear complexity of $(d_n)$ was motivated by heuristics of Caragiu et al. However, $(d_n)$ is not balanced and we show that a period of $(d_n)$ contains about $1/3$ zeros and $2/3$ ones if $p$ is sufficiently large. In contrast, $(t_n)$ is not only essentially balanced but also all longer patterns of length $s$ appear essentially equally often in the vector sequence $(t_n,t_{n+1},\ldots,t_{n+s-1})$, $n=0,1,\ldots,(p-5)/2$, for any fixed $s$ and sufficiently large $p$.