Binary Sequences Derived from Differences of Consecutive Primitive Roots

TL;DR

This paper investigates the pseudorandom properties of binary sequences derived from differences of primitive roots modulo a prime, analyzing their balance, complexity, and potential for cryptographic use.

Contribution

It provides new insights into the distribution, complexity bounds, and conditions for optimality of these sequences, highlighting their cryptographic relevance.

Findings

Sequences are often unbalanced for typical primes

Infinitely many primes yield well-balanced sequences

Lower bounds and conditions for maximum complexity are established

Abstract

Let $1<g_1<\ldots<g_{\varphi(p-1)}<p-1$ be the ordered primitive roots modulo~$p$. We study the pseudorandomness of the binary sequence $(s_n)$ defined by $s_n\equiv g_{n+1}+g_{n+2}\bmod 2$, $n=0,1,\ldots$. In particular, we study the balance, linear complexity and $2$-adic complexity of $(s_n)$. We show that for a typical $p$ the sequence $(s_n)$ is quite unbalanced. However, there are still infinitely many $p$ such that $(s_n)$ is very balanced. We also prove similar results for the distribution of longer patterns. Moreover, we give general lower bounds on the linear complexity and $2$-adic complexity of~$(s_n)$ and state sufficient conditions for attaining their maximums. Hence, for carefully chosen $p$, these sequences are attractive candidates for cryptographic applications.