Binary Sequences Derived from Differences of Consecutive Primitive Roots

Abstract

Let $1<g_1<\dots <g_{\phi (p-1)}<p-1$ be the ordered primitive roots modulo~$p$. We study the pseudorandomness of the binary sequence $(s_n)$ defined by $s_n\equiv g_{n+1}+g_{n+2}\mathrm{mod}2$, $n=0,1,\dots$. 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.

References in corpus (1)

• A linear complexity analysis of quadratic residues and primitive roots spacings