Binary sequences derived from differences of consecutive quadratic residues
This work addresses theoretical properties of sequences in number theory and cryptography, with incremental contributions to understanding linear complexity and randomness.
The paper tackles the problem of analyzing binary sequences derived from differences of consecutive quadratic residues modulo a prime, finding that one sequence attains maximal linear complexity but is unbalanced, while the other is balanced and exhibits uniform pattern distribution for large primes.
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 $p$.