Constructions of p‐variable 1‐resilient rotation symmetric functions over GF(p)

Abstract

Rotation symmetric Boolean functions have been extensively studied in the recent years because of their applications in cryptography. In this study, a novel method to construct p-variable 1-resilient rotation symmetric functions over GF(p) is proposed based on a Latin square with maximum cycle structure, which is not required to solve any equation system. And a lower bound on the number of p-variable 1-resilient rotation symmetric functions is given. At last, an equivalent characterization of p-variable 1-resilient rotation symmetric functions over GF(p) is demonstrated, as a direct corollary, the number of p-variable 1-resilient rotation symmetric functions is represented by all the solutions of the equation system.

Constructing 1-resilient rotation symmetric functions over <inline-formula><tex-math id="M1">\begin{document}$ {\mathbb F}_{p} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ {q} $\end{document}</tex-math></inline-formula> variables through special orthogonal arrays

“…A set S of vectors can be seen as a matrix whose row vectors are the elements of S, without consideration of the order of these elements. f −1 (l) is also called the l-value support table of f (x) [11].…”

“…Definition 1 [11,12,41] An w × n matrix C whose entries are from F p is called an orthogonal arrays of strength d, denoted by OA(w, n, p, d) for simplicity, if each vector in F d p occurs the same number of times in the sub-matrix of C composed of arbitrary d columns of C. [41,22,30] If f (x) ∈ B n,p , then f (x) is called a t-correlation immune function over F p if and only if all the vectors in f −1 (l) make up an OA(|f…”

“…Bent functions, perfect nonlinear functions and almost perfect nonlinear functions are good examples. In recent years, rotation symmetric functions over F p (RSFs for simplicity) have received more attention [7,28,29,27,18,19,11,12]. Symmetric functions are a special class of RSFs.…”

“…The authors also presented a lower bound on the number of balanced symmetric functions over F p and an equivalent characterization was proposed [27]. The enumeration of RSFs over F p was also investigated in [28,19,11,12]. New results on the resilient functions were also given in [45,13] recently.…”

“…Throughout this paper, we assume that p and q are different odd prime numbers. Motivated by [7,28,11,12], we devote to constructing a class of q-variable 1-resilient RSFs over F p in this study.…”