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On piece-wise permutation polynomials Fangmin Zhou1 © Instituto de Matemática e Estatística da Universidade de São Paulo 2020

Abstract In this paper we discuss piece-wise permutation polynomials (PP). The method is by combining the AGW lemma and module structure, and we get a generalized five lemma. Then we apply the results to additive and multiplicative structures of finite fields. This gives a unified treatment and a framework of extensive existing PPs in the literature. More precisely, we deal with PPs of the following forms q n −1

(1) [Multiplicative Structure] x a u(x d ), where a ∈ N, d|q n − 1, u(x) ∈ Fq n [x]. (2) [Additive Structure] L 1 + u(L 2 + δ), where L 1 , L 2 are linearized polynomials, u(x) ∈ Fq n [x], δ ∈ Fq n . If L 2 is the trace function Tr(x), this concerns both additive and multiplicative structures. Keywords Permutation polynomials · Finite field · Five lemma Mathematics Subject Classification 05A05 · 11T06 · 11T55

1 Introduction Let Fq denote the finite field of characteristic p with q elements, and Fq∗ be the set of nonzero elements of Fq . A permutation polynomial (PP) over Fq is a polynomial over Fq that induces a permutation on Fq with its evaluation on the field. A linearized n−1 i polynomial or q-polynomial [1] over Fq n has the form L(x) = i=0 ai x q ∈ Fq n [x]. Since L(x) is Fq -linear, it is easily seen that L(x) permutes Fq n if and only if the only root of L(x) in Fq n is the zero element. Permutation polynomials over finite fields have been an interesting subject of study for many years, and have applications in coding theory, cryptography, combinatorial design theory, and many other areas of mathe-

Communicated by Sergio R. López-Permouth.

B 1

Fangmin Zhou [email protected] School of Mathematics and Statistics, Zhaoqing University, Zhaoqing 526061, Guangdong, China

123

São Paulo Journal of Mathematical Sciences

matics and engineering. For properties and applications of permutation polynomials and linearized polynomials, please see [1]. For the convenient applications of PP, people are interested only in PPs of nice form, for example, sparse PPs or PPs having good algebraic structures. Since there are two algebraic structures in finite fields, that is, additive structure and multiplicative structure, there are two kinds of interesting PPs. PPs having additive stuctures often have the form L 1 (x) + u(L 2 (x)), where L 1 (x) and L 2 (x) are linearized polynomials. PPs having multiplicative stuctures often have the form x a H (x (q−1)/d ), where d|q − 1. Both kinds have been studied intensively in recent years, [2–7]. Li et al. k [8] presented some new permutation polynomial of the form (x p − x + δ)s + L(x) over Fq n , where L(x) is a linearized polynomial. Fernando and Hou [9] provided a piecewise construction of permutation polynomials over finite fields. Wang [10] used cyclotomy to construct new classes of permutation polynomials of large indices. For a comprehensive survey of present progress, please refer to [11] and the references therein. There are vast piecewise