Involutary pemutations over finite fields given by trinomials and quadrinomials

For all finite fields of $q$ elements where $q\equiv1\pmod4$ we have constructed permutation polynomials which have order 2 as permutations, and have 3 terms, or 4 terms as polynomials. Explicit formulas for their coefficients are given in terms of the primitive elements of the field. We also give polynomials providing involutions with larger number of terms but coefficients will be conveniently only two possible values. Our procedure gives at least $(q-1)/4$ trinomials, and $(q-1)/2$ quadrinomials, all yielding involutions with unique fixed points over a field of order $q$. Equal number of involutions with exactly $(q+1)/2$ fixed-points are provided as quadrinomials.Comment: 10 pages; comments welccom

Modifications of Bijective S-Boxes with Linear Structures

Various systematic modifications of vectorial Boolean functions have been used for finding new previously unknown classes of S-boxes with good or even optimal differential uniformity and nonlinearity. In this paper, a new general modification method is given that preserves the bijectivity property of the function in case the inverse of the function admits a linear structure. A previously known construction of such a modification based on bijective Gold functions in odd dimension is a special case of the new method

A note on constructions of bent functions from involutions

Bent functions are maximally nonlinear Boolean functions. They are important functions introduced by Rothaus and studied rstly by Dillon and next by many researchers for four decades. Since the complete classication of bent functions seems elusive, many researchers turn to design constructions of bent functions. In this note, we show that linear involutions (which are an important class of permutations) over nite elds give rise to bent functions in bivariate representations. In particular, we exhibit new constructions of bent functions involving binomial linear involutions whose dual functions are directly obtained without computation

On involutions of finite fields

International audienceIn this paper we study involutions over a finite field of order $\bf 2^n$. We present some classes, several constructions of involutions andwe study the set of their fixed points