The differential properties of certain permutation polynomials over finite fields

Finding functions, particularly permutations, with good differential properties has received a lot of attention due to their possible applications. For instance, in combinatorial design theory, a correspondence of perfect $c$-nonlinear functions and difference sets in some quasigroups was recently shown [1]. Additionally, in a recent manuscript by Pal and Stanica [20], a very interesting connection between the $c$-differential uniformity and boomerang uniformity when $c=-1$ was pointed out, showing that that they are the same for an odd APN permutations. This makes the construction of functions with low $c$-differential uniformity an intriguing problem. We investigate the $c$-differential uniformity of some classes of permutation polynomials. As a result, we add four more classes of permutation polynomials to the family of functions that only contains a few (non-trivial) perfect $c$-nonlinear functions over finite fields of even characteristic. Moreover, we include a class of permutation polynomials with low $c$-differential uniformity over the field of characteristic~$3$. As a byproduct, our proofs shows the permutation property of these classes. To solve the involved equations over finite fields, we use various techniques, in particular, we find explicitly many Walsh transform coefficients and Weil sums that may be of an independent interest

A Recursive Construction of Permutation Polynomials over Fq2\mathbb{F}_{q^2} with Odd Characteristic from R\'{e}dei Functions

In this paper, we construct two classes of permutation polynomials over $\mathbb{F}_{q^2}$ with odd characteristic from rational R\'{e}dei functions. A complete characterization of their compositional inverses is also given. These permutation polynomials can be generated recursively. As a consequence, we can generate recursively permutation polynomials with arbitrary number of terms. More importantly, the conditions of these polynomials being permutations are very easy to characterize. For wide applications in practice, several classes of permutation binomials and trinomials are given. With the help of a computer, we find that the number of permutation polynomials of these types is very large

Permutation polynomials of degree 8 over finite fields of odd characteristic

This paper provides an algorithmic generalization of Dickson's method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson's idea is to formulate from Hermite's criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree $d$. Previous classifications of PPs of degree at most $6$ were essentially deduced from manual analysis of these polynomial equations. However, these polynomials, needed for that purpose when $d>6$, are too complicated to solve. Our idea is to make them more solvable by calculating some radicals of ideals generated by them, implemented by a computer algebra system (CAS). Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree $8$ over an arbitrary finite field of odd order $q>8$. The main result is that for an odd prime power $q>8$, a PP $f$ of degree $8$ exists over the finite field of order $q$ if and only if $q\leqslant 31$ and $q\not\equiv 1\ (\mathrm{mod}\ 8)$, and $f$ is explicitly listed up to linear transformations.Comment: 15 page