Constructions and Necessities of Some Permutation Polynomials over Finite Fields

Xiaogang Liu,Li Haitao

doi:10.1155/2021/9409130

Abstract

Let F q denote the finite field with q elements. Permutation polynomials over finite fields have important applications in many areas of science and engineering such as coding theory, cryptography, and combinatorial design. The study of permutation polynomials has a long history, and many results are obtained in recent years. In this paper, we obtain some further results about the permutation properties of permutation polynomials. Some new classes of permutation polynomials are constructed, and the necessities of some permutation polynomials are studied.

Highlights

For a prime power q, let F q denote the finite field of order q, and F∗ q the multiplicative group.A polynomial f(x) ∈Fq is called a permutation polynomial (PP) over Fq, if the associated polynomial mapping f: c ⟶ f(c) is a permutation of F q. ey have applications in coding theory, cryptography, and combinatorial design theory [1, 2]. us, in both theoretical and applied aspects, finding new PPs is of great interest
Orthomorphisms map each maximal subgroup of the additive group of F q half into itself and half into its complement, and they have a single fixed point and are the same as Complete permutation polynomial (CPP) in even characteristic
In the study of CPPs, Li et al, found that certain polynomials over F 2n can have the same permutation properties as axk + bx over F2m for positive divisor m of n such that n/m is odd [36]. ey constructed some permutation binomials, and more permutation polynomials of the form a[Trnm(x)]k + u(c + x)(Trnm(x) + x) + bx can be obtained, and here a, b, c, u

Summary

Introduction

Permutation polynomials of the form x + cTrnm(xk) have been studied for special c ∈ Fqn, with even characteristic [32, 33], and for the case n 2, i.e., permutation trinomials. For the fourth class of permutation trinomials, a symbolic computation method related with resultants was used They found a new relation with a class of PPs of the form. In the study of CPPs, Li et al, found that certain polynomials over F 2n can have the same permutation properties as axk + bx over F2m for positive divisor m of n such that n/m is odd [36]. Ey studied permutation polynomials over F p2m of the form axpm + bx + h(xpm ± x), and Niho-type permutation trinomials over Fp2m were constructed.

Preliminaries

Permutation Polynomials That Can be Transformed to the Case of Monomials

Construction of Permutation Polynomials with Two or More terms

Necessities of Two Kinds of Permutation Polynomials

Conclusion

Disclosure