A General Construction of Permutation Polynomials of \Bbb F_q²

2022/04/04 by Xiang‐dong Hou, Hou, Xiang-dong, Vincenzo Pallozzi Lavorante +1
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #graph theory and CDMA systems #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2204.01545

Abstract

Let r be a positive integer, h(X)∈\Bbb F_q²[X], and μ_q+1 be the subgroup of order q+1 of \Bbb F_q²^*. It is well known that X^rh(X^q-1) permutes \Bbb F_q² if and only if gcd(r,q-1)=1 and X^rh(X)^q-1 permutes μ_q+1. There are many ad hoc constructions of permutation polynomials of \Bbb F_q² of this type such that h(X)^q-1 induces monomial functions on the cosets of a subgroup of μ_q+1. We give a general construction that can generate, through an algorithm, \em all permutation polynomials of \Bbb F_q² with this property, including many which are not known previously. The construction is illustrated explicitly for permutation binomials and trinomials.

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