On a Type of Permutation Rational Functions over Finite Fields

2019/10/26 by Xiang‐dong Hou, Hou, Xiang-dong, Christopher Sze +1
Computer Science · Mathematics · #11R58 #11T06 #12E12 #14H05 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1910.11989

openalex publication_date 2019/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be a prime and n be a positive integer. Let f_b(X)=X+(X^p-X+b)^-1, where b∈\Bbb F_pⁿ is such that Tr_pⁿ/p(b)≠ 0. In 2008, Yuan et al. \citeYuan-Ding-Wang-Pieprzyk-FFA-2008 showed that for p=2,3, f_b permutes \Bbb F_pⁿ for all n≥ 1. Using the Hasse-Weil bound, we show that when p>3 and n≥ 5, f does not permute \Bbb F_pⁿ. For p>3 and n=2, we prove that f_b permutes \Bbb F_p² if and only if Tr_p²/p(b)=± 1. We conjecture that for p>3 and n=3,4, f_b does not permute \Bbb F_pⁿ.

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