Permutation Polynomials of \Bbb F_q² of the form a\tt X+\tt X^r(q-1)+1

2015/10/01 by Xiang‐dong Hou, Xiang-dong Hou, Hou, Xiang-dong
Computer Science · Engineering · Mathematics · #11T06 #11T55 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #graph theory and CDMA systems #math.CO #math.NT #msc:11T06 #msc:11T55

paper · pdf · doi:10.48550/arxiv.1510.00437

24 pages

arxiv created 2015/10/01 · openalex publication_date 2015/10/01 · arxiv updated 2015/10/05 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let q be a prime power, 2≤ r≤ q, and f=a\tt X+\tt X^r(q-1)+1∈\Bbb F_q²[\tt X], where a≠ 0. The conditions on r,q,a that are necessary and sufficient for f to be a permutation polynomial (PP) of \Bbb F_q² are not known. (Such conditions are known under an additional assumption that a^q+1=1.) In this paper, we prove the following: (i) If f is a PP of \Bbb F_q², then gcd(r,q+1)>1 and (-a)^{(q+1)/gcd(r,q+1)}≠ 1. (ii) For a fixed r>2 and subject to the conditions that q+1≡ 0\pmod r and a^q+1≠ 1, there are only finitely many (q,a) for which f is a PP of \Bbb F_q². Combining (i) and (ii) confirms a recent conjecture regarding the type of permutation binomial considered here.

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