Determination of a Class of Permutation Trinomials in Characteristic Three

2018/11/29 by Xiang‐dong Hou, Hou, Xiang-dong, Ziran Tu +3 · 1 citation
Computer Science · Mathematics · #11T06 #11T55 #14H05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1811.11949

openalex publication_date 2018/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f(X)=X(1+aX^q(q-1)+bX^2(q-1))∈\Bbb F_q²[X], where a,b∈\Bbb F_q²^*. In a series of recent papers by several authors, sufficient conditions on a and b were found for f to be a permutation polynomial (PP) of \Bbb F_q² and, in characteristic 2, the sufficient conditions were shown to be necessary. In the present paper, we confirm that in characteristic 3, the sufficient conditions are also necessary. More precisely, we show that when char \Bbb F_q=3, f is a PP of \Bbb F_q² if and only if (ab)^q=a(b^q+1-a^q+1) and 1-(b/a)^q+1 is a square in \Bbb F_q^*.

Citations

• Permutation polynomials and group permutation polynomials

Cited by

• On the Tu-Zeng Permutation Trinomial of Type (1/4,3/4)

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