Functions which are PN on infiitely many extensions of Fp, p odd

2010/06/14 by Elodie Leducq, Leducq, Elodie · 2 citations
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Number Theory (math.NT) #cs.IT #math.IT #math.NT

paper · pdf · doi:10.48550/arxiv.1006.2610

openalex publication_date 2010/06/14 · arxiv created 2012/05/03 · arxiv updated 2012/05/04 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let p be an odd prime number. We prove that for m≡1\mod p, x^m is perfectly nonlinear over \mathbbF_pⁿ for infinitely many n if and only if m is of the form p^l+1, l∈ℕ. First, we study singularities of f(x,y)=((x+1)^m-x^m-(y+1)^m+y^m)/(x-y) and we use Bezout theorem to show that for m≠ 1+p^l, f(x,y) has an absolutely irreducible factor. Then by Weil theorem, f(x,y) has rationnal points such that x≠ y which means that x^m is not PN.

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