2022/07/08 by Christof Beierle, Beierle, Christof
Computer Science · Mathematics · #06E30 #11T06 #11T71 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Algebraic number #Binomial (polynomial) #Coding theory and cryptography #Combinatorics #Combinatorics (math.CO) #Computer science #Degree (music) #Discrete mathematics #FOS: Mathematics #Finite field #Function (biology) #Integer (computer science) #Mathematical analysis #Mathematics #Order (exchange) #Physics #Prime (order theory) #Statistics #Trinomial
paper · pdf · doi:10.48550/arxiv.2207.04087
openalex publication_date 2022/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let p>3 be a prime. We show that, for each integer d with p ≤ d ≤ 2(p-1), there exists a generalized almost perfect nonlinear (GAPN) binomial or trinomial over \mathbbFp2 of algebraic degree d. We start by deriving sufficient conditions for the function G \colon \mathbbFp2 → \mathbbFp2, X ↦ Xd1 + u Xd2 to be GAPN in the case where one of the terms of G is GAPN. We then give explicit constructions of GAPN binomials over \mathbbFp2 of any odd algebraic degree between p and 2(p-1) and, in the case where p is not a Mersenne prime, also of any even algebraic degree in this range. To obtain GAPN functions of even algebraic degree also in the general case, we finally show how to construct GAPN trinomials over \mathbbFp2 of any even algebraic degree between p and 2(p-1) by applying a characterization of a special form of GAPN binomials by Özbudak and Sălăgean. Our constructed functions are the first GAPN functions of even algebraic degree over extension fields of odd characteristic reported so far.