Generalized Almost Perfect Nonlinear Binomials and Trinomials Over Fields of Prime-Square Order

TL;DR

This paper constructs new generalized almost perfect nonlinear (GAPN) binomials and trinomials over finite fields of prime-square order, covering all degrees between p and 2(p-1), including the first known even-degree GAPN functions in such fields.

Contribution

It provides explicit constructions of GAPN binomials and trinomials over _{p^2} for all degrees in a specified range, including even degrees, expanding the known classes of GAPN functions.

Findings

Constructed GAPN binomials of all odd degrees between p and 2(p-1).

Constructed GAPN trinomials of all even degrees in the same range.

First known GAPN functions of even algebraic degree over extension fields of odd characteristic.

Abstract

Let $p>3$ be a prime. We show that, for each integer $d$ with $p \leq d \leq 2(p-1)$, there exists a generalized almost perfect nonlinear (GAPN) binomial or trinomial over $\mathbb{F}_{p^2}$ of algebraic degree $d$. We start by deriving sufficient conditions for the function $G \colon \mathbb{F}_{p^2} \rightarrow \mathbb{F}_{p^2}, X \mapsto X^{d_1} + u X^{d_2}$ to be GAPN in the case where one of the terms of $G$ is GAPN. We then give explicit constructions of GAPN binomials over $\mathbb{F}_{p^2}$ 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 $\mathbb{F}_{p^2}$ of any even algebraic degree between $p$ and $2(p-1)$ by applying a characterization of a special form of GAPN binomials by \"{O}zbudak and S\u{a}l\u{a}gean. Our constructed functions are the first GAPN functions of even algebraic degree over extension fields of odd characteristic reported so far.