Generalized Almost Perfect Nonlinear Binomials and Trinomials Over Fields of Prime-Square Order
Christof Beierle
TL;DR
This work addresses the existence of GAPN functions over extension fields of odd characteristic with even and odd algebraic degrees within $[p,2(p-1)]$ for odd primes $p>3$. It develops a framework to construct GAPN binomials over $\mathbb{F}_{p^2}$ by combining a GAPN monomial with a second term, establishing sufficient conditions when $d_2$ is odd or even and exploiting a key identity for $D_1^{(p-1)}M_d(X)$. Building on this, it provides explicit families of GAPN binomials of all odd degrees in the range and, for non-Mersenne $p$, GAPN binomials of even degrees; it further uses a characterization of gapn binomials of the same degree to produce GAPN trinomials of any even degree in the range. The results yield the first known GAPN functions of even algebraic degree over extension fields of odd characteristic, expanding the catalog of GAPN functions and contributing to their cryptographic and geometric applications.
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 Özbudak and Sălăgean. Our constructed functions are the first GAPN functions of even algebraic degree over extension fields of odd characteristic reported so far.