素数平方阶域上的广义几乎完美非线性二项式与三项式
Generalized Almost Perfect Nonlinear Binomials and Trinomials Over Fields of Prime-Square Order
AI总结：
该研究针对素数$p>3$的$\mathbb{F}_{p^2}$域，推导GAPN二项式的充分条件，显式构造出$p$到$2(p-1)$次的奇次GAPN二项式，结合已有刻画构造出对应范围的偶次GAPN三项式，填补了奇特征扩域偶次GAPN函数的空白。
摘要：
设$p>3$为素数。我们证明，对于每个满足$p \leq d \leq 2(p-1)$的整数$d$，有限域$\mathbb{F}_{p^2}$上存在代数次数为$d$的广义几乎完美非线性（generalized almost perfect nonlinear，简称GAPN）二项式或三项式。我们首先推导了函数$G \colon \mathbb{F}_{p^2} \rightarrow \mathbb{F}_{p^2}, X \mapsto X^{d_1} + u X^{d_2}$为GAPN函数的充分条件，其中$G$的其中一项本身是GAPN函数。随后，我们给出了$\mathbb{F}_{p^2}$上代数次数介于$p$和$2(p-1)$之间任意奇数次的GAPN二项式的显式构造；当$p$不是梅森素数（Mersenne prime）时，也构造了该范围内任意偶数次的GAPN二项式。为了在一般情形下也能得到偶数次代数次数的GAPN函数，我们最后借助Özbudak和Sălăgean提出的一类特殊形式GAPN二项式的刻画方法，展示了如何构造$\mathbb{F}_{p^2}$上代数次数介于$p$和$2(p-1)$之间任意偶数次的GAPN三项式。我们构造的函数是目前已报道的首批奇特征扩域上的偶数次代数次数GAPN函数。
英文摘要：
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.
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