论文标题
普遍的几乎完美的非线性二项式和三项元素优先级的阶段秩序
Generalized Almost Perfect Nonlinear Binomials and Trinomials Over Fields of Prime-Square Order
论文作者
论文摘要
令$ p> 3 $为素数。我们表明，对于带有$ p \ leq d \ leq 2（p-1）$的每个整数$ d $，存在一个概括性的非线性（GAPN）二项式或三项元素，超过$ \ MATHBB {f} _ {p} _ {p^2} $ n elgebraic degr $ d $ d $ d $ d $。我们首先为功能$ g \ colon \ mathbb {f} _ {p^2} \ rightArrow \ mathbb {f} _ {p^2}，x \ mapsto x^{d_1} + x^{d_2} $ gapn中的g $ g $ g $ g，首先为$ g \ mathbb {f} _ {p^2} \ rightArrow \ MathBb {f} _ {p^2}，x \ mapsto x^{d_1} + x^{d_1} + x^{d_2} $ gapn中的gapn in g $ gapn。然后，我们在$ \ mathbb {f} _ {p^2} $上对$ p $ p $ $ p $和$ 2（p-1）$的任何奇数代数的明确构造进行了明确的构造。在一般情况下，还要获得偶数级别的GAPN函数，我们最终展示了如何在$ p $ $ p $和$ p $和$ 2（p-1）$的任何代数学位的$ \ mathbb {f} _ {p^2} $上构建GAPN trinomials。我们构造的函数是到目前为止报道的奇数特征的扩展场上，即使是代数程度的第一个GAPN函数。
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.
