论文标题
$（Q，Q）$ - 双向APN功能的分类
Classification of $(q,q)$-biprojective APN functions
论文作者
论文摘要
在本文中，我们将$（q，q）$ - 双向主题分类几乎是完美的非线性（APN）功能，其功能超过$ \ mathbb {ll} \ times \ times \ mathbb {ll} $在$ \ mathrm {glm {gl}的自然左右动作下$ 2 $。这特别表明，唯一在$ \ MathBb {ll} \ times \ times \ Mathbb {ll} $上唯一的二次APN函数（最高CCZ-等效性）满足所谓的子场属性是Gold函数，是函数$κ$κ：是唯一已知的APN函数，等同于$ \ Mathbb {ll} \ Times \ Mathbb {ll} $上的排列函数与CCZ-Equivalence相当。 $κ$ - 功能是在（Browning，Dillon，McQuistan和Wolfe，2010年）中引入的。确定是否存在其他二次APN函数（可能与排列相同的CCZ等效），使子场属性或等效地将$κ$概括为更高维度是（Carlet，2015年）中列出的一个开放问题，因为它是密码功能上有趣的开放问题之一。
In this paper, we classify $(q,q)$-biprojective almost perfect nonlinear (APN) functions over $\mathbb{LL} \times \mathbb{LL}$ under the natural left and right action of $\mathrm{GL}(2,\mathbb{LL})$ where $\mathbb{LL}$ is a finite field of characteristic $2$. This shows in particular that the only quadratic APN functions (up to CCZ-equivalence) over $\mathbb{LL} \times \mathbb{LL}$ that satisfy the so-called subfield property are the Gold functions and the function $κ: \mathbb{F}_{64} \to \mathbb{F}_{64}$ which is the only known APN function that is equivalent to a permutation over $\mathbb{LL} \times \mathbb{LL}$ up to CCZ-equivalence. The $κ$-function was introduced in (Browning, Dillon, McQuistan, and Wolfe, 2010). Deciding whether there exist other quadratic APN functions (possibly CCZ-equivalent to permutations) that satisfy subfield property or equivalently, generalizing $κ$ to higher dimensions was an open problem listed for instance in (Carlet, 2015) as one of the interesting open problems on cryptographic functions.
