An almost perfect non-linear (APN) function over \(\mathbb {F}_{2^n}\) is called exceptional APN if it remains APN over infinitely many extensions of \(\mathbb {F}_{2^n}\). Exceptional APN functions have attracted attention of many researchers in the last decades. While the classification of exceptional APN monomials has been done by Hernando and McGuire, it has been conjectured by Aubry, McGuire and Rodier that up to equivalence, the only exceptional APN functions are the Gold and the Kasami–Welch monomial functions. Since then, many partial results have been on classifying non-exceptional APN polynomials. In this paper, for the classification of the exceptional property of APN functions, we introduce a new method that uses techniques from curves over finite fields. Then, we apply the method with Eisenstein’s irreducibility criterion and Kummer’s theorem to obtain new non-exceptional APN functions.

如果在\(\mathbb {F}_{2^n}\)上几乎完美的非线性 (APN) 函数在无限多个\(\mathbb {F}_{2^)上保持 APN，则该函数被称为异常 APN n}\)。过去几十年来，卓越的 APN 功能引起了许多研究人员的关注。虽然 Hernando 和 McGuire 对特殊 APN 单项式进行了分类，但 Aubry、McGuire 和 Rodier 推测，在等价性范围内，唯一特殊的 APN 函数是 Gold 和 Kasami-Welch 单项式函数。从那时起，许多部分结果都是关于非例外 APN 多项式的分类。在本文中，为了对 APN 函数的特殊性质进行分类，我们引入了一种使用有限域曲线技术的新方法。然后，我们应用爱森斯坦不可约性准则和库默尔定理的方法来获得新的非例外APN函数。