Recently, many cryptographic primitives such as homomorphic encryption (HE), multi-party computation (MPC) and zero-knowledge (ZK) protocols have been proposed in the literature which operate on the prime field \({\mathbb {F}}_p\) for some large prime p. Primitives that are designed using such operations are called arithmetization-oriented primitives. As the concept of arithmetization-oriented primitives is new, a rigorous cryptanalysis of such primitives is yet to be done. In this paper, we investigate arithmetization-oriented APN functions. More precisely, we investigate APN permutations in the CCZ-classes of known families of APN power functions over the prime field \({\mathbb {F}}_p\). Moreover, we present a class of binomial permutation having differential uniformity at most 5 defined via the quadratic character over finite fields of odd characteristic. Computationally it is confirmed that the latter family contains new APN permutations for some small parameters. We conjecture it to contain an infinite subfamily of APN permutations.

最近，文献中提出了许多密码原语，例如同态加密（HE）、多方计算（MPC）和零知识（ZK）协议，它们在素数域上运行\({\mathbb {F}}_p \)对于一些大素数p 。使用此类操作设计的原语称为面向算术化的原语。由于面向算术化的原语的概念是新的，因此尚未对此类原语进行严格的密码分析。在本文中，我们研究了面向算术化的 APN 函数。更准确地说，我们研究素数域\({\mathbb {F}}_p\)上已知 APN 幂函数族的 CCZ 类中的 APN 排列。此外，我们提出了一类二项式排列，其微分均匀性最多为 5，通过奇数特征的有限域上的二次特征定义。计算证实后一个家族包含一些小参数的新 APN 排列。我们推测它包含 APN 排列的无限子族。