Paper 2023/1081

ARITHMETIZATION-ORIENTED APN FUNCTIONS

Abstract

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 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 prime field $\mathbb{F}_p$. Moreover, we present a new class of APN binomials over $\mathbb{F}_q$ obtained by modifying the planar function $x^2$ over $\mathbb{F}_q$. We also present a class of binomials having differential uniformity at most $5$ defined via the quadratic character over finite fields of odd characteristic. We give sufficient conditions for which this family of binomials is permutation. Computationally it is confirmed that the latter family contains new APN functions for some small parameters. We conjecture it to contain an infinite subfamily of APN functions.