More infinite classes of APN-like Power Functions

TL;DR

This paper introduces new infinite classes of APN-like power functions that are not APN, including locally-APN functions with optimal boomerang uniformity and 0-APN functions, expanding the understanding of cryptographic function classes.

Contribution

The paper constructs the first known infinite classes of locally-APN but not APN functions in over a decade and introduces seven new infinite classes of 0-APN functions.

Findings

Two infinite classes of locally-APN but not APN functions over GF(2^{2m})

The class F_1 has boomerang uniformity 2 and differential uniformity greater than boomerang uniformity

Seven new infinite classes of 0-APN but not APN functions

Abstract

In the literature, there are many APN-like functions that generalize the APN properties or are similar to APN functions, e.g. locally-APN functions, 0-APN functions or those with boomerang uniformity 2. In this paper, we study the problem of constructing infinite classes of APN-like but not APN power functions. For one thing, we find two infinite classes of locally-APN but not APN power functions over $\gf_{2^{2m}}$ with $m$ even, i.e., $\mathcal{F}_1(x)=x^{j(2^m-1)}$ with $\gcd(j,2^m+1)=1$ and $\mathcal{F}_2(x)=x^{j(2^m-1)+1}$ with $j = \frac{2^m+2}{3}$. As far as the authors know, our infinite classes of locally-APN but not APN functions are the only two discovered in the last eleven years. Moreover, we also prove that this infinite class $\mathcal{F}_1$ is not only with the optimal boomerang uniformity $2$, but also has an interesting property that its differential uniformity is strictly greater than its boomerang uniformity. For another thing, using the multivariate method, including the above infinite class $\mathcal{F}_1$, we construct seven new infinite classes of 0-APN but not APN power functions.