The number of almost perfect nonlinear functions grows exponentially

TL;DR

This paper demonstrates that the number of almost perfect nonlinear (APN) functions, crucial for cryptography, grows exponentially, significantly improving known lower bounds and analyzing their automorphism groups.

Contribution

It provides a new exponential lower bound on the number of inequivalent APN functions using Taniguchi's construction and determines their automorphism group.

Findings

Exponential growth of inequivalent APN functions.

Improved lower bounds for even and odd m.

Automorphism group characterization of Taniguchi's APN functions.

Abstract

Almost perfect nonlinear (APN) functions play an important role in the design of block ciphers as they offer the strongest resistance against differential cryptanalysis. Despite more than 25 years of research, only a limited number of APN functions are known. In this paper, we show that a recent construction by Taniguchi provides at least $\frac{\varphi(m)}{2}\left\lceil \frac{2^m+1}{3m} \right\rceil$ inequivalent APN functions on the finite field with ${2^{2m}}$ elements, where $\varphi$ denotes Euler's totient function. This is a great improvement of previous results: for even $m$, the best known lower bound has been $\frac{\varphi(m)}{2}\left(\lfloor \frac{m}{4}\rfloor +1\right)$, for odd $m$, there has been no such lower bound at all. Moreover, we determine the automorphism group of Taniguchi's APN functions.