Millions of inequivalent quadratic APN functions in eight variables

TL;DR

This paper computationally constructs over 3.7 million inequivalent quadratic APN functions in eight variables, providing evidence that their total number exceeds six million, significantly advancing understanding of their diversity.

Contribution

The authors computationally enumerate a vast number of inequivalent quadratic APN functions in dimension 8, supporting conjectures about their total quantity and diversity.

Findings

Constructed 3,775,599 inequivalent quadratic APN functions

Estimated total number of such functions to be about 6 million

Supported conjectures on the abundance of quadratic APN functions

Abstract

The only known example of an almost perfect nonlinear (APN) permutation in even dimension was obtained by applying CCZ-equivalence to a specific quadratic APN function. Motivated by this result, there have been numerous recent attempts to construct new quadratic APN functions. Currently, 32,892 quadratic APN functions in dimension 8 are known and two recent conjectures address their possible total number. The first, proposed by Y. Yu and L. Perrin (Cryptogr. Commun. 14(6): 1359-1369, 2022), suggests that there are more than 50,000 such functions. The second, by A. Polujan and A. Pott (Proc. 7th Int. Workshop on Boolean Functions and Their Applications, 2022), argues that their number exceeds that of inequivalent quadratic (8,4)-bent functions, which is 92,515. We computationally construct 3,775,599 inequivalent quadratic APN functions in dimension 8 and estimate the total number to be about 6 million.