Millions of inequivalent quadratic APN functions in eight variables

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 in dimension six. Motivated by this result, there have been numerous attempts to construct new quadratic APN functions. Prior to this work, $32\,892$ quadratic APN functions in dimension eight are known and two recent conjectures address their possible total number. The first, proposed by Y. Yu and L. Perrin, suggests that there are more than $50\,000$ such functions. The second, by A. Polujan and A. Pott, 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 eight and estimate the total number to be about six million.