Millions of inequivalent quadratic APN functions in eight variables
math.CO, cs.CR, cs.DM, cs.IT, math.IT
Submitted: 2025-08-06
Updated: 2026-08-27
Comments: Links for the associated Zenodo data sets: https://doi.org/10.5281/zenodo.16752428 and https://zenodo.org/records/22010376
Code: https://github.com/lpp-crypto/sboxU
License: http://creativecommons.org/licenses/by/4.0/
The gist: 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.
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.
Sources
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