Millions of inequivalent quadratic APN functions in eight variables

August 06, 2025 · The Ethereal · 🏛 arXiv.org

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Authors Christof Beierle, Philippe Langevin, Gregor Leander, Alexandr Polujan, Shahram Rasoolzadeh arXiv ID 2508.04644 Category math.CO: Combinatorics Cross-listed cs.CR, cs.DM, cs.IT Citations 0 Venue arXiv.org Last Checked 3 months ago

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.