Millions of inequivalent quadratic APN functions in eight variables

2025-08-06

AI Summary

This work addresses the exact enumeration of quadratic APN functions in dimension eight. Resolving prior conflicting conjectures—suggesting either 50,000 or over 92,515 such functions—the authors introduce a large-scale parallel search framework integrating CCZ-equivalence class enumeration with computational algebra. For the first time, they systematically construct and rigorously classify 3,775,599 pairwise CCZ-inequivalent quadratic APN functions in dimension eight, leading to an estimated total count of approximately six million—orders of magnitude larger than previous conjectures. This result not only sets a new record for the number of known quadratic APN functions but also establishes a novel paradigm for efficient high-dimensional APN function construction and equivalence testing. It provides the richest known library of candidate functions for designing even-dimensional APN permutations and optimizing cryptographic S-boxes.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

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.

Problem

• Constructing new quadratic APN functions in dimension 8
• Estimating total number of inequivalent quadratic APN functions
• Addressing conjectures on APN function counts in even dimension

Innovation

• Construct millions of quadratic APN functions
• Use computational methods for estimation
• Focus on eight-variable functions