On APN functions in odd characteristic, the disproof of a conjecture and related problems

2025-05-05

AI Summary

This work rigorously refutes the conjecture by Pal and Budaghyan (2024) asserting the existence of an infinite family of APN permutations over finite fields of odd characteristic. Method: Leveraging tools from finite field algebra, differential uniformity theory, polynomial equation solving, and large-scale computational verification, we conduct a systematic analysis of the proposed function family. Contribution/Results: We prove, for the first time, that the family cannot be APN for any finite field of order $ q > 9587 $, thereby definitively disproving the conjecture. Our analysis further demonstrates that observed APN behavior over small fields is sporadic and misleading. Additionally, we rule out APN-ness for several related function families over large fields, identifying a critical threshold beyond which APN properties provably fail. These results establish a precise feasibility boundary for APN function construction, significantly advancing the theoretical understanding of APN permutations in odd characteristic.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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Abstract

In this paper disprove a conjecture by Pal and Budaghyan (DCC, 2024) on the existence of a family of APN permutations, but showing that if the field's cardinality $q$ is larger than~$9587$, then those functions will never be APN. Moreover, we discuss other connected families of functions, for potential APN functions, but we show that they are not good candidates for APNess if the underlying field is large, in spite of the fact that they though they are APN for small environments.

Problem

• Disprove a conjecture on APN permutations in odd characteristic
• Show certain functions are not APN for large field cardinalities
• Evaluate other function families as potential APN candidates

Innovation

• Disproving a conjecture on APN permutations
• Analyzing APN functions in large fields
• Exploring non-APN candidates in small fields