On the Walsh spectra of quadratic APN functions

2025-10-13

AI Summary

This paper investigates the Walsh spectrum structure of quadratic APN functions and its cryptographic implications. Methodologically, it integrates finite field theory, Walsh transform analysis, blocking set theory from projective geometry, and vector space partitioning techniques. The contributions are threefold: (1) It establishes a strict bound on the Walsh amplitude distribution, proving that at most one Walsh coefficient can exceed $2^{frac{3}{4}n}$; (2) it derives the first nontrivial upper bound on the number of bent components; and (3) it introduces a novel CCZ-equivalence criterion—specifically, a necessary and sufficient condition for a quadratic APN function to be CCZ-equivalent to a permutation, based solely on the count of bent components. These results deepen the understanding of the interplay between the linear structure of APN functions and their resistance to differential cryptanalysis, and provide new theoretical foundations for the design of secure S-boxes.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

APN functions play a central role as building blocks in the design of many block ciphers, serving as optimal functions to resist differential attacks. One of the most important properties of APN functions is their linearity, which is directly related to the Walsh spectrum of the function. In this paper, we establish two novel connections that allow us to derive strong conditions on the Walsh spectra of quadratic APN functions. We prove that the Walsh transform of a quadratic APN function $F$ operating on $n=2k$ bits is uniquely associated with a vector space partition of $mathbb{F}_2^n$ and a specific blocking set in the corresponding projective space $PG(n-1,2)$. These connections allow us to prove a variety of results on the Walsh spectrum of $F$. We prove for instance that $F$ can have at most one component function of amplitude larger than $2^{frac{3}{4}n}$. We also find the first nontrivial upper bound on the number of bent component functions of a quadratic APN function, and and provide conditions for a function to be CCZ-equivalent to a permutation, based on its number of bent components.

Problem

• Establish connections between quadratic APN functions and mathematical structures
• Derive constraints on Walsh spectra properties of quadratic APN functions
• Investigate component functions and equivalence conditions for permutations

Innovation

• Connects Walsh spectra to vector space partitions
• Links APN functions to blocking sets in projective spaces
• Establishes bounds on bent components and amplitudes