The combinatorial structure and value distributions of plateaued functions

2024-10-01

arXiv.org

AI Summary

This study systematically characterizes the combinatorial structure and value distribution of plateaued functions, focusing on deep interrelations among their Walsh spectra, nonlinearity, differential uniformity, and image/preimage set sizes. Methodologically, it integrates Walsh analysis, finite field function theory, combinatorial counting, and algebraic geometry techniques. Key contributions include: (i) establishing necessary and sufficient existence criteria for d-to-1 plateaued functions within the “almost balanced” class—those admitting only two distinct nonzero preimage sizes; (ii) revealing an exact correspondence between the Walsh spectrum and value distribution of plateaued APN functions; (iii) deriving closed-form expressions for nonlinearity and tight upper bounds on differential uniformity; (iv) proving that plateaued monomials exist only for a very restricted set of exponents d; and (v) providing a complete classification of preimage size patterns for almost balanced plateaued functions. These results advance the structural understanding of plateaued functions and resolve several open problems in Boolean and vectorial function theory.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Structured Learning

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study combinatorial properties of plateaued functions. All quadratic functions, bent functions and most known APN functions are plateaued, so many cryptographic primitives rely on plateaued functions as building blocks. The main focus of our study is the interplay of the Walsh transform and linearity of a plateaued function, its differential properties, and their value distributions, i.e., the sizes of image and preimage sets. In particular, we study the special case of ``almost balanced'' plateaued functions, which only have two nonzero preimage set sizes, generalizing for instance all monomial functions. We achieve several direct connections and (non)existence conditions for these functions, showing for instance that plateaued $d$-to-$1$ functions (and thus plateaued monomials) only exist for a very select choice of $d$, and we derive for all these functions their linearity as well as bounds on their differential uniformity. We also specifically study the Walsh transform of plateaued APN functions and their relation to their value distribution.

Problem

• Study combinatorial properties of plateaued functions in cryptography
• Analyze Walsh transform and linearity of plateaued functions
• Investigate value distributions and differential properties of plateaued functions

Innovation

• Analyzes plateaued functions' Walsh transform and linearity
• Studies differential properties and value distributions
• Examines almost balanced plateaued functions' existence