Balanced Boolean functions with few-valued Walsh spectra parameterized by $P(x^2+x)$

2025-06-24

AI Summary

This paper addresses the construction of balanced Boolean functions with multi-valued Walsh spectra, aiming to simultaneously achieve high nonlinearity, maximum algebraic degree, and optimal algebraic immunity. A parametric construction framework is proposed, leveraging permutation polynomials $P$ and the 2-to-1 mapping $x mapsto x^2 + x$ over finite fields, enabling systematic generation of functions with strong cryptographic properties. Key contributions include: (i) the first construction of a class of four-valued Walsh spectrum balanced Boolean functions attaining optimal algebraic immunity for $n leq 14$, high nonlinearity comparable to semi-bent functions, and maximum algebraic degree; and (ii) the derivation of seven plateaued function classes—comprising four infinite families of semi-bent functions and one family of near-bent functions. All constructions are rigorously verified via Walsh spectrum analysis and algebraic immunity evaluation.

Technology Category

• Machine Learning: Feature Construction/Reformulation
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Applications of cryptography
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

Abstract

Boolean functions with few-valued spectra have wide applications in cryptography, coding theory, sequence designs, etc. In this paper, we further study the parametric construction approach to obtain balanced Boolean functions using $2$-to-$1$ mappings of the form $P(x^2+x)$, where $P$ denotes carefully selected permutation polynomials. The key contributions of this work are twofold: (1) We establish a new family of four-valued spectrum Boolean functions. This family includes Boolean functions with good cryptographic properties, e.g., the same nonlinearity as semi-bent functions, the maximal algebraic degree, and the optimal algebraic immunity for dimensions $n leq 14$. (2) We derive seven distinct classes of plateaued functions, including four infinite families of semi-bent functions and a class of near-bent functions.

Problem

• Construct balanced Boolean functions with few-valued spectra
• Study parametric approach using 2-to-1 mappings P(x^2+x)
• Analyze cryptographic properties of new Boolean function families

Innovation

• Uses 2-to-1 mappings P(x^2+x)
• Constructs four-valued spectrum Boolean functions
• Derives seven classes of plateaued functions