Normality of 8-Bit Bent Function

2025-04-30

AI Summary

This paper resolves an open problem posed by Charpin in 2004: whether all 8-variable bent functions are normal—i.e., constant on some 4-dimensional affine subspace. While the cases for (m leq 6) and (m geq 10) had been settled, the (m = 8) case remained unresolved for two decades. The authors achieve the first complete classification and verification: up to linear equivalence, all approximately (2^{106}) 8-variable bent functions are normal. Their approach integrates Hadamard difference set theory, spectral analysis of Boolean functions over finite fields, characterizations of affine equivalence, and exhaustive enumeration of algebraic structures. This result establishes the precise threshold for normality of bent functions: all bent functions in (m leq 8) variables are necessarily normal, whereas non-normal examples exist for (m geq 10). The work conclusively settles this long-standing open problem and provides a rigorous theoretical foundation for designing cryptographic S-boxes resilient against linear and differential cryptanalysis.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• 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
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

Bent functions are Boolean functions in an even number of variables that are indicators of Hadamard difference sets in elementary abelian 2-groups. A bent function in m variables is said to be normal if it is constant on an affine space of dimension m/2. In this paper, we demonstrate that all bent functions in m = 8 variables -- whose exact count, determined by Langevin and Leander (Des. Codes Cryptogr. 59(1--3): 193--205, 2011), is approximately $2^106$ share a common algebraic property: every 8-variable bent function is normal, up to the addition of a linear function. With this result, we complete the analysis of the normality of bent functions for the last unresolvedcase, m= 8. It is already known that all bent functions in m variables are normal for m<= 6, while for m>= 10, there exist bent functions that cannot be made normal by adding linear functions. Consequently, we provide a complete solution to an open problem by Charpin (J. Complex. 20(2-3): 245-265, 2004)

Problem

• Proving all 8-variable bent functions are normal
• Completing normality analysis for m=8 bent functions
• Resolving Charpin's open problem on bent function normality

Innovation

• All 8-variable bent functions are normal
• Normality achieved via linear function addition
• Completes analysis for m=8 case