Work

Normality of 8-bit Bent Functions

Year: 2026

Type: article

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 functionfinmvariables is said to be normal if it is constant on an affine space of dimensionm/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 2106, 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 unresolved case,m= 8. It is already known that all bent functions inmvariables are normal form≤ 6, while form≥ 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).

Source: IEEE Transactions on Information Theory

DOI: 10.1109/tit.2026.3685370

Authors: Valérie Gillot, Philippe Langevin, Alexandr Polujan

Institutions: Université de Toulon, Institut de Mathématiques de Toulon, Otto-von-Guericke-Universität Magdeburg

Language: English

Topic: Coding theory and cryptography

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,645

Funder: Agence Nationale de la Recherche