Normality of 8-Bit Bent Function
Valérie Gillot, Philippe Langevin, Alexandr Polujan
TL;DR
The paper addresses whether bent functions on $\mathbb{F}_2^m$ are normal or weakly normal in the critical case $m=8$, resolving Charpin's open problem by proving all $8$-variable bent functions are normal or weakly normal (up to affine adjustments). It introduces the relative degree $\deg_{a+V}(f)$ as a generalization of normality and develops a computational framework combining a sieve for abnormality with bent expansions of near-bent halves via duals. Key contributions include determining $D^ abla_r(k,m)$ for $m=5,6,7$, establishing the existence of abnormal 7-bit quartics, and, crucially, demonstrating that all $8$-bit bent functions arise from expansions that are normal or weakly normal, after enumerating about $2^{24}$ cases. The results provide a unified algebraic property for all $8$-bit bent functions and introduce methods (relative degree, bent-expansion) that may extend to higher even dimensions, with open questions remaining for $m\ge 10$ and for specific quartic classes. The work advances the understanding of the structure of bent functions and offers computational tools for exploring normality without full classifications at higher dimensions.
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)