Normality of 8-bit Bent Functions
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).
Related
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BibTeX
@article{gillot2026normality,
title = {{Normality of 8-bit Bent Functions}},
author = {Valérie Gillot and Philippe Langevin and Alexandr Polujan},
journal = {IEEE Transactions on Information Theory},
year = {2026},
month = oct,
doi = {10.1109/tit.2026.3685370},
url = {https://hal.science/hal-05774356v1},
note = {CC BY},
}
RIS
TY - JOUR
TI - Normality of 8-bit Bent Functions
AU - Valérie Gillot
AU - Philippe Langevin
AU - Alexandr Polujan
PY - 2026
DA - 2026/10/01
JO - IEEE Transactions on Information Theory
DO - 10.1109/tit.2026.3685370
UR - https://hal.science/hal-05774356v1
AB - 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).
SN - 0018-9448
ER -