Normality of 8-Bit Bent Function
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918005461483520 |
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| author | Gillot, Valérie Langevin, Philippe Polujan, Alexandr |
| author_facet | Gillot, Valérie Langevin, Philippe Polujan, Alexandr |
| contents | 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) |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21779 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normality of 8-Bit Bent Function Gillot, Valérie Langevin, Philippe Polujan, Alexandr Discrete Mathematics 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) |
| title | Normality of 8-Bit Bent Function |
| topic | Discrete Mathematics |
| url | https://arxiv.org/abs/2504.21779 |