Balanced Boolean functions with few-valued Walsh spectra parameterized by $P(x^2+x)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912446941233152 |
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| author | Zhang, Qiancheng Li, Kangquan Qu, Longjiang |
| author_facet | Zhang, Qiancheng Li, Kangquan Qu, Longjiang |
| contents | Boolean functions with few-valued spectra have wide applications in cryptography, coding theory, sequence designs, etc. In this paper, we further study the parametric construction approach to obtain balanced Boolean functions using $2$-to-$1$ mappings of the form $P(x^2+x)$, where $P$ denotes carefully selected permutation polynomials. The key contributions of this work are twofold: (1) We establish a new family of four-valued spectrum Boolean functions. This family includes Boolean functions with good cryptographic properties, e.g., the same nonlinearity as semi-bent functions, the maximal algebraic degree, and the optimal algebraic immunity for dimensions $n \leq 14$. (2) We derive seven distinct classes of plateaued functions, including four infinite families of semi-bent functions and a class of near-bent functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19521 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Balanced Boolean functions with few-valued Walsh spectra parameterized by $P(x^2+x)$ Zhang, Qiancheng Li, Kangquan Qu, Longjiang Information Theory Boolean functions with few-valued spectra have wide applications in cryptography, coding theory, sequence designs, etc. In this paper, we further study the parametric construction approach to obtain balanced Boolean functions using $2$-to-$1$ mappings of the form $P(x^2+x)$, where $P$ denotes carefully selected permutation polynomials. The key contributions of this work are twofold: (1) We establish a new family of four-valued spectrum Boolean functions. This family includes Boolean functions with good cryptographic properties, e.g., the same nonlinearity as semi-bent functions, the maximal algebraic degree, and the optimal algebraic immunity for dimensions $n \leq 14$. (2) We derive seven distinct classes of plateaued functions, including four infinite families of semi-bent functions and a class of near-bent functions. |
| title | Balanced Boolean functions with few-valued Walsh spectra parameterized by $P(x^2+x)$ |
| topic | Information Theory |
| url | https://arxiv.org/abs/2506.19521 |