Ternary Binomial and Trinomial Bent Functions in the Completed Maiorana-McFarland Class
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912505246253056 |
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| author | Helleseth, Tor Kholosha, Alexander Spithaki, Niki |
| author_facet | Helleseth, Tor Kholosha, Alexander Spithaki, Niki |
| contents | Two classes of ternary bent functions of degree four with two and three terms in the univariate representation that belong to the completed Maiorana-McFarland class are found. Binomials are mappings $\F_{3^{4k}}\mapsto\fthree$ given by $f(x)=\Tr_{4k}\big(a_1 x^{2(3^k+1)}+a_2 x^{(3^k+1)^2}\big)$, where $a_1$ is a nonsquare in $\F_{3^{4k}}$ and $a_2$ is defined explicitly by $a_1$. Particular subclasses of the binomial bent functions we found can be represented by exceptional polynomials over $\fthreek$. Bent trinomials are mappings $\F_{3^{2k}}\mapsto\fthree$ given by $f(x)=\Tr_n\big(a_1 x^{2\cdot3^k+4} + a_2 x^{3^k+5} + a_3 x^2\big)$ with coefficients explicitly defined by the parity of $k$. The proof is based on a new criterion that allows checking bentness by analyzing first- and second-order derivatives of $f$ in the direction of a chosen $n/2$-dimensional subspace. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20715 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ternary Binomial and Trinomial Bent Functions in the Completed Maiorana-McFarland Class Helleseth, Tor Kholosha, Alexander Spithaki, Niki Discrete Mathematics Two classes of ternary bent functions of degree four with two and three terms in the univariate representation that belong to the completed Maiorana-McFarland class are found. Binomials are mappings $\F_{3^{4k}}\mapsto\fthree$ given by $f(x)=\Tr_{4k}\big(a_1 x^{2(3^k+1)}+a_2 x^{(3^k+1)^2}\big)$, where $a_1$ is a nonsquare in $\F_{3^{4k}}$ and $a_2$ is defined explicitly by $a_1$. Particular subclasses of the binomial bent functions we found can be represented by exceptional polynomials over $\fthreek$. Bent trinomials are mappings $\F_{3^{2k}}\mapsto\fthree$ given by $f(x)=\Tr_n\big(a_1 x^{2\cdot3^k+4} + a_2 x^{3^k+5} + a_3 x^2\big)$ with coefficients explicitly defined by the parity of $k$. The proof is based on a new criterion that allows checking bentness by analyzing first- and second-order derivatives of $f$ in the direction of a chosen $n/2$-dimensional subspace. |
| title | Ternary Binomial and Trinomial Bent Functions in the Completed Maiorana-McFarland Class |
| topic | Discrete Mathematics |
| url | https://arxiv.org/abs/2507.20715 |