Some more constructions of $n-$cycle permutation polynomials
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| Format: | Preprint |
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2025
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| _version_ | 1866912453427724288 |
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| author | Jarali, Varsha Poojary, Prasanna R, Vadiraja Bhatta G. |
| author_facet | Jarali, Varsha Poojary, Prasanna R, Vadiraja Bhatta G. |
| contents | $ n-$cycle permutation polynomials with small n have the advantage that their compositional inverses are efficient in terms of implementation. These permutation polynomials have significant applications in cryptography and coding theory. In this article, we propose criteria for the construction of $ n-$cycle permutation using linearized polynomial $ L(x) $ for larger $ n $. Furthermore, we investigate and generalize certain novel forms of $ n-$cycle permutation polynomials. Finally, we demonstrate our approach by constructing explicit $ n-$cycle permutation of the form $ L(x)+γh(Tr_{q^{m}/q}(x)) $, and $ G(x)+γf(x) $ with a Boolean function $ f(x) $. The polynomial $ x^{d}+γf(x) $ with $ f(x) $ being a Boolean function is shown to be quadruple and quintuple permutation polynomials. Moreover, linear binomial triple-cycle permutation polynomials are constructed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_21936 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some more constructions of $n-$cycle permutation polynomials Jarali, Varsha Poojary, Prasanna R, Vadiraja Bhatta G. Rings and Algebras Information Theory Combinatorics 2010: 05A05 11T06 $ n-$cycle permutation polynomials with small n have the advantage that their compositional inverses are efficient in terms of implementation. These permutation polynomials have significant applications in cryptography and coding theory. In this article, we propose criteria for the construction of $ n-$cycle permutation using linearized polynomial $ L(x) $ for larger $ n $. Furthermore, we investigate and generalize certain novel forms of $ n-$cycle permutation polynomials. Finally, we demonstrate our approach by constructing explicit $ n-$cycle permutation of the form $ L(x)+γh(Tr_{q^{m}/q}(x)) $, and $ G(x)+γf(x) $ with a Boolean function $ f(x) $. The polynomial $ x^{d}+γf(x) $ with $ f(x) $ being a Boolean function is shown to be quadruple and quintuple permutation polynomials. Moreover, linear binomial triple-cycle permutation polynomials are constructed. |
| title | Some more constructions of $n-$cycle permutation polynomials |
| topic | Rings and Algebras Information Theory Combinatorics 2010: 05A05 11T06 |
| url | https://arxiv.org/abs/2506.21936 |