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A new class of permutation trinomials constructed from Niho exponents

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Abstract

Permutation polynomials over finite fields are an interesting subject due to their important applications in the areas of mathematics and engineering. In this paper, we investigate the trinomial f(x) = x (p−1)q+1 + x pq − x q+(p−1) over the finite field \(\mathbb {F}_{q^{2}}\), where p is an odd prime and q = p k with k being a positive integer. It is shown that when p = 3 or 5, f(x) is a permutation trinomial of \(\mathbb {F}_{q^{2}}\) if and only if k is even. This property is also true for a more general class of polynomials g(x) = x (q+1)l+(p−1)q+1 + x (q+1)l + pq − x (q+1)l + q+(p−1), where l is a nonnegative integer and \(\gcd (2l+p,q-1)=1\). Moreover, we also show that for p = 5 the permutation trinomials f(x) proposed here are new in the sense that they are not multiplicative equivalent to previously known ones of similar form.

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Acknowledgments

T. Bai and Y. Xia were supported in part by National Natural Science Foundation of China under Grant 61771021 and Grant 11301552, and in part by Natural Science Foundation of Hubei Province under Grant 2017CFB425. T. Bai was also supported by Graduate Innovation Fund of South-Central University for Nationalities.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, 430074, China

    Tao Bai & Yongbo Xia

  2. Hubei Key Laboratory of Intelligent Wireless Communications, South-Central University for Nationalities, Wuhan, 430074, China

    Yongbo Xia

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  1. Tao Bai
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  2. Yongbo Xia
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Correspondence to Yongbo Xia.

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Bai, T., Xia, Y. A new class of permutation trinomials constructed from Niho exponents. Cryptogr. Commun. 10, 1023–1036 (2018). https://doi.org/10.1007/s12095-017-0263-4

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  • DOI: https://doi.org/10.1007/s12095-017-0263-4

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