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Register a new userSeveral Classes of Permutation Trinomials From Niho Exponents
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Added by
Nian Li
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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Motivated by recent results on the constructions of permutation polynomials with few terms over the finite field $mathbb{F}_{2^n}$, where $n$ is a positive even integer, we focus on the construction of permutation trinomials over $mathbb{F}_{2^n}$ from Niho exponents. As a consequence, several new classes of permutation trinomials over $mathbb{F}_{2^n}$ are constructed from Niho exponents based on some subtle manipulation of solving equations with low degrees over finite fields.
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In this paper, a class of permutation trinomials of Niho type over finite fields with even characteristic is further investigated. New permutation trinomials from Niho exponents are obtained from linear fractional polynomials over finite fields, and it is shown that the presented results are the generalizations of some earlier works.
Let $mathbb{F}_{p^{n}}$ be the finite field with $p^n$ elements and $operatorname{Tr}(cdot)$ be the trace function from $mathbb{F}_{p^{n}}$ to $mathbb{F}_{p}$, where $p$ is a prime and $n$ is an integer. Inspired by the works of Mesnager (IEEE Trans. Inf. Theory 60(7): 4397-4407, 2014) and Tang et al. (IEEE Trans. Inf. Theory 63(10): 6149-6157, 2017), we study a class of bent functions of the form $f(x)=g(x)+F(operatorname{Tr}(u_1x),operatorname{Tr}(u_2x),cdots,operatorname{Tr}(u_{tau}x))$, where $g(x)$ is a function from $mathbb{F}_{p^{n}}$ to $mathbb{F}_{p}$, $taugeq2$ is an integer, $F(x_1,cdots,x_n)$ is a reduced polynomial in $mathbb{F}_{p}[x_1,cdots,x_n]$ and $u_iin mathbb{F}^{*}_{p^n}$ for $1leq i leq tau$. As a consequence, we obtain a generic result on the Walsh transform of $f(x)$ and characterize the bentness of $f(x)$ when $g(x)$ is bent for $p=2$ and $p>2$ respectively. Our results generalize some earlier works. In addition, we study the construction of bent functions $f(x)$ when $g(x)$ is not bent for the first time and present a class of bent functions from non-bent Gold functions.
Let $mathbb{F}_q$ denote the finite fields with $q$ elements. The permutation behavior of several classes of infinite families of permutation polynomials over finite fields have been studied in recent years. In this paper, we continue with their studies, and get some further results about the permutation properties of the permutation polynomials. Also, some new classes of permutation polynomials are constructed. For these, we alter the coefficients, exponents or the underlying fields, etc.
In this paper, let $n=2m$ and $d=3^{m+1}-2$ with $mgeq2$ and $gcd(d,3^n-1)=1$. By studying the weight distribution of the ternary Zetterberg code and counting the numbers of solutions of some equations over the finite field $mathbb{F}_{3^n}$, the correlation distribution between a ternary $m$-sequence of period $3^n-1$ and its $d$-decimation sequence is completely determined. This is the first time that the correlation distribution for a non-binary Niho decimation has been determined since 1976.
Permutation polynomials over finite fields have important applications in many areas of science and engineering such as coding theory, cryptography, combinatorial design, etc. In this paper, we construct several new classes of permutation polynomials, and the necessities of some permutation polynomials are studied.
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