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Register a new userA class of APcN power functions over finite fields of even characteristic
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Ziran Tu
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In this paper, we investigate the power functions $F(x)=x^d$ over the finite field $mathbb{F}_{2^{4n}}$, where $n$ is a positive integer and $d=2^{3n}+2^{2n}+2^{n}-1$. It is proved that $F(x)=x^d$ is APcN at certain $c$s in $mathbb{F}_{2^{4n}}$, and it is the second class of APcN power functions over finite fields of even characteristic. Further, the $c$-differential spectrum of these power functions is also determined.
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Differential uniformity is a significant concept in cryptography as it quantifies the degree of security of S-boxes respect to differential attacks. Power functions of the form $F(x)=x^d$ with low differential uniformity have been extensively studied in the past decades due to their strong resistance to differential attacks and low implementation cost in hardware. In this paper, we give an affirmative answer to a recent conjecture proposed by Budaghyan, Calderini, Carlet, Davidova and Kaleyski about the differential uniformity of $F(x)=x^d$ over $mathbb{F}_{2^{4n}}$, where $n$ is a positive integer and $d=2^{3n}+2^{2n}+2^{n}-1$, and we completely determine its differential spectrum.
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.
In this paper, we produce new classes of MDS self-dual codes via (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. Among our constructions, there are many MDS self-dual codes with new parameters which have never been reported. For odd prime power $q$ with $q$ square, the total number of lengths for MDS self-dual codes over $mathbb{F}_q$ presented in this paper is much more than those in all the previous results.
For the finite field $mathbb{F}_{2^{3m}}$, permutation polynomials of the form $(x^{2^m}+x+delta)^{s}+cx$ are studied. Necessary and sufficient conditions are given for the polynomials to be permutation polynomials. For this, the structures and properties of the field elements are analyzed.
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