Nian Li

Hubei University · Faculty of Mathematics and Statistics

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. Most previous results obtained so far were for cyclic codes with no more than three zeroes. Inspired by the works of Li et al. (Sci China Math 53:3279–3286, 2010; IEEE Trans Inf Theory 60:3903–3912, 2014), we study two families of cyclic co...

In this paper, a class of functions from Niho exponents with four-valued Walsh transform is obtained for any prime by a uniform method, and the distribution of the Walsh transform values is also completely determined. In particular, this class of functions is proven to be bent for a special case. Although it is shown that the obtained bent function...

Several new classes of binary and p-ary regular bent functions are obtained in this paper. The bentness of all these functions is determined by some exponential sums over finite fields, most of which have close relations with the well-known Kloosterman sums.

Cyclic codes are an important subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics. In this paper, two families of optimal ternary cyclic codes are presented. The first family of cyclic codes has parameters $[3^m-1, 3^m-1-2m, 4]$ and contains a class of conjectured cyclic codes...

Three classes of binary sequences of period with optimalautocorrelationvalue/magnitudehavebeenconstructedby Tang and Gong based on interleaving certain kinds of sequences of period , i.e., the Legendre sequence, twin-prime sequence and generalized GMW sequence. In this paper, by means of sequence polynomials of the underlying sequences, the propert...

Substitution boxes (S-boxes) play a significant role in ensuring the resistance of block ciphers against various attacks. The Difference Distribution Table (DDT), the Feistel Boomerang Connectivity Table (FBCT), the Feistel Boomerang Difference Table (FBDT) and the Feistel Boomerang Extended Table (FBET) of a given S-box are crucial tools to analyz...

In this paper, we construct a large family of projective linear codes over ${\mathbb F}_{q}$ from the general simplicial complexes of ${\mathbb F}_{q}^m$ via the defining-set construction, which generalizes the results of [IEEE Trans. Inf. Theory 66(11):6762-6773, 2020]. The parameters and weight distribution of this class of codes are completely d...

This paper deals with Niho functions which are one of the most important classes of functions thanks to their close connections with a wide variety of objects from mathematics, such as spreads and oval polynomials or from applied areas, such as symmetric cryptography, coding theory and sequences. In this paper, we investigate specifically the $c$-d...

Boolean functions with n variables are functions from to . They play an important role in both cryptographic and error correcting coding activities. The important information about the cryptographic properties of Boolean functions can be obtained from the study of the Walsh transform. Generally speaking, it is difficult to construct functions with...

In this paper, we construct two families of linear codes over the ring \({\mathbb F}_{q}+u{\mathbb F}_{q}\) by the defining set approach, where q is a prime power and \(u^2=0\). We completely determine their Lee weight distributions, which shows that these codes have few Lee weights. Via the Gray map, we obtain a family of near Griesmer codes over...

The switching method is a powerful method to construct bent functions. In this paper, using this method, we present two generic constructions of piecewise bent functions from known ones, which generalize some earlier works. Further, based on these two generic constructions, we obtain several infinite families of bent functions from quadratic bent f...

In this paper, we construct a large family of projective linear codes over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> q </sub> from the general simplicial complexes of F <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> m </sup> <sub xmlns:mml="http...

Starting with the multiplication of elements in $\mathbb{F}_{q}^2$ which is consistent with that over $\mathbb{F}_{q^2}$, where $q$ is a prime power, via some identification of the two environments, we investigate the $c$-differential uniformity for bivariate functions $F(x,y)=(G(x,y),H(x,y))$. By carefully choosing the functions $G(x,y)$ and $H(x,...

Let F2n be a finite field with 2n elements and fc_(x)=c0x2m(2k+1)+c1x2m+k+1+c2x2m+2k+c3x2k+1∈F2n[x], where n, m and k are positive integers with n=2m and gcd⁡(m,k)=e. In this paper, motivated by a recent work of Li, Xiong and Zeng (Li et al. (2021) [12]), we further study the boomerang uniformity of fc_(x) by using similar ideas and carrying out pa...

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(f1(x),f2(x),…,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy...

In this article, we focus on the concept of locally-APN-ness (``APN" is the abbreviation of the well-known notion of Almost Perfect Nonlinear) introduced by Blondeau, Canteaut, and Charpin, which makes the corpus of S-boxes somehow larger regarding their differential uniformity and, therefore, possibly, more suitable candidates against the differen...

In this paper, we construct four families of linear codes over finite fields from the complements of either the union of subfields or the union of cosets of a subfield, which can produce infinite families of optimal linear codes, including infinite families of (near) Griesmer codes. We also characterize the optimality of these four families of line...

In this paper, we study the boomerang spectrum of the power mapping $F(x)=x^{k(q-1)}$ over ${\mathbb F}_{q^2}$, where $q=p^m$, $p$ is a prime, $m$ is a positive integer and $\gcd(k,q+1)=1$. We first determine the differential spectrum of $F(x)$ and show that $F(x)$ is locally-APN. This extends a result of [IEEE Trans. Inf. Theory 57(12):8127-8137,...

Very recently, a class of cryptographically strong permutations with boomerang uniformity 4 and the best known nonlinearity is constructed from the closed butterfly structure in Li et al. (Des Codes Cryptogr 89(4):737–761, 2021). In this note, we provide two additional results concerning these permutations. We first represent the conditions of thes...

Let \begin{document}$ {\mathcal{C}}_{(u, v)} $\end{document} denote the ternary cyclic code with two zeros \begin{document}$ \alpha^u $\end{document} and \begin{document}$ \alpha^v $\end{document}, where \begin{document}$ \alpha $\end{document} is a generator of \begin{document}$ {\mathbb F}_{3^m}^{*} $\end{document} and \begin{document}$ 1\leq u,...

In this paper, we investigate the power function F ( x ) = x^d over the finite field F_{2^{4 n}}, where n is a positive integer and d = 2^{3 n} + 2^{2 n} + 2ⁿ - 1. We prove that this power function is AP c N with respect to all c ϵ F_{2^{4 n}} \ {1} satisfying c <sup>2<sup>2...

Motivated by a recent work of Zhang and Yan on the \begin{document}$ c $\end{document}-differential spectrum of some power functions over finite fields, we further study an AP\begin{document}$ c $\end{document}N function and express its \begin{document}$ c $\end{document}-differential spectrum in terms of \begin{document}$ (i, j, k)_2 $\end{documen...

This article focuses on the so-called locally-APN power functions introduced by Blondeau, Canteaut and Charpin, which generalize the well-known notion of APN functions and possibly more suitable candidates against differential attacks. Specifically, given two coprime positive integers $m$ and $k$ such that $\gcd (2^{m}+1,2^{k}+1)=1$ , we inve...

In this paper, permutation polynomials of the form xd+L(xs) over finite fields Fq3 are investigated, where q=2m and m is a positive integer. By means of the iterative method, the multivariate method and the resultant elimination, six classes of permutation trinomials are obtained from certain integers d, s and linearized polynomials L(x). It is als...

Linear codes with few weights have applications in data storage systems, secret sharing schemes and authentication codes. In this paper, inspired by the butterfly structure [6], [29] and the works of Li, Yue and Fu [21] and Jian, Lin and Feng [19], we introduce a new defining set with the form of the closed butterfly structure and consequently we o...

Functions with low c-differential uniformity were proposed in 2020 and attracted lots of attention, especially the PcN and APcN functions, due to their applications in cryptography. The objective of this paper is to study PcN and APcN functions. As a consequence, we propose two classes of PcN functions and three classes of APcN functions by using t...

In this paper, we construct four families of linear codes over finite fields from the complements of either the union of subfields or the union of cosets of a subfield, which can produce infinite families of optimal linear codes, including infinite families of (near) Griesmer codes. We also characterize the optimality of these four families of line...

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...

Extending previous results, we study a class of general quadrinomials over the field of size 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m</sup> with odd m and characterize conditions under which they are permutations with 4-uniform BCT, a new and important parameter related to boomerang-style a...

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 c...

Functions with low $c$-differential uniformity were proposed in $2020$ and attracted lots of attention, especially the P$c$N and AP$c$N functions, due to their applications in cryptography. The objective of this paper is to study P$c$N and AP$c$N functions. As a consequence, we propose a class of P$c$N functions and four classes of AP$c$N functions...

Linear codes have a wide range of applications in the data storage systems, communication systems, consumer electronics products since their algebraic structure can be analyzed and they are easy to implement in hardware. How to construct linear codes with excellent properties to meet the demands of practical systems becomes a research topic, and it...

A class of quadratic vectorial bent functions having the form F(x)=Trmn(ax2s1+1)+Tr1n(bx2s2+1) is investigated, where n; m; s1; s2 are positive integers and the coefficients a, b belong to the finite field F2ⁿ. Through some discussions on the permutation property of certain linearized polynomials over F2ⁿ, several classes of quadratic vectorial ben...

Linear codes with few weights constructed from defining sets have been extensively studied due to their applications in data storage systems, secret sharing schemes and authentication codes. In this paper, inspired by the works of Li et al. (Appl Algebra Eng Commun Comput 28(1):11–30, 2017) and Jian et al. (Finite Fields Appl 57:92–107, 2019), we p...

Linear codes with few weights have applications in data storage systems, secret sharing schemes and authentication codes. In this paper, inspired by the works of Heng and Yue (2016) [14] and Tan, Zhou, Tang and Helleseth (2017) [25], we extend Tan, Zhou, Tang and Helleseth's work to obtain a class of optimal 1-weight binary linear codes, new classe...

In Eurocrypt'18, Cid et al. proposed a new cryptanalysis tool called Boomerang Connectivity Table (BCT), to evaluate S-boxes of block ciphers. Later, Boura and Canteaut further investigated the new parameter Boomerang uniformity for cryptographic S-boxes. It is of great interest to find new S-boxes with low Boomerang uniformity for even dimensions....

As a generalization of Dillon's APN permutation, butterfly structure and generalizations have been of great interest since they generate permutations with the best known differential and nonlinear properties over the field of size $2^{4k+2}$. Complementary to these results, we show in this paper that butterfly structure, more precisely the closed b...

We study a class of general quadrinomials over the field of size $2^{2m}$ with odd $m$ and characterize conditions under which they are permutations with the best boomerang uniformity, a new and important parameter related to boomerang-style attacks. This vastly extends previous results from several recent papers.

In this paper, a class of p-ary 3-weight linear codes and a class of binary 2-weight linear codes are proposed respectively by virtue of the properties of the perfect nonlinear functions over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(m)</sub> and (m, s)-bent functions from F <sub xmlns:mml="h...

An involution over finite fields is a permutation polynomial whose inverse is itself. Owing to this property, involutions over finite fields have been widely used in applications such as cryptography and coding theory. Following the idea in [28] to characterize the involutory behavior of the generalized cyclotomic mappings, this paper gives a more...

The Niho exponent was introduced by Yoji Niho, who investigated the cross-correlation function between an m-sequence and its decimation sequence in 1972. Since then, Niho exponents have been used in other research areas such as in cryptography and coding theory. In this paper, we introduce some research problems related to Niho exponents and survey...

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.

Rotation symmetric bent functions are a special class of Boolean functions, and their construction is of theoretical and practical interest. In this paper, we propose a generic construction of rotation symmetric bent functions by modifying the support of a known class of quadratic rotation symmetric bent functions, which generalizes some earlier wo...

Three classes of (balanced) Boolean functions with few Walsh transform values derived from bent functions, Gold functions and the product of linearized polynomials are obtained in this paper. Further, the value distributions of their Walsh transform are also determined by virtue of the property of bent functions, the Walsh transform property of Gol...

In this paper, based on the theory of \begin{document}$ \mathbb{Z}_{4} $\end{document}-valued quadratic forms we propose several classes of generalized Boolean bent functions over \begin{document}$ \mathbb{Z}_{4} $\end{document}, and new families of codebooks are constructed from these functions. The codebooks constructed in this paper are nearly o...

An involution over finite fields is a permutation polynomial whose inverse is itself. Owing to this property, involutions over finite fields have been widely used in applications such as cryptography and coding theory. As far as we know, there are not many involutions, and there isn't a general way to construct involutions over finite fields. This...

In this paper, by analyzing the quadratic factors of an $11$-th degree polynomial over the finite field $\ftwon$, a conjecture on permutation trinomials over $\ftwon[x]$ proposed very recently by Deng and Zheng is settled, where $n=2m$ and $m$ is a positive integer with $\gcd(m,5)=1$.

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. Most previous results obtained so far were for cyclic codes with no more than three zeroes. Inspired by the works \cite{Li-Zeng-Hu} and \cite{gegeng2}, we study two families of cyclic codes over $\mathbb{F}_p$ with arbitrary number of zeroe...

Motivated by recent results on the constructions of permutation polynomials with few terms over the finite field 𝔽2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mat...

We study the problem of existence of APN functions of algebraic degree n over F2n. We characterize such functions by means of derivatives and power moments of the Walsh transform. We deduce several non-existence results which imply, in particular, that for most of the known APN functions F over F2n the function x2n−1+F(x) is not APN, and changing a...

Permutation polynomials having the form over finite fields of even characteristic are investigated in this paper. Eight classes of such polynomials are proved to be permutations.

A family IP8 of sequences over the 8-ary quadrature-pulse amplitude modulation (Q-PAM) constellation with asymptotically optimal correlation property was presented by Anand and Kumar in 2008. This is the only known family of sequences over the quadrature amplitude modulation (QAM) constellation whose correlation magnitude asymptotically achieves th...

In this paper, we determine the Walsh spectra of three classes of quadratic APN functions and we prove that the class of quadratic trinomial APN functions constructed by Göloğlu is affine equivalent to Gold functions.

The construction of permutation trinomials over finite fields attracts people’s interest recently due to their simple form and some additional properties. Motivated by some results on the construction of permutation trinomials with Niho exponents, in this paper, by constructing some new fractional polynomials that permute the set of the (q + 1)-th...

In this paper, a class of binary cyclic codes with three generalized Niho-type nonzeros is introduced. Based on some techniques in solving certain equations over finite fields, the proposed cyclic codes are shown to have six nonzero weights and the weight distribution is also completely determined.

In this paper, let $n=2m$ and $d=3^{m+1}-2$ with $m\geq2$ 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 seq...

Linear codes with few weights have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, several classes of p-ary linear codes with two or three weights are constructed from quadratic Bent functions over the finite field Fp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackag...

Permutation polynomials with few terms attracts researchers' interest in recent years due to their simple algebraic form and some additional extraordinary properties. In this paper, by analyzing the quadratic factors of a fifth-degree polynomial and a seventh-degree polynomial over the finite field $\mathbb{F}_{3^{2k}}$, two conjectures on permutat...

In this paper, let n = 2k and d = 3 · 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> - 2 with k ≥ 3 and gcd(d, 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> - 1) = 1. Based on some analysis of certain equations over finite fields and...

Negabent functions as a class of generalized bent functions have attracted a lot of attention recently due to their applications in cryptography and coding theory. In this paper, we consider the constructions of negabent functions over finite fields. First, by using the compositional inverses of certain binomial and trinomial permutations, we prese...

Linear codes with a few weights have applications in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. This paper first generalizes the method of constructing two-weight and three-weight linear codes of Ding et al. and Zhou et al. to general weakly regul...

Cyclic codes have been an important topic of both mathematics and engineering for decades. They have been widely used in consumer electronics, data transmission technologies, broadcast systems, and computer applications as they have efficient encoding and decoding algorithms. The objective of this paper is to provide a survey of three-weight cyclic...

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Let $m=2\ell+1$ for an integer $\ell\geq 1$ and $\pi$ be a generator of $\gf(3^m)^*$. In this paper, a class of cyclic codes $\C_{(u,v)}$ over $\gf(3)$ wi...

Cyclic codes are an important class of linear codes and have been widely used in many areas such as consumer electronics, data storage and communication systems. Let C(1,e) denote the cyclic code with generator polynomial mα(x)mαe (x), where α is a primitive element of F3m and mαi (x) denotes the minimal polynomial of αi over F3 for 1 ≤ i ≤ 3m − 1....

A class of quaternary sequences Sλ had been proven to be optimal for some special values of λ. In this note, Sλ is investigated for all λ by virtue of the ℤ4-valued quadratic forms over Galois rings. As a consequence, a new class of quaternary sequences with low correlation is obtained and the correlation distribution is also completely determined....

In this paper we extend the works \cite{gegeng2,XLZD} further in two directions and compute the weight distribution of these cyclic codes under more relaxed conditions. It is interesting to note that many cyclic codes in the family are optimal and have only a few non-zero weights. Besides using similar ideas from \cite{gegeng2,XLZD}, we carry out s...

By using a powerful criterion for permutation polynomials, we give several classes of complete permutation polynomials over finite fields. First, two classes of complete permutation monomials whose exponents are of Niho type are presented. Second, for any odd prime p, we give a sufficient and necessary condition for a −1x d to be a complete permuta...

In this paper, a construction of codebooks based on a set of bent functions satisfying certain conditions is introduced. It includes some earlier constructions of codebooks meeting the Levenstein bound as special cases. With this construction, two new families of codebooks achieving the Levenstein bound are obtained. The codebooks constructed in th...

Let p and q be two odd primes with p=Mf+1 and M is even. A new construction of M -ary sequences of period pq with low periodic autocorrelation is presented in this paper based on interleaving the M -ary power residue sequence of period p according to the quadratic residue with respect to q. This construction can generate the well-known twin-prime s...

New quadratic bent functions in polynomial form are constructed in this paper. The constructions give new Boolean bent, generalized Boolean bent and (p) -ary bent functions. Based on (boldsymbol {mathbb {Z}}_{4}) -valued quadratic forms, a simple method provides several new constructions of generalized Boolean bent functions. From these generalized...

In this paper, four classes of complete permutation polynomials over finite fields of characteristic two are presented. To consider the permutation property of the first three classes, Dickson polynomials play a key role. The fourth class is a generalization of a known result. In addition, we also calculate the inverses of these bijective monomials...

In this paper, by using a powerful criterion for permutation polynomials given by Zieve, and some low degree permutation polynomials over $\F_{q}$, we give several classes of complete permutation monomials over $\F_{q^r}$. In addition, we present a class of complete permutation multinomials, which is a generalization of recent work.

Let $m\geq 3$ be an odd integer and $p$ be an odd prime. % with $p-1=2^rh$, where $h$ is an odd integer. In this paper, many classes of three-weight cyclic codes over $\mathbb{F}_{p}$ are presented via an examination of the condition for the cyclic codes $\mathcal{C}_{(1,d)}$ and $\mathcal{C}_{(1,e)}$, which have parity-check polynomials $m_1(x)m_d...

A class of permutation polynomials with given form over finite fields is investigated in this paper, which is a further study on a recent work of Zha and Hu. Based on some particular techniques over finite fields, two results obtained by Zha and Hu are improved and new permutation polynomials are also obtained.

Bipolar complementary sequence pairs of Types II and III are defined, enumerated for n ≤ 28, and classified. Type-II pairs are shown to exist only at lengths 2 m , and necessary conditions for Type-III pairs lead to a non-existence conjecture regarding their length.

New quadratic bent functions in polynomial forms are constructed in this paper. The constructions give new boolean bent and generalized boolean bent functions. Based on Z4-valued quadratic forms, a simple method provides several new constructions of generalized boolean bent functions. From these generalized boolean bent functions a method is presen...

Let m and k be positive integers with m/gcd(m,k) being odd, for a ∈ R and b ∈ L, the exponential sum Σx∈LiTr(ax+2bx2k+1) is studied systematically in this paper, where i=√(-1), R = GR (4,m) is a Galois ring, L is the Teichmüller set of R and Tr(·) is the trace function from the Galois ring R to Z4. Through the discussions on the solutions of certai...

Recently, new optimal Families S and U of quaternary sequences have been presented, and the optimal binary sequence Family V obtained from Family S under Gray map has been investigated as well. The two sequence Families U and V are optimal with respect to the well-known Sidelnikov bound and Welch bound, but their exact correlation distributions are...

In this paper, we follow the recent work of Helleseth, Kholosha, Johanssen and Ness to study the cross correlation between an $m$-sequence of period $2^m-1$ and the $d$-decimation of an $m$-sequence of shorter period $2^{n}-1$ for an even number $m=2n$. Assuming that $d$ satisfies $d(2^l+1)=2^i({\rm mod} 2^n-1)$ for some $l$ and $i$, we prove the c...

In this paper, for an even integer n ≥ 4 and any positive integer k with gcd(n/2,k) = gcd(n/2 − k,2k) = d being odd, a class of p-ary codes Ck\mathcal{C}^k is defined and the weight distribution is completely determined, where p is an odd prime. A class of nonbinary sequence families is constructed from these codes, and the correlation distributio...

Based on cyclic difference sets, sequences with two-valued autocorrelation can be constructed. Using these constructed sequences, two classes of binary constant weight codes are presented. Some codes proposed in this paper are proven to be optimal.

In this paper, for an even integer $n\geq 4$ and any positive integer $k$ with ${\rm gcd}(n/2,k)={\rm gcd}(n/2-k,2k)=d$ being odd, a class of $p$-ary codes $\mathcal{C}^k$ is defined and their weight distribution is completely determined, where $p$ is an odd prime. As an application, a class of nonbinary sequence families is constructed from these...