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67

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Introduction

## Publications

Publications (67)

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...

Using tools developed in a recent work by Shen and the second author, in this paper we carry out an in-depth study on the average decoding error probability of the random matrix ensemble over the erasure channel under three decoding principles, namely unambiguous decoding, maximum likelihood decoding and list decoding. We obtain explicit formulas f...

Let $n$ be a positive integer and $p$ a prime. The power mapping $x^{p^n-3}$ over $\mathbb{F}_{p^n}$ has desirable differential properties, and its differential spectra for $p=2,\,3$ have been determined. In this paper, for any odd prime $p$, by investigating certain quadratic character sums and some equations over $\mathbb{F}_{p^n}$, we determine...

In this paper, we provide two complementary results for cyclic codes of composite length. First, we give a general construction of cyclic codes of length
$nr$
from cyclic codes of length
$n$
and a lower bound on the minimum distance. Numerical data show that many cyclic codes of composite length with the best parameters can be obtained in this...

Extending previous results, 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 4-uniform BCT, a new and important parameter related to boomerang-style attacks. These permutations are known to have the best known nonlinearity. Numerical data al...

Let m be an even positive integer. A Boolean bent function f on F2m-1 ×F2 is called a cyclic bent function if for any a ≠ b ∈ F2m-1 and ε ∈ F2, f(ax1,x2)+f(bx1,x2+) is always bent, where x1 ∈ F2m-1,x2 ∈ F2. Cyclic bent functions look extremely rare. This paper focuses on cyclic bent functions on F2m-1 ×F2 and their applications. The first objective...

It is known that the empirical spectral distribution of random matrices obtained from linear codes of increasing length converges to the well-known Marchenko-Pastur law, if the Hamming distance of the dual codes is at least 5. In this paper, we prove that the convergence rate in probability is at least of the order
$n^{-1/4}$
where
$n$
is the l...

At Eurocrypt’18, Cid, Huang, Peyrin, Sasaki, and Song introduced a new tool called Boomerang Connectivity Table (BCT) for measuring the resistance of a block cipher against the boomerang attack which is an important cryptanalysis technique introduced by Wagner in 1999 against block ciphers. Next, Boura and Canteaut introduced an important parameter...

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.

Two group randomness tests have been investigated in the literature for linear codes over finite fields, based on the Marchenko-Pastur and Wigner's semicircle laws respectively. Authors proved that linear codes with dual distance at least 5 perform well in both tests. In this paper, we prove that a large collection of mutually unbiased bases over $...

Boolean functions, coding theory and
$t$
-designs have close connections and interesting interplay. A standard approach to constructing
$t$
-designs is the use of linear codes with certain regularity. The Assmus-Mattson Theorem and the automorphism groups are two ways for proving that a code has sufficient regularity for supporting
$t$
-desig...

Coding theory and t-designs have close connections and interesting interplay. In this paper, we first introduce a class of ternary linear codes and study their parameters. We then focus on their three-weight subcodes with a special weight distribution. We determine the weight distributions of some shortened codes and punctured codes of these three-...

Special functions, coding theory and $t$-designs have close connections and interesting interplay. A standard approach to constructing $t$-designs is the use of linear codes with certain regularity. The Assmus-Mattson Theorem and the automorphism groups are two ways for proving that a code has sufficient regularity for supporting $t$-designs. Howev...

Recently we considered a new normalization of matrices obtained by choosing distinct codewords at random from linear codes over finite fields and proved that under some natural algebraic conditions their empirical spectral distribution converges to Wigner's semicircle law as the length of the codes goes to infinity. One of the conditions is that th...

In this paper we consider a new normalization of matrices obtained by choosing distinct codewords at random from linear codes over finite fields and find that under some natural algebraic conditions of the codes their empirical spectral distribution converges to Wigner’s semicircle law as the length of the codes goes to infinity. One such condition...

In this paper we first study deviations of the complete weight distribution of a linear code from that of a random code. Then we consider a large family of subfield subcodes of algebraic-geometric codes over prime fields which include BCH codes and Goppa codes and prove that the complete weight distribution is close to that of a random code if the...

At Eurocrypt'18, Cid, Huang, Peyrin, Sasaki, and Song introduced a new tool called Boomerang Connectivity Table (BCT) for measuring the resistance of a block cipher against the boomerang attack (which is an important cryptanalysis technique introduced by Wagner in 1999 against block ciphers). Next, Boura and Canteaut introduced an important paramet...

It is known that the empirical spectral distribution of random matrices obtained from linear codes of increasing length converges to the well-known Marchenko-Pastur law, if the Hamming distance of the dual codes is at least 5. In this paper, we prove that the convergence in probability is at least in the order of $n^{-1/4}$ where $n$ is the length...

Coding theory and $t$-designs have close connections and interesting interplay. In this paper, we first introduce a class of ternary linear codes and study their parameters. We then focus on their three-weight subcodes with a special weight distribution. We determine the weight distributions of some shortened codes and punctured codes of these thre...

Let $m$ be an even positive integer. A Boolean bent function $f$ on $\GF{m-1} \times \GF {}$ is called a \emph{cyclic bent function} if for any $a\neq b\in \GF {m-1}$ and $\epsilon \in \GF{}$, $f(ax_1,x_2)+f(bx_1,x_2+\epsilon)$ is always bent, where $x_1\in \GF {m-1}, x_2 \in \GF {}$. Cyclic bent functions look extremely rare. This paper focuses on...

In this paper we consider a new normalization of matrices obtained by choosing distinct codewords at random from linear codes over finite fields and find that under some natural algebraic conditions of the codes their empirical spectral distribution converges to Wigner's semicircle law as the length of the codes goes to infinity. One such condition...

Let \(m \ge 5\) be an odd integer. For \(d=2^m+2^{(m+1)/2}+1\) or \(d=2^{m+1}+3\), Blondeau et al. conjectured that the power function \(F_d=x^d\) over \(\mathrm {GF}(2^{2m})\) is differentially 8-uniform in which all values \(0, \, 2, \, 4,\, 6,\, 8\) appear. In this paper, we confirm this conjecture and compute the differential spectrum of \(F_d\...

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...

Let be a power function over the finite field , where k is a positive integer and . It is known by a recent result of Bracken and Leander that is a highly nonlinear differentially 4-uniform function. Building upon this work, in this paper we determine the differential spectrum of .

Cyclic codes are widely employed in communication systems, storage devices and consumer electronics, as they have efficient encoding and decoding algorithms. BCH codes, as a special subclass of cyclic codes, are in most cases among the best cyclic codes. A subclass of good BCH codes are the narrow-sense BCH codes over GF(q) with length n = (qm–1)=(...

In an interesting paper Professor Cunsheng Ding provided three constructions of cyclic codes of length being a product of two primes. Numerical data shows that many codes from these constructions are best cyclic codes of the same length and dimension over the same finite field. However, not much is known about these codes. In this paper we explain...

Constacyclic codes which generalize the classical cyclic codes have played important roles in recent constructions of many new quantum maximum distance separable (MDS) codes. However, the mathematical mechanism may not have been fully understood. In this paper, we use pseudo-cyclic codes, which is a further generalization of constacyclic codes, to...

Cyclic codes over finite fields are widely employed in communication systems, storage devices and consumer electronics, as they have efficient encoding and decoding algorithms. BCH codes, as a special subclass of cyclic codes, are in most cases among the best cyclic codes. A subclass of good BCH codes are the narrow-sense BCH codes over $\gf(q)$ wi...

Maximum-distance separable (MDS) convolutional codes form an optimal family
of convolutional codes, the study of which is of great importance. There are
very few general algebraic constructions of MDS convolutional codes. In this
paper, we construct a large family of unit-memory MDS convolutional codes over
$\F$ with flexible parameters. Compared w...

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. So far, most of previous results obtained were for cyclic codes with no more than three nonzeros. Recently, the authors of [37] constructed a class of cyclic codes with arbitrary number of nonzeros, and computed the weight distribution for...

The generalized Hamming weights (GHWs) of linear codes are fundamental
parameters, the knowledge of which is of great interest in many applications.
However, to determine the GHWs of linear codes is difficult in general. In this
paper, we study the GHWs for a family of reducible cyclic codes and obtain the
complete weight hierarchy in several cases...

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...

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...

Zero-difference balanced (ZDB) functions integrate a number of subjects in
combinatorics and algebra, and have many applications in coding theory,
cryptography and communications engineering. In this paper, three new families
of ZDB functions are presented. The first construction, inspired by the recent
work \cite{Cai13}, gives ZDB functions define...

Extending the idea of \cite{dab2} and using the 2-descent method, we provide
three general families of elliptic curves over Q such that a positive
proportion of prime-twists of such elliptic curves have rank zero
simultaneously.

We study the distribution of the size of Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with a non-trivial \$2\$-torsion point over \$\mathbb {Q}\$. This complements the work [Xiong and Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math.
219...

For a function field $k$ over a finite field with $\mathbb{F}_q$ as the field
of constant, and a finite abelian group $G$ whose exponent is divisible by
$q-1$, we study the distribution of zeta zeroes for a random $G$-extension of
$k$, ordered by the degree of conductors. We prove that when the degree goes to
infinity, the number of zeta zeroes lyi...

Cyclic codes are a subclass of linear codes and have wide applications in
consumer electronics, data storage systems, and communication systems due to
their efficient encoding and decoding algorithms. Cyclic codes with many zeros
and their dual codes have been a subject of study for many years. However,
their weight distributions are known only for...

Cyclic codes have been widely used in digital communication systems and
consume electronics as they have efficient encoding and decoding algorithms.
The weight distribution of cyclic codes has been an important topic of study
for many years. It is in general hard to determine the weight distribution of
linear codes. In this paper, a class of cyclic...

We study the distribution of the size of Selmer groups and Tate–Shafarevich groups arising from a 2-isogeny and its dual 2-isogeny for elliptic curves E
n
:y
2=x
3−n
3. We show that the 2-ranks of these groups all follow the same distribution. The result also implies that the mean value of the 2-rank of the corresponding Tate–Shafarevich groups fo...

Let C be a smooth projective curve of genus g >= 1 over a finite field F-q of cardinality q. Denote by #J(C) the size of the Jacobian of C over F-q. We first obtain an estimate on #J(C) when F-q(C)/F-q(X) is a geometric Galois extension, which improves a general result of Shparlinski [19]. Then we study the behavior of the quantity #J(C) as C varie...

Recently, the weight distributions of the duals of the cyclic codes with two
zeros have been obtained for several cases. In this paper we provide a slightly
different approach toward the general problem and use it to solve one more
special case. We make extensive use of standard tools in number theory such as
characters of finite fields, the Gauss...

In the paper, we generalize a result of P. Shiu on the number of square-full integers between successive squares. We extend, via a different approach, the result to k-full integers, and we also obtain explicit error terms in the process.

Let $ B=(B-{Q}\!)-{{Q \in {\mathbb N}}} $ be an increasing sequence of positive square free integers satisfying the condition that $ B-{{Q-1}}\vert B-{{Q-2}} $ whenever Q1 < Q2. For any subinterval I ⊂ [0, 1], let \documentclass{article}\begin{document}$$ {\mathscr{F}-{{B}\!,-Q}(I)}=\left\lbrace a/q \in I: 1 \le a \le q \le Q, \gcd (a,q)=\gcd (q,B-...

Digital signals are complex-valued functions on Z
n
. Signal sets with certain properties are required in various communication systems. Traditional signal sets consider only the time distortion during transmission. Recently, signal sets taking care of both the time and phase distortion have been studied, and are called time-phase signal sets. Seve...

Let $C$ be a smooth projective curve of genus $g \ge 1$ over a finite field $\F$ of cardinality $q$. In this paper, we first study $\#\J_C$, the size of the Jacobian of $C$ over $\F$ in case that $\F(C)/\F(X)$ is a geometric Galois extension. This improves results of Shparlinski \cite{shp}. Then we study fluctuations of the quantity $\log \#\J_C-g...

For each positive integer Q, let F Q denote the Farey sequence of order Q. We prove the existence of the pair correlation measure associated to the sum F Q + F Q modulo 1, as Q tends to infinity, and compute the corresponding limiting pair correlation function.

Let 𝔽q be a finite field of cardinality q and l ≥ 2 be a prime number such that q ≡ 1 (mod l). Extending the work of Faifman and Rudnick [6] on hyperelliptic curves, we study the distribution of zeros of zeta functions of curves over 𝔽q varying over the moduli spaces of cyclic l-fold covers of ℙ1(𝔽q) in the limit of large genus. The zeros all lie o...

Let l ≥ 2 be a positive integer, Fq a finite field of cardinality q with q ≡ 1 (mod l). In this paper, inspired by [6, 3, 4] and using a slightly different method, we study the fluctuations in the number of Fq-points on the curve CF given by the affine model CF : Y l = F(X), where F is drawn at random uniformly from the set of all monic l-th power-...

For the finite field Fq of q elements (q odd) and a quadratic non-residue (that is, a non-square) α ∈ Fq, we define the distance functionδ (u + v sqrt(α), x + y sqrt(α)) = frac((u - x)2 - α (v - y)2, v y) on the upper half plane Hq = {x + y sqrt(α) | x ∈ Fq, y ∈ Fq*} ⊆ Fq2. For two sets E, F ⊂ Hq with # E = E, # F = F and a non-trivial additive cha...

We prove the existence of the pair correlation measure associated to torsion points on the real locus E(R) of an elliptic curve E and provide an explicit formula for the limiting pair correlation function.

We prove two versions of the Erdos-Kac type theorem for polynomials of several variables on some varieties arising from translation and affine linear transformation.

Given a positive integer n n , a finite field F q \mathbb F_q of q q elements ( q q odd), and a non-degenerate quadratic form Q Q on F q n \mathbb {F}_q^n , in this paper we study the largest possible cardinality of subsets E ⊆ F q n \mathcal {E} \subseteq \mathbb {F}_q^n with pairwise integral Q Q -distances; that is, for any two vectors {x} = ( x...

We study the distribution of the sizes of the Selmer groups arising from the three 2-isogenies and their dual 2-isogenies for the elliptic curve E n : y2 = x3 - n2x. We show that three of them are almost always trivial, while the 2-rank of the other three follows a Gaussian distribution. It implies three almost always trivial Tate-Shafarevich group...

We study the distribution of the size of the Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with full 2-torsion points in Q. We show that one of these Selmer groups is almost always bounded, while the 2-rank of the other follows a Gaussian distribution. This provides us with a small Tate–Shafar...

We prove that the pair correlation of the sequence of rational numbers in the unit interval with prime denominators is Poissonian. The result is also true for rational numbers with denominators having exactly two distinct prime factors.

Inspired by Riemann’s work on certain quotients of the Dedekind Eta function, in this paper we investigate the value distribution of quotients of values of the Dedekind Eta function in the complex plane, using the form \({\frac{\eta(A_jz)} {\eta(A_{j-1}z)}}\) , where A
j-1 and A
j
are matrices whose rows are the coordinates of consecutive visible l...

In this paper we present a bias phenomenon on the behavior of Dedekind sums at visible points in a dilated region. Our results indicate that in more than three quarters of the time the Dedekind sum increases as one moves from one visible point to the next.

Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums \(\frac{1}{\varphi(N)} \sum_{\mathop{\mathop{ 0 \le m< N}}\limits_{\gcd(m,N)=1}} \vert S(m,N)\vert\), as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form \(A_{h}(Q)=\frac{1}{\sum_{\f...

We investigate local spacing problems along curves via smooth maps. Moreover we provide explicit formulas for the nearest neighbor spacing distribution function of torsion points on elliptic curves over R, and of rational points on the unit circle.

In this paper we determine all elliptic curves En:y2=x3−n2x with the smallest 2-Selmer groups Sn=Sel2(En(Q))={1} and Sn′=Sel2(En′(Q))={±1,±n}(En′:y2=x3+4n2x) based on the 2-descent method. The values of n for such curves En are described in terms of graph-theory language. It is well known that the rank of the group En(Q) for such curves En is zero,...

## Projects

Project (1)