arXiv subjects

Maosheng Xiong

Publications and source records attributed to Maosheng Xiong.

At least 37 records · Page 2Linked to original sources

arXiv2021

The Differential Spectrum of the Power Mapping $x^{p^n-3}$

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 the differential spectrum of $x^{p^n-3}$ with a unified approach. The obtained result shows that for any given odd prime $p$, the differential spectrum can be expressed explicitly in terms of $n$. Compared with previous results, a special elliptic curve over $\mathbb{F}_{p}$ plays an important role in our computation for the general case $p \ge 5$.

cs.IT↗

arXiv2020

Convergence Rate of Empirical Spectral Distribution of Random Matrices from Linear Codes

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 of the order $n^{-1/4}$ where $n$ is the length of the code.

math.PR↗

arXiv2020

Spectral distribution of random matrices from Mutually Unbiased Bases

We consider the random matrix obtained by picking vectors randomly from a large collection of mutually unbiased bases of $\mathbb{C}^n$, and prove that the spectral distribution converges to the Marchenko-Pastur law. This shows that vectors in mutually unbiased bases behave like random vectors. This phenomenon is similar to that of binary linear codes of dual distance at least 5, which was studied in previous work.

math.PR↗

arXiv2020

Random Matrices from Linear Codes and Wigner's Semicircle Law II

Recently we considered a class of random 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 the dual distance of the codes is at least 5. In this paper, employing more advanced techniques related to Stieltjes transform, we show that the dual distance being at least 5 is sufficient to ensure the convergence, and the convergence rate is of the form $n^{-β}$ for some $0 < β< 1$, where $n$ is the length of the code.

math.PR↗

arXiv2020

On permutation quadrinomials and $4$-uniform BCT

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.

cs.IT↗

arXiv2020

$4$-uniform BCT permutations from generalized butterfly structure

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 butterfly also yields permutations with the best boomerang uniformity, a new and important parameter related to boomerang-style attacks. This is the sixth known infinite family of permutations in the literature with the best known boomerang uniformity over such fields.

cs.IT↗

arXiv2019

Codes, differentially $δ$-uniform functions and $t$-designs

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. However, some linear codes hold $t$-designs, although they do not satisfy the conditions in the Assmus-Mattson Theorem and do not admit a $t$-transitive or $t$-homogeneous group as a subgroup of their automorphisms. The major objective of this paper is to develop a theory for explaining such codes and obtaining such new codes and hence new $t$-designs. To this end, a general theory for punctured and shortened codes of linear codes supporting $t$-designs is established, a generalized Assmus-Mattson theorem is developed, and a link between $2$-designs and differentially $δ$-uniform functions and $2$-designs is built. With these general results, binary codes with new parameters and known weight distributions are obtained, new $2$-designs and Steiner system $S(2, 4, 2^n)$ are produced in this paper.

cs.IT↗

arXiv2019

On the boomerang uniformity of (quadratic) permutations over $F_{2^n}$

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 (related to the BCT) for cryptographic Sboxes called boomerang uniformity. In this context, we present a brief state-of-the-art on the notion of boomerang uniformity of vectorial functions (or Sboxes) and provide new results. More specifically, we present a slightly different (and more convenient) formulation of the boomerang uniformity and show that the row sum and the column sum of the boomerang connectivity table can be expressed in terms of the zeros of the second-order derivative of the permutation or its inverse. Most importantly, we specialize our study of boomerang uniformity to quadratic permutations in even dimension and generalize the previous results on quadratic permutation with optimal BCT (optimal means that the maximal value in the Boomerang Connectivity Table equals the lowest known differential uniformity). As a consequence of our general result, we prove that the boomerang uniformity of the binomial differentially $4$-uniform permutations presented by Bracken, Tan, and Tan equals $4$. This result gives rise to a new family of optimal Sboxes.

cs.CR↗

arXiv2019

Steiner systems $S(2, 4, \frac{3^m-1}{2})$ and $2$-designs from ternary linear codes of length $\frac{3^m-1}{2}$

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-weight subcodes. These shortened and punctured codes contain some codes that have the same parameters as the best ternary linear codes known in the database maintained by Markus Grassl at http://www.codetables.de/. These three-weight subcodes with a special weight distribution do not satisfy the conditions of the Assmus-Mattson theorem and do not admit $2$-transitive or $2$-homogeneous automorphism groups in general. By employing the theory of projective geometries and projective generalized Reed-Muller codes, we prove that they still hold $2$-designs. We also determine the parameters of these $2$-designs. This paper mainly confirms some recent conjectures of Ding and Li regarding Steiner systems and $2$-designs from a special type of ternary projective codes.

cs.IT↗

arXiv2018

Cyclic bent functions and their applications in codes, codebooks, designs, MUBs and sequences

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 $ε\in \GF{}$, $f(ax_1,x_2)+f(bx_1,x_2+ε)$ is always bent, where $x_1\in \GF {m-1}, x_2 \in \GF {}$. Cyclic bent functions look extremely rare. This paper focuses on cyclic bent functions on $\GF {m-1} \times \GF {}$ and their applications. The first objective of this paper is to construct a new class of cyclic bent functions, which includes all known constructions of cyclic bent functions as special cases. The second objective is to use cyclic bent functions to construct good mutually unbiased bases (MUBs), codebooks and sequence families. The third objective is to study cyclic semi-bent functions and their applications. The fourth objective is to present a family of binary codes containing the Kerdock code as a special case, and describe their support designs. The results of this paper show that cyclic bent functions and cyclic semi-bent functions have nice applications in several fields such as symmetric cryptography, quantum physics, compressed sensing and CDMA communication.

cs.IT↗

arXiv2018

Random Matrices from Linear Codes and Wigner's semicircle law

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 is that the dual distance of the codes is at least 5. This is analogous to previous work on the empirical spectral distribution of similar matrices obtained in this fashion that converges to the Marchenko-Pastur law.

cs.IT↗

arXiv2018

Weight distribution of cyclic codes with arbitrary number of generalized Niho type zeroes with corrigendum

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 zeroes of generalized Niho type, more precisely $\ca$ (for $p=2$) of $t+1$ zeroes, and $\cb$ (for any prime $p$) of $t$ zeroes for any $t$. We find that the first family has at most $(2t+1)$ non-zero weights, and the second has at most $2t$ non-zero weights. Their weight distribution are also determined in the paper.

cs.IT↗

arXiv2017

On cyclic codes of composite length and the minimal distance

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 some of the mysteries of the numerical data by developing a general method on cyclic codes of composite length and on estimating the minimal distance. Inspired by the new method, we also provide a general construction of cyclic codes of composite length. Numerical data shows that it produces many best cyclic codes as well. Finally, we point out how these cyclic codes can be used to construct convolutional codes with large free distance.

cs.IT↗

arXiv2016

Narrow-Sense BCH Codes over $\gf(q)$ with Length $n=\frac{q^m-1}{q-1}$

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 $\gf(q)$ with length $n=(q^m-1)/(q-1)$. Little is known about this class of BCH codes when $q>2$. The objective of this paper is to study some of the codes within this class. In particular, the dimension, the minimum distance, and the weight distribution of some ternary BCH codes with length $n=(3^m-1)/2$ are determined in this paper. A class of ternary BCH codes meeting the Griesmer bound is identified. An application of some of the BCH codes in secret sharing is also investigated.

cs.IT↗

arXiv2015

Construction of Unit-Memory MDS Convolutional Codes

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 with previous works, the field size $q$ required to define these codes is much smaller. The construction also leads to many new strongly-MDS convolutional codes, an important subclass of MDS convolutional codes proposed and studied in \cite{GL2}. Many examples are presented at the end of the paper.

cs.IT↗

arXiv2015

The Weight Hierarchy of Some Reducible Cyclic Codes

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. This is achieved by extending the idea of \cite{YLFL} into higher dimension and by employing some interesting combinatorial arguments. It shall be noted that these cyclic codes may have arbitrary number of nonzeroes.

cs.IT↗

arXiv2015

Optimal cyclic codes with generalized Niho type zeroes and the weight distribution

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 some subtle manipulation of certain exponential sums.

cs.IT↗

arXiv2014

Weight Distributions of a Class of Cyclic Codes with Arbitrary Number of Zeros II

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 zeros. Recently, \cite{Y-X-D12} constructed a class of cyclic codes with arbitrary number of zeros, and computed the weight distributions for several cases. In this paper, we determine the weight distribution for a new family of such codes. This is achieved by certain new methods, such as the theory of Jacobi sums over finite fields and subtle treatment of some complicated combinatorial identities.

math.NT↗