arXiv subjects

Chaoping Xing

Publications and source records attributed to Chaoping Xing.

At least 37 records · Page 2Linked to original sources

arXiv2022

Binary sequences with a low correlation via cyclotomic function fields with odd characteristics

Sequences with a low correlation have very important applications in communications, cryptography, and compressed sensing. In the literature, many efforts have been made to construct good sequences with various lengths where binary sequences attracts great attention. As a result, various constructions of good binary sequences have been proposed. However, most of the known constructions made use of the multiplicative cyclic group structure of finite field $\mathbb{F}_{p^n}$ for a prime $p$ and a positive integer $n$. In fact, all $p^n+1$ rational places including the place at infinity of the rational function field over $\mathbb{F}_{p^n}$ form a cyclic structure under an automorphism of order $p^n+1$. In this paper, we make use of this cyclic structure to provide an explicit construction of binary sequences with a low correlation of length $p^n+1$ via cyclotomic function fields over $\mathbb{F}_{p^n}$ for any odd prime $p$. Each family of binary sequences has size $p^n-2$ and its correlation is upper bounded by $4+\lfloor 2\cdot p^{n/2}\rfloor$. To the best of our knowledge, this is the first construction of binary sequences with a low correlation of length $p^n+1$ for odd prime $p$. Moreover, our sequences can be constructed explicitly and have competitive parameters.

cs.IT↗

arXiv2022

A new construction of nonlinear codes via algebraic function fields

In coding theory, constructing codes with good parameters is one of the most important and fundamental problems. Though a great many of good codes have been produced, most of them are defined over alphabets of sizes equal to prime powers. In this paper, we provide a new explicit construction of $(q+1)$-ary nonlinear codes via algebraic function fields, where $q$ is a prime power. Our codes are constructed by evaluations of rational functions at all rational places of the algebraic function field. Compared with algebraic geometry codes, the main difference is that we allow rational functions to be evaluated at pole places. After evaluating rational functions from a union of Riemann-Roch spaces, we obtain a family of nonlinear codes over the alphabet $\mathbb{F}_{q}\cup \{\infty\}$. It turns out that our codes have better parameters than those obtained from MDS codes or good algebraic geometry codes via code alphabet extension and restriction.

cs.IT↗

arXiv2022

Good locally repairable codes via propagation rules

In classical coding theory, it is common to construct new codes via propagation rules. There are various propagation rules to construct classical block codes. However, propagation rules have not been extensively explored for constructions of locally repairable codes. In this paper, we introduce a few propagation rules to construct good locally repairable codes. To our surprise, these simple propagation rules produce a few interesting results. Firstly, by concatenating a locally repairable code as an inner code with a classical block code as an outer code, we obtain quite a few dimension-optimal binary locally repairable codes. Secondly, from this concatenation, we explicitly build a family of locally repairable codes that exceeds the Zyablov-type bound. Thirdly, by a lengthening propagation rule that adds some rows and columns from a parity-check matrix of a given linear code, we are able to produce a family of dimension-optimal binary locally repairable codes from the extended Hamming codes, and to convert a classical maximum distance separable (MDS) code into a Singleton-optimal locally repairable code. Furthermore, via the lengthening propagation rule, we greatly simplify the construction of a family of locally repairable codes in \cite[Theorem 5]{MX20} that breaks the asymptotic Gilbert-Varshamov bound. In addition, we make use of three other propagation rules to produce more dimension-optimal binary locally repairable codes. Finally, one of phenomena that we observe in this paper is that some trivial propagation rules in classical block codes do not hold anymore for locally repairable codes.

cs.IT↗

arXiv2022

Upper bounds on maximum lengths of Singleton-optimal locally repairable codes

A locally repairable code is called Singleton-optimal if it achieves the Singleton-type bound. Such codes are of great theoretic interest in the study of locally repairable codes. In the recent years there has been a great amount of work on this topic. One of the main problems in this topic is to determine the largest length of a q-ary Singleton-optimal locally repairable code for given locality and minimum distance. Unlike classical MDS codes, the maximum length of Singleton? Optimal locally repairable codes are very sensitive to minimum distance and locality. Thus, it is more challenging and complicated to investigate the maximum length of Singleton-optimal locally repairable codes. In literature, there has been already some research on this problem. However, most of work is concerned with some specific parameter regime such as small minimum distance and locality, and rely on the constraint that (r + 1)|n and recovery sets are disjoint, where r is locality and n is the code length. In this paper we study the problem for large range of parameters including the case where minimum distance is proportional to length. In addition, we also derive some upper bounds on the maximum length of Singleton-optimal locally repairable codes with small minimum distance by removing this constraint. It turns out that even without the constraint we still get better upper bounds for codes with small locality and distance compared with known results. Furthermore, based on our upper bounds for codes with small distance and locality and some propagation rule that we propose in this paper, we are able to derive some upper bounds for codes with relatively large distance and locality assuming that (r + 1)|n and recovery sets are disjoint.

cs.IT↗

arXiv2021

Bounds and Constructions for Insertion and Deletion Codes

The present paper mainly studies limits and constructions of insertion and deletion (insdel for short) codes. The paper can be divided into two parts. The first part focuses on various bounds, while the second part concentrates on constructions of insdel codes. Although the insdel-metric Singleton bound has been derived before, it is still unknown if there are any nontrivial codes achieving this bound. Our first result shows that any nontrivial insdel codes do not achieve the insdel-metric Singleton bound. The second bound shows that every $[n,k]$ Reed-Solomon code has insdel distance upper bounded by $2n-4k+4$ and it is known in literature that an $[n,k]$ Reed-Solomon code can have insdel distance $2n-4k+4$ as long as the field size is sufficiently large. The third bound shows a trade-off between insdel distance and code alphabet size for codes achieving the Hamming-metric Singleton bound. In the second part of the paper, we first provide a non-explicit construction of nonlinear codes that can approach the insdel-metric Singleton bound arbitrarily when the code alphabet size is sufficiently large. The second construction gives two-dimensional Reed-Solomon codes of length $n$ and insdel distance $2n-4$ with field size $q=O(n^5)$.

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arXiv2021

Maximally recoverable local reconstruction codes from subspace direct sum systems

Maximally recoverable local reconstruction codes (MR LRCs for short) have received great attention in the last few years. Various constructions have been proposed in literatures. The main focus of this topic is to construct MR LRCs over small fields. An $(N=nr,r,h,\Gd)$-MR LRC is a linear code over finite field $\F_\ell$ of length $N$, whose codeword symbols are partitioned into $n$ local groups each of size $r$. Each local group can repair any $\Gd$ erasure errors and there are further $h$ global parity checks to provide fault tolerance from more global erasure patterns. MR LRCs deployed in practice have a small number of global parities such as $h=O(1)$. In this parameter setting, all previous constructions require the field size $\ell =Ω_h (N^{h-1-o(1)})$. It remains challenging to improve this bound. In this paper, via subspace direct sum systems, we present a construction of MR LRC with the field size $\ell= O(N^{h-2+\frac1{h-1}-o(1)})$. In particular, for the most interesting cases where $h=2,3$, we improve previous constructions by either reducing field size or removing constraints. In addition, we also offer some constructions of MR LRCs for larger global parity $h$ that have field size incomparable with known upper bounds. The main techniques used in this paper is through subspace direct sum systems that we introduce. Interestingly, subspace direct sum systems are actually equivalent to $\F_q$-linear codes over extension fields. Based on various constructions of subspace direct sum systems, we are able to construct several classes of MR LRCs.

cs.IT↗

arXiv2021

Binary sequences with low correlation via cyclotomic function fields

Due to wide applications of binary sequences with low correlation to communications, various constructions of such sequences have been proposed in literature. However, most of the known constructions via finite fields make use of the multiplicative cyclic group of $\F_{2^n}$. It is often overlooked in this community that all $2^n+1$ rational places (including "place at infinity") of the rational function field over $\F_{2^n}$ form a cyclic structure under an automorphism of order $2^n+1$. In this paper, we make use of this cyclic structure to provide an explicit construction of families of binary sequences of length $2^n+1$ via the finite field $\F_{2^n}$. Each family of sequences has size $2^n-1$ and its correlation is upper bounded by $\lfloor 2^{(n+2)/2}\rfloor$. Our sequences can be constructed explicitly and have competitive parameters. In particular, compared with the Gold sequences of length $2^n-1$ for even $n$, we have larger length and smaller correlation although the family size of our sequences is slightly smaller.

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arXiv2021

Leakage-Resilient Secret Sharing with Constant Share Size

We consider the leakage resilience of AG code-based ramp secret sharing schemes extending the leakage resilience of linear threshold secret sharing schemes over prime fields done by Benhamouda et al. Since there is not any explicit efficient construction of AG codes over prime fields, we consider constructions over prime fields with the help of concatenation method and those over field extensions. Extending the Fourier analysis done by Benhamouda et al., concatenated algebraic geometric codes over prime fields do produce some nice leakage-resilient secret sharing schemes. One natural and curious question is whether AG codes over extension fields produce better leakage-resilient secret sharing schemes than the construction based on concatenated AG codes. Such construction provides several advantages compared to the construction over prime fields using concatenation method. First, AG codes over extension fields give secret sharing schemes with smaller reconstruction for a fixed privacy parameter t. Second, concatenated AG codes do not enjoy strong multiplicity and hence they are not applicable to secure MPC schemes. It is also confirmed that indeed AG codes over extension fields have stronger leakage-resilience under some reasonable assumptions. These three advantages strongly motivate the study of secret sharing schemes from AG codes over extension fields. The current paper has two main contributions: 1, we obtain leakage-resilient secret sharing schemes with constant share sizes and unbounded numbers of players. Like Shamir secret scheme, our schemes enjoy multiplicity and hence can be applied to MPC. 2, via a sophisticated Fourier Analysis, we analyze the leakage-resilience of secret sharing schemes from codes over extension fields. This is of its own theoretical interest independent of its application to secret sharing schemes from algebraic geometric codes over extension fields.

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arXiv2021

Secure Weighted Aggregation for Federated Learning

The pervasive adoption of Internet-connected digital services has led to a growing concern in the personal data privacy of their customers. On the other hand, machine learning (ML) techniques have been widely adopted by digital service providers to improve operational productivity and customer satisfaction. ML inevitably accesses and processes users' personal data, which could potentially breach the relevant privacy protection regulations if not performed carefully. The situation is exacerbated by the cloud-based implementation of digital services when user data are captured and stored in distributed locations, hence aggregation of the user data for ML could be a serious breach of privacy regulations. In this backdrop, Federated Learning (FL) is an emerging area that allows ML on distributed data without the data leaving their stored location. However, depending on the nature of the digital services, data captured at different locations may carry different significance to the business operation, hence a weighted aggregation will be highly desirable for enhancing the quality of the FL-learned model. Furthermore, to prevent leakage of user data from the aggregated gradients, cryptographic mechanisms are needed to allow secure aggregation of FL. In this paper, we propose a privacy-enhanced FL scheme for supporting secure weighted aggregation. Besides, by devising a verification protocol based on Zero-Knowledge Proof (ZKP), the proposed scheme is capable of guarding against fraudulent messages from FL participants. Experimental results show that our scheme is practical and secure. Compared to existing FL approaches, our scheme achieves secure weighted aggregation with an additional security guarantee against fraudulent messages with an affordable 1.2 times runtime overheads and 1.3 times communication costs.

cs.CR↗

arXiv2021

An improvement of the Hasse-Weil bound for Artin-Schreier curves via cyclotomic function fields

The corresponding Hasse-Weil bound was a major breakthrough in history of mathematics. It has found many applications in mathematics, coding theory and theoretical computer science. In general, the Hasse-Weil bound is tight and cannot be improved. However, the Hasse-Weil bound is no longer tight when it is applied to some specific classes of curves. One of the examples where the Hasse-Weil bound is not tight is the family of Artin-Schreier curves. Due to various applications of Artin-Schreier curves to coding, cryptography and theoretical computer science, researchers have made great effort to improve the Hasse-Weil bound for Artin-Schreier curves. In this paper, we focus on the number of rational places of the Artin-Schreier curve defined by $y^p-y=f(x)$ over the finite field $\mathbb{F}_q$ of characteristic $p$, where $f(x)$ is a polynomial in $\mathbb{F}_q[x]$. Our road map for attacking this problem works as follows. We first show that the function field $E_f:=\mathbb{F}_q(x,y)$ of the Artin-Schreier curve $y^p-y=f(x)$ is a subfield of some cyclotomic function field. We then make use of the class field theory to prove that the number of points of the curve is upper bounded by a function of a minimum distance of a linear code. By analyzing the minimum distance of this linear code, we can improve the Hasse-Weil bound and Serre bound for Artin-Schreier curves.

math.NT↗

arXiv2021

MPC-enabled Privacy-Preserving Neural Network Training against Malicious Attack

The application of secure multiparty computation (MPC) in machine learning, especially privacy-preserving neural network training, has attracted tremendous attention from the research community in recent years. MPC enables several data owners to jointly train a neural network while preserving the data privacy of each participant. However, most of the previous works focus on semi-honest threat model that cannot withstand fraudulent messages sent by malicious participants. In this paper, we propose an approach for constructing efficient $n$-party protocols for secure neural network training that can provide security for all honest participants even when a majority of the parties are malicious. Compared to the other designs that provide semi-honest security in a dishonest majority setting, our actively secure neural network training incurs affordable efficiency overheads of around 2X and 2.7X in LAN and WAN settings, respectively. Besides, we propose a scheme to allow additive shares defined over an integer ring $\mathbb{Z}_N$ to be securely converted to additive shares over a finite field $\mathbb{Z}_Q$, which may be of independent interest. Such conversion scheme is essential in securely and correctly converting shared Beaver triples defined over an integer ring generated in the preprocessing phase to triples defined over a field to be used in the calculation in the online phase.

cs.CR↗

arXiv2020

Optimal rate list decoding over bounded alphabets using algebraic-geometric codes

We give new constructions of two classes of algebraic code families which are efficiently list decodable with small output list size from a fraction $1-R-ε$ of adversarial errors where $R$ is the rate of the code, for any desired positive constant $ε$. The alphabet size depends only $ε$ and is nearly-optimal. The first class of codes are obtained by folding algebraic-geometric codes using automorphisms of the underlying function field. The list decoding algorithm is based on a linear-algebraic approach, which pins down the candidate messages to a subspace with a nice "periodic" structure. The list is pruned by precoding into a special form of "subspace-evasive" sets, which are constructed pseudorandomly. Instantiating this construction with the Garcia-Stichtenoth function field tower yields codes list-decodable up to a $1-R-ε$ error fraction with list size bounded by $O(1/ε)$, matching the existential bound up to constant factors. The parameters we achieve are thus quite close to the existential bounds in all three aspects: error-correction radius, alphabet size, and list-size. The second class of codes are obtained by restricting evaluation points of an algebraic-geometric code to rational points from a subfield. Once again, the linear-algebraic approach to list decoding to pin down candidate messages to a periodic subspace. We develop an alternate approach based on "subspace designs" to precode messages. Together with the subsequent explicit constructions of subspace designs, this yields a deterministic construction of an algebraic code family of rate $R$ with efficient list decoding from $1-R-ε$ fraction of errors over a constant-sized alphabet. The list size is bounded by a very slowly growing function of the block length $N$; in particular, it is at most $O(\log^{(r)} N)$ (the $r$'th iterated logarithm) for any fixed integer $r$.

cs.CC↗

arXiv2020

On List Decoding of Insertion and Deletion Errors

Insdel errors occur in communication systems caused by the loss of positional information of the message. Since the work by Guruswami and Wang, there have been some further investigations on the list decoding of insertion codes, deletion codes and insdel codes. However, unlike classical Hamming metric or even rank-metric, there are still many unsolved problems on list decoding of insdel codes. The contributions of this paper mainly consist of two parts. Firstly, we analyze the list decodability of random insdel codes. We show that list decoding of random insdel codes surpasses the Singleton bound when there are more insertion errors than deletion errors and the alphabet size is sufficiently large. Furthermore, our results reveal the existence of an insdel code that can be list decoded against insdel errors beyond its minimum insdel distance while still having polynomial list size. This provides a more complete picture on the list decodability of insdel codes when both insertion and deletion errors happen. Secondly, we construct a family of explicit insdel codes with efficient list decoding algorithm. As a result, we derive a Zyablov-type bound for insdel errors. Recently, after our results appeared, Guruswami et al. provided a complete solution for another open problem on list decoding of insdel codes. In contrast to the problems we considered, they provided a region containing all possible insertion and deletion errors that are still list decodable by some q-ary insdel codes of non-zero rate. More specifically, for a fixed number of insertion and deletion errors, while our paper focuses on maximizing the rate of a code that is list decodable against that amount of insertion and deletion errors, Guruswami et al. focuses on finding out the existence of a code with asymptotically non-zero rate which is list decodable against this amount of insertion and deletion errors.

cs.IT↗

arXiv2020

The group structures of automorphism groups of elliptic function fields over finite fields and their applications to optimal locally repairable codes

The automorphism group of an elliptic curve over an algebraically closed field is well known. However, for various applications in coding theory and cryptography, we usually need to apply automorphisms defined over a finite field. Although we believe that the automorphism group of an elliptic curve over a finite field is well known in the community, we could not find this in the literature. Nevertheless, in this paper we show the group structure of the automorphism group of an elliptic curve over a finite field. More importantly, we characterize subgroups and abelian subgroups of the automorphism group of an elliptic curve over a finite field. Despite of theoretical interest on this topic, our research is largely motivated by constructions of optimal locally repairable codes. The first research to make use of automorphism group of function fields to construct optimal locally repairable codes was given in a paper \cite{JMX20} where automorphism group of a projective line was employed. The idea was further generated to an elliptic curve in \cite{MX19} where only automorphisms fixing the point at infinity were used. Because there are at most $24$ automorphisms of an elliptic curve fixing the point at infinity, the locality of optimal locally repairable codes from this construction is upper bounded by $23$. One of the main motivation to study subgroups and abelian subgroups of the automorphism group of an elliptic curve over a finite field is to remove the constraints on locality.

math.NT↗

arXiv2020

A Construction of New Quantum MDS Codes

It has been a great challenge to construct new quantum MDS codes. In particular, it is very hard to construct quantum MDS codes with relatively large minimum distance. So far, except for some sparse lengths, all known $q$-ary quantum MDS codes have minimum distance less than or equal to $q/2+1$. In the present paper, we provide a construction of quantum MDS codes with minimum distance bigger than $q/2+1$. In particular, we show existence of $q$-ary quantum MDS codes with length $n=q^2+1$ and minimum distance $d$ for any $d\le q-1$ and $d= q+1$(this result extends those given in \cite{Gu11,Jin1,KZ12}); and with length $(q^2+2)/3$ and minimum distance $d$ for any $d\le (2q+2)/3$ if $3|(q+1)$. Our method is through Hermitian self-orthogonal codes. The main idea of constructing Hermitian self-orthogonal codes is based on the solvability in $\F_q$ of a system of homogenous equations over $\F_{q^2}$.

cs.IT↗

arXiv2020

Secure Federated Transfer Learning

Machine learning relies on the availability of a vast amount of data for training. However, in reality, most data are scattered across different organizations and cannot be easily integrated under many legal and practical constraints. In this paper, we introduce a new technique and framework, known as federated transfer learning (FTL), to improve statistical models under a data federation. The federation allows knowledge to be shared without compromising user privacy, and enables complimentary knowledge to be transferred in the network. As a result, a target-domain party can build more flexible and powerful models by leveraging rich labels from a source-domain party. A secure transfer cross validation approach is also proposed to guard the FTL performance under the federation. The framework requires minimal modifications to the existing model structure and provides the same level of accuracy as the non-privacy-preserving approach. This framework is very flexible and can be effectively adapted to various secure multi-party machine learning tasks.

cs.LG↗

arXiv2020

Symmetric uncoded caching schemes with low subpacketization levels

Caching is a commonly used technique in content-delivery networks which aims to deliver information from hosting servers to users in the most efficient way. In 2014, Maddah-Ali and Niessen formulated caching into a formal information theoretic problem and it has gained a lot of attention since then. It is known that the caching schemes proposed by Ali-Niesen and Yu et. al. are optimal, that is, they require the least number of transmissions from the server to satisfy all users' demands. However for these schemes to work, each file needs to be partitioned into $F^*$ subfiles ($F^*$ is called the subpacketization level of files) with $F^*$ growing exponentially in the number $K$ of users. As a result, it is problematic to apply these schemes in practical situations, where $K$ tends to be very large. There rise the following questions: (1) are there optimal schemes in which each file is partitioned into $F$ subfiles, where $F$ is not exponential, say polynomial for example, in $K$? (2) if the answer to this question is no, is there a near-optimal scheme, a scheme which is as asymptotically good as the one in \cite{ali1,yu}, with $F$ polynomial in $K$? Both these questions are open. Our main contribution in this paper is to provide answers to above questions. Firstly, we prove that under some mild restriction on user's cache rate, there are no optimal schemes with $F$ smaller than $F^*$. Moreover, we give necessary and sufficient conditions for the existence of optimal schemes in this case. Secondly, we provide an affirmative answer to the second question raised above by an explicit construction and a detailed performance analysis.

cs.IT↗

arXiv2019

A new construction of nonlinear codes via rational function fields

It is well known that constructing codes with good parameters is one of the most important and fundamental problems in coding theory. Though a great many of good codes have been produced, most of them are defined over alphabets of sizes equal to prime powers. In this paper, we provide a new explicit construction of (q+1)-ary nonlinear codes via rational function fields, where q is a prime power. Our codes are constructed by evaluations of rational functions at all the rational places (including the place of "infinity") of the rational function field. Compared to the rational algebraic geometry codes, the main difference is that we allow rational functions to be evaluated at pole places. After evaluating rational functions from a union of Riemann-Roch spaces, we obtain a family of nonlinear codes with length q+1 over the alphabet $\mathbb{F}_{q}\cup \{\infty\}$. As a result, our codes have reasonable parameters as they are very close to the Singleton bound. Furthermore, our codes have better parameters than those obtained from MDS codes via code alphabet restriction or extension.

cs.IT↗