arXiv subjects

Zhuqing Jia

Publications and source records attributed to Zhuqing Jia.

12 recordsLinked to original sources

arXiv2025

Robust Dynamic Coded Distributed Storage with Partially Storage Constrained Servers

We consider the problem of Robust Dynamic Coded Distributed Storage (RDCDS) with partially storage constrained servers where the goal is to enable robust (resilient to server dropouts) and efficient (as measured by the communication costs) read and update operations, subject to the constraint that the storage at $S$ out of $N$ servers is limited by $1/K_c$ the size of the message. Building upon previously established converse arguments and achievability schemes by Jia et al., in this work we develop a set of new converse arguments and coding designs that enable us to completely characterize the fundamental limits of RDCDS with partially storage constrained servers, i.e., the minimum number of available servers for feasible update operation and the minimum communication costs for read and update operations across various server dropout scenarios.

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arXiv2024

How to Read and Update Coded Distributed Storage Robustly and Optimally?

We consider the problem of robust dynamic coded distributed storage (RDCDS) that is associated with the coded distributed storage of a message with $N$ servers where 1) it suffices to recover the message from the storage at any $R_r$ servers; and 2) each of the servers stores a coded portion of the message that is at most $\frac{1}{K_c}$ the size of the message. The goal is to enable two main functionalities: the read operation and the update operation of the message. Specifically, at time slot $t$, the user may execute either the read operation or the update operation, where the read operation allows the user to recover the message from the servers, and the update operation allows the user to update the message to the servers in the form of an additive increment so that any up to $X^{(t)}$ colluding servers reveal nothing about the increment. The two functionalities are robust if at any time slot $t$ 1) they tolerate temporarily dropout servers up to certain thresholds (the read threshold is $R_r$ and the update threshold is denoted as $R_u^{(t)}$); and 2) the user may remain oblivious to prior server states. The communication efficiency is measured by the download cost $C_r^{(t)}$ of the read operation and the upload cost $C_u^{(t)}$ of the update operation. Given $K_c$ and $R_r$, we are curious about the optimal $(R_u^{(t)},C_r^{(t)},C_u^{(t)})$ tuple. In this work, we settle the fundamental limits of RDCDS. In particular, denoting the number of dropout servers at time slot $t$ as $|\mathcal{D}^{(t)}|$, we first show that 1) $R_u^{(t)}\geq N-R_r+\lceil K_c\rceil+X^{(t)}$; and 2) $C_r^{(t)}\geq \frac{N-|\mathcal{D}^{(t)}|}{N-R_r+\lceil K_c\rceil-|\mathcal{D}^{(t)}|}, C_u^{(t)}\geq \frac{N-|\mathcal{D}^{(t)}|}{R_r-X^{(t)}-|\mathcal{D}^{(t)}|}$. Then, inspired by the idea of staircase codes, we construct an RDCDS scheme that simultaneously achieves the above lower bounds.

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arXiv2023

The Asymptotic Capacity of $X$-Secure $T$-Private Linear Computation with Graph Based Replicated Storage

The problem of $X$-secure $T$-private linear computation with graph based replicated storage (GXSTPLC) is to enable the user to retrieve a linear combination of messages privately from a set of $N$ distributed servers where every message is only allowed to store among a subset of servers subject to an $X$-security constraint, i.e., any groups of up to $X$ colluding servers must reveal nothing about the messages. Besides, any groups of up to $T$ servers cannot learn anything about the coefficients of the linear combination retrieved by the user. In this work, we completely characterize the asymptotic capacity of GXSTPLC, i.e., the supremum of average number of desired symbols retrieved per downloaded symbol, in the limit as the number of messages $K$ approaches infinity. Specifically, it is shown that a prior linear programming based upper bound on the asymptotic capacity of GXSTPLC due to Jia and Jafar is tight by constructing achievability schemes. Notably, our achievability scheme also settles the exact capacity (i.e., for finite $K$) of $X$-secure linear combination with graph based replicated storage (GXSLC). Our achievability proof builds upon an achievability scheme for a closely related problem named asymmetric $\mathbf{X}$-secure $\mathbf{T}$-private linear computation with graph based replicated storage (Asymm-GXSTPLC) that guarantees non-uniform security and privacy levels across messages and coefficients. In particular, by carefully designing Asymm-GXSTPLC settings for GXSTPLC problems, the corresponding Asymm-GXSTPLC schemes can be reduced to asymptotic capacity achieving schemes for GXSTPLC. In regard to the achievability scheme for Asymm-GXSTPLC, interesting aspects of our construction include a novel query and answer design which makes use of a Vandermonde decomposition of Cauchy matrices, and a trade-off among message replication, security and privacy thresholds.

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arXiv2021

On the Capacity of Secure Distributed Batch Matrix Multiplication

The problem of secure distributed batch matrix multiplication (SDBMM) studies the communication efficiency of retrieving a sequence of desired matrix products ${\bf AB}$ $=$ $({\bf A}_1{\bf B}_1,$ ${\bf A}_2{\bf B}_2,$ $\cdots,$ ${\bf A}_S{\bf B}_S)$ from $N$ distributed servers where the constituent matrices ${\bf A}=({\bf A}_1, {\bf A}_2, \cdots, {\bf A}_S)$ and ${\bf B}=({\bf B}_1, {\bf B}_2,\cdots,{\bf B}_S)$ are stored in $X$-secure coded form, i.e., any group of up to $X$ colluding servers learn nothing about ${\bf A, B}$. It is assumed that ${\bf A}_s\in\mathbb{F}_q^{L\times K}, {\bf B}_s\in\mathbb{F}_q^{K\times M}, s\in\{1,2,\cdots, S\}$ are uniformly and independently distributed and $\mathbb{F}_q$ is a large finite field. The rate of an SDBMM scheme is defined as the ratio of the number of bits of desired information that is retrieved, to the total number of bits downloaded on average. The supremum of achievable rates is called the capacity of SDBMM. In this work we explore the capacity of SDBMM, as well as several of its variants, e.g., where the user may already have either ${\bf A}$ or ${\bf B}$ available as side-information, and/or where the security constraint for either ${\bf A}$ or ${\bf B}$ may be relaxed. We obtain converse bounds, as well as achievable schemes for various cases of SDBMM, depending on the $L, K, M, N, X$ parameters, and identify parameter regimes where these bounds match. A remarkable aspect of our upper bounds is a connection between SDBMM and a form of private information retrieval (PIR) problem, known as multi-message $X$-secure $T$-private information retrieval (MM-XSTPIR). Notable features of our achievable schemes include the use of cross-subspace alignment and a transformation argument that converts a scalar multiplication problem into a scalar addition problem, allowing a surprisingly efficient solution.

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arXiv2021

Price of Precision in Coded Distributed Matrix Multiplication: A Dimensional Analysis

Coded distributed matrix multiplication (CDMM) schemes, such as MatDot codes, seek efficient ways to distribute matrix multiplication task(s) to a set of $N$ distributed servers so that the answers returned from any $R$ servers are sufficient to recover the desired product(s). For example, to compute the product of matrices ${\bf U, V}$, MatDot codes partition each matrix into $p>1$ sub-matrices to create smaller coded computation tasks that reduce the upload/storage at each server by $1/p$, such that ${\bf UV}$ can be recovered from the answers returned by any $R=2p-1$ servers. An important concern in CDMM is to reduce the recovery threshold $R$ for a given storage/upload constraint. Recently, Jeong et al. introduced Approximate MatDot (AMD) codes that are shown to improve the recovery threshold by a factor of nearly $2$, from $2p-1$ to $p$. A key observation that motivates our work is that the storage/upload required for approximate computing depends not only on the dimensions of the (coded) sub-matrices that are assigned to each server, but also on their precision levels -- a critical aspect that is not explored by Jeong et al. Our main contribution is a dimensional analysis of AMD codes inspired by the Generalized Degrees of Freedom (GDoF) framework previously developed for wireless networks, which indicates that for the same upload/storage, once the precision levels of the task assignments are accounted for, AMD codes surprisingly fall short in all aspects to even the trivial replication scheme which assigns the full computation task to every server. Indeed, the trivial replication scheme has a much better recovery threshold of $1$, better download cost, better computation cost, and much better encoding/decoding (none required) complexity than AMD codes. The dimensional analysis is supported by simple numerical experiments.

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arXiv2021

$X$-Secure $T$-Private Federated Submodel Learning with Elastic Dropout Resilience

Motivated by recent interest in federated submodel learning, this work explores the fundamental problem of privately reading from and writing to a database comprised of $K$ files (submodels) that are stored across $N$ distributed servers according to an $X$-secure threshold secret sharing scheme. One after another, various users wish to retrieve their desired file, locally process the information and then update the file in the distributed database while keeping the identity of their desired file private from any set of up to $T$ colluding servers. The availability of servers changes over time, so elastic dropout resilience is required. The main contribution of this work is an adaptive scheme, called ACSA-RW, that takes advantage of all currently available servers to reduce its communication costs, fully updates the database after each write operation even though the database is only partially accessible due to server dropouts, and ensures a memoryless operation of the network in the sense that the storage structure is preserved and future users may remain oblivious of the past history of server dropouts. The ACSA-RW construction builds upon CSA codes that were originally introduced for XSTPIR and have been shown to be natural solutions for secure distributed matrix multiplication problems. ACSA-RW achieves the desired private read and write functionality with elastic dropout resilience, matches the best results for private-read from PIR literature, improves significantly upon available baselines for private-write, reveals a striking symmetry between upload and download costs, and exploits redundant storage dimensions to accommodate arbitrary read and write dropout servers up to certain threshold values. It also answers in the affirmative an open question by Kairouz et al. by exploiting synergistic gains from the joint design of private read and write operations.

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arXiv2021

Double Blind $T$-Private Information Retrieval

Double blind $T$-private information retrieval (DB-TPIR) enables two users, each of whom specifies an index ($θ_1, θ_2$, resp.), to efficiently retrieve a message $W(θ_1,θ_2)$ labeled by the two indices, from a set of $N$ servers that store all messages $W(k_1,k_2), k_1\in\{1,2,\cdots,K_1\}, k_2\in\{1,2,\cdots,K_2\}$, such that the two users' indices are kept private from any set of up to $T_1,T_2$ colluding servers, respectively, as well as from each other. A DB-TPIR scheme based on cross-subspace alignment is proposed in this paper, and shown to be capacity-achieving in the asymptotic setting of large number of messages and bounded latency. The scheme is then extended to $M$-way blind $X$-secure $T$-private information retrieval (MB-XS-TPIR) with multiple ($M$) indices, each belonging to a different user, arbitrary privacy levels for each index ($T_1, T_2,\cdots, T_M$), and arbitrary level of security ($X$) of data storage, so that the message $W(θ_1,θ_2,\cdots, θ_M)$ can be efficiently retrieved while the stored data is held secure against collusion among up to $X$ colluding servers, the $m^{th}$ user's index is private against collusion among up to $T_m$ servers, and each user's index $θ_m$ is private from all other users. The general scheme relies on a tensor-product based extension of cross-subspace alignment and retrieves $1-(X+T_1+\cdots+T_M)/N$ bits of desired message per bit of download.

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arXiv2020

GCSA Codes with Noise Alignment for Secure Coded Multi-Party Batch Matrix Multiplication

A secure multi-party batch matrix multiplication problem (SMBMM) is considered, where the goal is to allow a master to efficiently compute the pairwise products of two batches of massive matrices, by distributing the computation across S servers. Any X colluding servers gain no information about the input, and the master gains no additional information about the input beyond the product. A solution called Generalized Cross Subspace Alignment codes with Noise Alignment (GCSA-NA) is proposed in this work, based on cross-subspace alignment codes. The state of art solution to SMBMM is a coding scheme called polynomial sharing (PS) that was proposed by Nodehi and Maddah-Ali. GCSA-NA outperforms PS codes in several key aspects - more efficient and secure inter-server communication, lower latency, flexible inter-server network topology, efficient batch processing, and tolerance to stragglers. The idea of noise alignment can also be combined with N-source Cross Subspace Alignment (N-CSA) codes and fast matrix multiplication algorithms like Strassen's construction. Moreover, noise alignment can be applied to symmetric secure private information retrieval to achieve the asymptotic capacity.

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arXiv2019

Cross Subspace Alignment Codes for Coded Distributed Batch Computation

Coded distributed batch computation distributes a computation task, such as matrix multiplication, $N$-linear computation, or multivariate polynomial evaluation, across $S$ servers through a coding scheme, such that the response from any $R$ servers ($R$ is called the recovery threshold) is sufficient for the user to recover the desired computed value. Current approaches are based on either exclusively matrix-partitioning (Entangled Polynomial (EP) Codes for matrix multiplication), or exclusively batch processing (Lagrange Coded Computing (LCC)). We present three related classes of codes, based on the idea of Cross-Subspace Alignment (CSA) which was introduced originally in the context of private information retrieval. CSA codes are characterized by a Cauchy-Vandermonde matrix structure that facilitates interference alignment along Vandermonde terms, while the desired computations remain resolvable along the Cauchy terms. These codes unify, generalize and improve upon the state-of-art codes for distributed computing. First we introduce CSA codes for matrix multiplication, which yield LCC codes as a special case, and are shown to outperform LCC codes in general over strictly download-limited settings. Next, we introduce Generalized CSA (GCSA) codes for matrix multiplication that bridge the extremes of matrix-partitioning and batch processing approaches. Finally, we introduce $N$-CSA codes for $N$-linear distributed batch computations and multivariate batch polynomial evaluations. $N$-CSA codes include LCC codes as a special case, and are in general capable of achieving significantly lower downloads than LCC codes due to cross-subspace alignment. Generalizations of $N$-CSA codes to include $X$-secure data and $B$-byzantine servers are also obtained.

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arXiv2019

$X$-secure $T$-private Information Retrieval from MDS Coded Storage with Byzantine and Unresponsive Servers

The problem of $X$-secure $T$-private information retrieval from MDS coded storage is studied in this paper, where the user wishes to privately retrieve one out of $K$ independent messages that are distributed over $N$ servers according to an MDS code. It is guaranteed that any group of up to $X$ colluding servers learn nothing about the messages and that any group of up to $T$ colluding servers learn nothing about the identity of desired message. A lower bound of achievable rates is proved by presenting a novel scheme based on \emph{cross-subspace alignment} and a successive decoding with interference cancellation strategy. For large number of messages $(K\rightarrow\infty)$ the achieved rate, which we conjecture to be optimal, improves upon the best known rates previously reported in the literature by Raviv and Karpuk, and generalizes an achievable rate for MDS-TPIR previously found by Freij-Hollanti et al. that is also conjectured to be asymptotically optimal. The setting is then expanded to allow unresponsive and Byzantine servers. Finally, the scheme is applied to find a new lower convex hull of (download, upload) pairs of secure and private distributed matrix multiplication that generalizes, and in certain asymptotic settings strictly improves upon the best known previous results.

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arXiv2019

On the Asymptotic Capacity of $X$-Secure $T$-Private Information Retrieval with Graph Based Replicated Storage

The problem of private information retrieval with graph-based replicated storage was recently introduced by Raviv, Tamo and Yaakobi. Its capacity remains open in almost all cases. In this work the asymptotic (large number of messages) capacity of this problem is studied along with its generalizations to include arbitrary $T$-privacy and $X$-security constraints, where the privacy of the user must be protected against any set of up to $T$ colluding servers and the security of the stored data must be protected against any set of up to $X$ colluding servers. A general achievable scheme for arbitrary storage patterns is presented that achieves the rate $(ρ_{\min}-X-T)/N$, where $N$ is the total number of servers, and each message is replicated at least $ρ_{\min}$ times. Notably, the scheme makes use of a special structure inspired by dual Generalized Reed Solomon (GRS) codes. A general converse is also presented. The two bounds are shown to match for many settings, including symmetric storage patterns. Finally, the asymptotic capacity is fully characterized for the case without security constraints $(X=0)$ for arbitrary storage patterns provided that each message is replicated no more than $T+2$ times. As an example of this result, consider PIR with arbitrary graph based storage ($T=1, X=0$) where every message is replicated at exactly $3$ servers. For this $3$-replicated storage setting, the asymptotic capacity is equal to $2/ν_2(G)$ where $ν_2(G)$ is the maximum size of a $2$-matching in a storage graph $G[V,E]$. In this undirected graph, the vertices $V$ correspond to the set of servers, and there is an edge $uv\in E$ between vertices $u,v$ only if a subset of messages is replicated at both servers $u$ and $v$.

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arXiv2019

Cross Subspace Alignment and the Asymptotic Capacity of $X$-Secure $T$-Private Information Retrieval

$X$-secure and $T$-private information retrieval (XSTPIR) is a form of private information retrieval where data security is guaranteed against collusion among up to $X$ servers and the user's privacy is guaranteed against collusion among up to $T$ servers. The capacity of XSTPIR is characterized for arbitrary number of servers $N$, and arbitrary security and privacy thresholds $X$ and $T$, in the limit as the number of messages $K\rightarrow\infty$. Capacity is also characterized for any number of messages if either $N=3, X=T=1$ or if $N\leq X+T$. Insights are drawn from these results, about aligning versus decoding noise, dependence of PIR rate on field size, and robustness to symmetric security constraints. In particular, the idea of cross subspace alignment, i.e., introducing a subspace dependence between Reed-Solomon code parameters, emerges as the optimal way to align undesired terms while keeping desired terms resolvable.

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