arXiv subjects

Syed A. Jafar

Publications and source records attributed to Syed A. Jafar.

At least 37 records · Page 2Linked to original sources

arXiv2025

On the Utility of Quantum Entanglement for Joint Communication and Instantaneous Detection

Entanglement is known to significantly improve the performance (separately) of communication and detection schemes that utilize quantum resources. This work explores the simultaneous utility of quantum entanglement for (joint) communication and detection schemes, over channels that are convex combinations of identity, depolarization and erasure operators, both with perfect and imperfect entanglement assistance. The channel state is binary, rapidly time-varying and unknown to the transmitter. While the communication is delay-tolerant, allowing the use of arbitrarily long codewords to ensure reliable decoding, the channel state detection is required to be instantaneous. The detector is neither co-located with the transmitter, nor able to wait for the decoding in order to learn the transmitted waveform. The results of this work appear in the form of communication-rate vs instantaneous-detection-error tradeoffs, with and without quantum entanglement. Despite the challenges that place the two tasks at odds with each other, the results indicate that quantum entanglement can indeed be simultaneously and significantly beneficial for joint communication and instantaneous detection.

quant-ph↗

arXiv2024

The Capacity of 3 User Linear Computation Broadcast

The $K$ User Linear Computation Broadcast (LCBC) problem is comprised of $d$ dimensional data (from $\mathbb{F}_q$), that is fully available to a central server, and $K$ users, who require various linear computations of the data, and have prior knowledge of various linear functions of the data as side-information. The optimal broadcast cost is the minimum number of $q$-ary symbols to be broadcast by the server per computation instance, for every user to retrieve its desired computation. The reciprocal of the optimal broadcast cost is called the capacity. The main contribution of this paper is the exact capacity characterization for the $K=3$ user LCBC for all cases, i.e., for arbitrary finite fields $\mathbb{F}_q$, arbitrary data dimension $d$, and arbitrary linear side-informations and demands at each user. A remarkable aspect of the converse is that unlike the $2$ user LCBC whose capacity was determined previously, the entropic formulation (where the entropies of demands and side-informations are specified, but not their functional forms) is insufficient to obtain a tight converse for the $3$ user LCBC. Instead, the converse exploits functional submodularity. Notable aspects of achievability include a decomposition of the users' collective signal space into subspaces that allow different degrees of efficiency in broadcast cost, revealing a tradeoff that leads to a constrained water-filling solution. Random coding arguments are invoked to resolve compatibility issues that arise as each user has a different view of these subspaces, conditioned on its own side-information.

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arXiv2024

The Capacity of Classical Summation over a Quantum MAC with Arbitrarily Distributed Inputs and Entanglements

The $Σ$-QMAC problem is introduced, involving $S$ servers, $K$ classical ($\mathbb{F}_d$) data streams, and $T$ independent quantum systems. Data stream ${\sf W}_k, k\in[K]$ is replicated at a subset of servers $\mathcal{W}(k)\subset[S]$, and quantum system $\mathcal{Q}_t, t\in[T]$ is distributed among a subset of servers $\mathcal{E}(t)\subset[S]$ such that Server $s\in\mathcal{E}(t)$ receives subsystem $\mathcal{Q}_{t,s}$ of $\mathcal{Q}_t=(\mathcal{Q}_{t,s})_{s\in\mathcal{E}(t)}$. Servers manipulate their quantum subsystems according to their data and send the subsystems to a receiver. The total download cost is $\sum_{t\in[T]}\sum_{s\in\mathcal{E}(t)}\log_d|\mathcal{Q}_{t,s}|$ qudits, where $|\mathcal{Q}|$ is the dimension of $\mathcal{Q}$. The states and measurements of $(\mathcal{Q}_t)_{t\in[T]}$ are required to be separable across $t\in[T]$ throughout, but for each $t\in[T]$, the subsystems of $\mathcal{Q}_{t}$ can be prepared initially in an arbitrary (independent of data) entangled state, manipulated arbitrarily by the respective servers, and measured jointly by the receiver. From the measurements, the receiver must recover the sum of all data streams. Rate is defined as the number of dits ($\mathbb{F}_d$ symbols) of the desired sum computed per qudit of download. The capacity of $Σ$-QMAC, i.e., the supremum of achievable rates is characterized for arbitrary data replication and entanglement distribution maps $\mathcal{W}, \mathcal{E}$. Coding based on the $N$-sum box abstraction is optimal in every case. Notably, for every $S\neq 3$ there exists an instance of the $Σ$-QMAC where $S$-party entanglement is necessary to achieve the fully entangled capacity.

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arXiv2022

The Extremal GDoF Gain of Optimal versus Binary Power Control in $K$ User Interference Networks Is $Θ(\sqrt{K})$

Using ideas from Generalized Degrees of Freedom (GDoF) analyses and extremal network theory, this work studies the extremal gain of optimal power control over binary (on/off) power control, especially in large interference networks, in search of new theoretical insights. Whereas numerical studies have already established that in most practical settings binary power control is close to optimal, the extremal analysis shows not only that there exist settings where the gain from optimal power control can be quite significant, but also bounds the extremal values of such gains from a GDoF perspective. As its main contribution, this work explicitly characterizes the extremal GDoF gain of optimal over binary power control as $Θ\left(\sqrt{K}\right)$ for all $K$. In particular, the extremal gain is bounded between $\lfloor \sqrt{K}\rfloor$ and $2.5\sqrt{K}$ for every $K$. For $K=2,3,4,5,6$ users, the precise extremal gain is found to be $1, 3/2, 2, 9/4$ and $41/16$, respectively. Networks shown to achieve the extremal gain may be interpreted as multi-tier heterogeneous networks. It is worthwhile to note that because of their focus on asymptotic analysis, the sharp characterizations of extremal gains are valuable primarily from a theoretical perspective, and not as contradictions to the conventional wisdom that binary power control is generally close to optimal in practical, non-asymptotic settings.

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arXiv2021

Flexible Distributed Matrix Multiplication

The distributed matrix multiplication problem with an unknown number of stragglers is considered, where the goal is to efficiently and flexibly obtain the product of two massive matrices by distributing the computation across N servers. There are up to N - R stragglers but the exact number is not known a priori. Motivated by reducing the computation load of each server, a flexible solution is proposed to fully utilize the computation capability of available servers. The computing task for each server is separated into several subtasks, constructed based on Entangled Polynomial codes by Yu et al. The final results can be obtained from either a larger number of servers with a smaller amount of computation completed per server or a smaller number of servers with a larger amount of computation completed per server. The required finite field size of the proposed solution is less than 2N. Moreover, the optimal design parameters such as the partitioning of the input matrices is discussed. Our constructions can also be generalized to other settings such as batch distributed matrix multiplication and secure distributed matrix multiplication.

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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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arXiv2021

Multilevel Topological Interference Management: A TIM-TIN Perspective

The robust principles of treating interference as noise (TIN) when it is sufficiently weak, and avoiding it when it is not, form the background of this work. Combining TIN with the topological interference management (TIM) framework that identifies optimal interference avoidance schemes, we formulate a TIM-TIN problem for multilevel topological interference management, wherein only a coarse knowledge of channel strengths and no knowledge of channel phases is available to transmitters. To address the TIM-TIN problem, we first propose an analytical baseline approach, which decomposes a network into TIN and TIM components, allocates the signal power levels to each user in the TIN component, allocates signal vector space dimensions to each user in the TIM component, and guarantees that the product of the two is an achievable number of signal dimensions available to each user in the original network. Next, a distributed numerical algorithm called ZEST is developed. The convergence of the algorithm is demonstrated, leading to the duality of the TIM-TIN problem (in terms of GDoF). Numerical results are also provided to demonstrate the superior sum-rate performance and fast convergence of ZEST.

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arXiv2020

Secure GDoF of the Z-channel with Finite Precision CSIT: How Robust are Structured Codes?

Under the assumption of perfect channel state information at the transmitters (CSIT), it is known that structured codes offer significant advantages for secure communication in an interference network, e.g., structured jamming signals based on lattice codes may allow a receiver to decode the sum of the jamming signal and the signal being jammed, even though they cannot be separately resolved due to secrecy constraints, subtract the aggregate jammed signal, and then proceed to decode desired codewords at lower power levels. To what extent are such benefits of structured codes fundamentally limited by uncertainty in CSIT? To answer this question, we explore what is perhaps the simplest setting where the question presents itself -- a Z interference channel with secure communication. Using sum-set inequalities based on Aligned Images bounds we prove that the GDoF benefits of structured codes are lost completely under finite precision CSIT. The secure GDoF region of the Z interference channel is obtained as a byproduct of the analysis.

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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

Towards an Extremal Network Theory -- Robust GDoF Gain of Transmitter Cooperation over TIN

Significant progress has been made recently in Generalized Degrees of Freedom (GDoF) characterizations of wireless interference channels (IC) and broadcast channels (BC) under the assumption of finite precision channel state information at the transmitters (CSIT), especially for smaller or highly symmetric network settings. A critical barrier in extending these results to larger and asymmetric networks is the inherent combinatorial complexity of such networks. Motivated by other fields such as extremal combinatorics and extremal graph theory, we explore the possibility of an extremal network theory, i.e., a study of extremal networks within particular regimes of interest. As our test application, we study the GDoF benefits of transmitter cooperation in a $K$ user IC over the simple scheme of power control and treating interference as Gaussian noise (TIN) for three regimes of interest -- a TIN regime where TIN was shown to be GDoF optimal for the $K$ user interference channel, a CTIN regime where the GDoF region achievable by TIN is convex without time-sharing, and an SLS regime where a simple layered superposition (SLS) scheme is shown to be optimal in the $K$ user MISO BC, albeit only for $K\leq 3$. As our first result, we show that under finite precision CSIT, TIN is GDoF optimal for the $K$ user IC throughout the CTIN regime. Furthermore, under finite precision CSIT, appealing to extremal network theory we obtain the following results. In both TIN and CTIN regimes, we show that the extremal GDoF gain from transmitter cooperation over TIN is bounded regardless of the number of users: the gain is exactly a factor of $3/2$ in the TIN regime, and $2-1/K$ in the CTIN regime, for arbitrary number of users $K>1$. However, in the SLS regime, the gain is $Θ(\log_2(K))$, i.e., it scales logarithmically with the number of users.

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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

GDoF of Interference Channel with Limited Cooperation under Finite Precision CSIT

The Generalized Degrees of Freedom (GDoF) of the two user interference channel are characterized for all parameter regimes under the assumption of finite precision channel state information at the transmitters (CSIT), when a limited amount of (half-duplex or full-duplex) cooperation is allowed between the transmitters in the form of $π$ DoF of shared messages. In all cases, the number of over-the-air bits that each cooperation bit buys is shown to be equal to either $0, 1, 1/2$ or $1/3$. The most interesting aspect of the result is the $1/3$ slope, which appears only under finite precision CSIT and strong interference, and as such has not been encountered in previous studies that invariably assumed perfect CSIT. Indeed, the achievability and converse for the parameter regimes with $1/3$ slope are the most challenging aspects of this work. In particular, the converse relies on non-trivial applications of Aligned Images bounds.

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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

On the Capacity of Computation Broadcast

The two-user computation broadcast problem is introduced as the setting where User $1$ wants message $W_1$ and has side-information $W_1'$, User $2$ wants message $W_2$ and has side-information $W_2'$, and $(W_1, W_1', W_2, W_2')$ may have arbitrary dependencies. The rate of a computation broadcast scheme is defined as the ratio $H(W_1,W_2)/H(S)$, where $S$ is the information broadcast to both users to simultaneously satisfy their demands. The supremum of achievable rates is called the capacity of computation broadcast $C_{CB}$. It is shown that $C_{CB}\leq H(W_1,W_2)/\left[H(W_1|W_1')+H(W_2|W_2')-\min\Big(I(W_1; W_2, W_2'|W_1'), I(W_2; W_1, W_1'|W_2')\Big)\right]$. For the linear computation broadcast problem, where $W_1, W_1', W_2, W_2'$ are comprised of arbitrary linear combinations of a basis set of independent symbols, the bound is shown to be tight. For non-linear computation broadcast, it is shown that this bound is not tight in general. Examples are provided to prove that different instances of computation broadcast that have the same entropic structure, i.e., the same entropy for all subsets of $\{W_1,W_1',W_2,W_2'\}$, can have different capacities. Thus, extra-entropic structure matters even for two-user computation broadcast. The significance of extra-entropic structure is further explored through a class of non-linear computation broadcast problems where the extremal values of capacity are shown to correspond to minimally and maximally structured problems within that class.

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