# Your search: "author:Jafar, Syed Ali"

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## Scholarly Works (9 results)

Originating from the construction of the asymptotic-capacity achieving scheme for X-secure T-private information retrieval (XSTPIR), the technique of cross-subspace alignment (CSA) emerges as the natural solution to secure and private information retrieval, secure distributed matrix multiplication, and coded distributed batch computation. Characterized by a Cauchy-Vandermonde structure that facilitates interference alignment along Vandermonde terms, while the desired signals remain resolvable along the Cauchy terms, the idea of CSA is shown to be the essential ingredient in the optimal/asymptotically optimal/state-of-art approaches that minimize the download and/or communication cost of these independently introduced but closely related problems.

In this dissertation we will first introduce the idea of CSA to the applications of XSTPIR, XSTPIR with graph-based replicated storage (GXSTPIR) and XSTPIR with MDS coded storage (MDS-XSTPIR). The CSA solution to XSTPIR exploits the Cauchy-Vandermonde structured answer strings that align interference symbols guaranteeing T-privacy and X-security to achieve the asymptotic capacity. The achievability scheme for GXSTPIR reveals a non-trivial generalization of CSA that takes the advantage of a special structure inspired by dual Generalized Reed Solomon (GRS) codes to allow interference alignment for arbitrary storage patterns. For MDS-XSTPIR, we propose a novel scheme based on confluent Cauchy-Vandermonde storage structure and a strategy of successive decoding with interference cancellation. Next, by characterizing a connection between a form of XSTPIR problem known as multi-message XSTPIR and the problem of secure distributed matrix multiplication (SDMM), we characterize a series of capacity regions of SDMM, as well as several of its variants. The idea of CSA serves as an essential component of the construction of several achievability schemes. Given the insights from all these results, next, we construct CSA/GCSA codes based on (confluent) Cauchy-Vandermonde structures for coded distributed batch computation (CDBC) that unify, generalize and improve upon the state-of-art codes for distributed computing such as LCC codes for multivariate polynomial evaluations and EP codes for matrix multiplication. Finally, we study the problem of X-secure T-private federated submodel learning (XSTPFSL), which is a non-trivial generalization of the XSTPIR problem where private writes are needed. The proposed ACSA-RW scheme achieves the desired private read and write functionality with elastic dropout resilience, takes the advantage of the synergistic gain from the joint design of private read and write for low communication costs, and sheds light on a striking symmetry between upload and download costs. Intuitively, private read and write functionalities, as well as their synergistic gain, rely on secure distributed matrix multiplication so that the idea of CSA emerges as the core of the solution.

The problem of weakly-private information retrieval, is a variant of private information

retrieval, where the user wants to retrieve 1 out of K messages from a distributed storage

system with N servers that stores all K messages, and is willing to leak some information

of the identity of the desired message. In this work, we study the problem of weakly-private

information retrieval. A novel information leakage metric is proposed, and the capacity for

the setting of N = 2 servers, and arbitrary number of messages K is characterized. In

particular, in the capacity achieving scheme, designing the distribution of the non-uniformly

distributed noise Z turns out to be the key to achieve the capacity.

The emergence of ‘Aligned Images Sum-set (AIS) Inequalities’ has spurred much progress in Generalized Degrees of Freedom (GDoF) characterizations of wireless networks under robust assumptions that limit the channel state information at transmitters (CSIT). Much of this progress is limited to small networks, and to larger networks under highly symmetric parameter values. Extending these results to larger networks with asymmetric parameters is challenging because of the inherent curse of dimensionality, and also because the scope of AIS bounds is far from well understood. Making progress in this direction is the main goal of this dissertation.We first explore the feasibility of an extremal network theory, i.e., a study of extremal networks within particular parameter regimes of interest. In particular, we quantify the extremal gain in sum GDoF brought about by transmitter cooperation in K user interference channels in various parameter regimes that correspond to weak interference. Specifically, with the robust scheme of ‘Treating Interference as Noise (TIN)’ as the baseline, we find the extremal gain in a large parameter regime (known as the Simple Layered Superposition, or SLS, regime) to be Θ(log2 K), which scales logarithmically with the number of users. As our next contribution we explore robust GDoF characterizations for large networks in the presence of security constraints. We identify surprisingly broad new regimes for both interference and broadcast networks, where robust secure GDoF are fully characterized for arbitrary number of users. The unifying feature of these regimes is the optimality of TIN along with wiretap coding, power control and jamming. Continuing with the security constraint, in the final part of this dissertation we study the robustness of structured codes for secure GDoF characterizations under limited CSIT. In particular, structured jamming based on lattice codes is known to offer significant advantages by allowing receivers to decode and remove the sum of jamming and message signals in aggregate. However, we show that such advantages are completely lost in GDoF under the robust assumption of limited CSIT. The complete robust GDoF characterization of 2 user secure Z channels comes as a byproduct of the analysis. Limitations of existing AIS bounds are identified that stand in the way of generalizations to larger networks in this case.

We investigate the optimality of linear interference

alignment (allowing symbol extensions) for Â 3-user

$M_T\times M_R$ MIMO interference channel where $M_T$ and $M_R$

denote the number of antennas at each transmitter and each receiver,

respectively, and the \emph{channel coefficients are held constant}. Recently, Wang et al. have conjectured that interference alignment based on linear beamforming using only proper Gaussian codebooks and possibly with symbol extensions, is sufficient to achieve the information theoretic DoF outer bound for all $M_T, M_R$ values except

if $|M_T-M_R|=1$, $\min(M_T,M_R)\geq 2$. A partial proof of the conjecture is provided by Wang et al. for arbitrary $M_T, M_R$ values subject to a final numerical evaluation step that needs to be performed for each $M_T, M_R$ setting to complete the proof. The numerical evaluation step is also carried out explicitly by Wang et al. to settle the conjecture for all Â $M_T, M_R$ values up to 10. For $|M_T-M_R|=1$, $\min(M_T,M_R)\geq 2$, Wang et al. show that interference alignment schemes based on linear beamforming with proper Gaussian signaling and symbol extensions are not sufficient to achieve the information-theoretic DoF outer bonds. In contrast, in this note we show, for all $M_T, M_R$ values up to 10, that interference alignment schemes based on linear beamforming over symbol extensions are enough to achieve the information theoretic DoF outer bounds for constant channels, if \emph{asymmetric complex signaling} is utilized. Based on this new insight, we conjecture that linear interference alignment is optimal for achieving the information theoretic DoF outer bounds for all $M_T, M_R$ values in the 3 user $M_T\times M_R$ MIMO interference channel with constant channel coefficients, except for the case $M_T=M_R=1$ where it is known that either time/frequency-varying channels or non-linear (e.g., rational alignment) schemes are required.

We characterize the degrees of freedom (DoF) of MIMO interference networks with rank-

deficient channel matrices. For the 2-user rank deficient MIMO interference channel, we prove

the optimality of previously known achievable DoF in the symmetric case and generalize the re-

sult to fully asymmetric settings. For the K-user rank deficient interference channel, we improve

the previously known achievable DoF and provide a tight outer bound to establish optimality

in symmetric settings. In particular, we show that for the K-user rank deficient interference

channel, when all nodes have M antennas, all direct channels have rank D0, all cross chan-

nels are of rank D, and the channels are otherwise generic, the optimal DoF value per user is

min(D0, M −min(M,(K−1)D)). For 2-user and 3-user rank deficient channels, achievable schemes2

are for both constant and time-varying channels, while for K-user rank deficient channels, we present schemes for time-varying channels and note that the insights would act as stepping stones for constant channels. Notably for interference channels, the rank-deficiency of direct channels does not help and the rank-deficiency of cross-channels does not hurt. The main technical challenge is to account for the spatial dependencies introduced by rank deficiencies in the interference alignment schemes that typically rely on the independence of channel coefficients.