arXiv subjects

Mohamed Nomeir

Publications and source records attributed to Mohamed Nomeir.

2 recordsLinked to original sources

arXiv2026

Quantum Private Distributed Matrix Multiplication: Extending the Classical Codes and Limitations

In this paper, we explore how quantum resources can be used to increase the rate of private distributed matrix multiplication (PDMM). In PDMM, a user who has two high-dimensional matrices, A and B, and lacks the computational capabilities to apply matrix multiplication locally, divides the matrices A and B into K and L sub-blocks, respectively. Then, the user sends them to N servers to apply the required multiplication \emph{privately}, i.e., any $T$ colluding servers cannot get any information about the user's matrices. The goal is to reduce the number of servers needed to perform the required matrix multiplication, thereby decreasing the communication cost. First, in the high-privacy regime, the state-of-the-art classical code is called the gap additive secure polynomial (GASP) code. We define a feasibility requirement in the quantum setting for the GASP code such that the highest performance is achieved when the requirement is satisfied. Thus, super-dense coding gain is achieved when the feasibility condition is satisfied. We show that when $T \geq KL-K+1$, the feasibility condition is always satisfied and the GASP code can be extended to the quantum version. In the case of $T < KL-K+1$, the feasibility can still be satisfied. To further examine this behavior, we numerically study how the minimum privacy requirement depends on the matrix dimensions and provide a quadratic estimate for this relation. The results suggest that feasibility can be achieved when $T \sim 0.5 KL$. Second, in the low-privacy regime, the recently developed cyclic-addition degree tables (CAT) and discretely optimized GASP (DOG) codes are among the most efficient known classical constructions for PDMM. We show that the feasibility condition developed for GASP can be adopted for both CAT and DOG codes as well, thus unifying the feasibility framework for multiple classical PDMM coding schemes.

cs.IT↗

arXiv2026

Local Private Information Retrieval for Graph-Based Replicated Systems

We rethink the definition of privacy in multi-server, graph-replicated private information retrieval (PIR) systems, by introducing a novel setting where the user's privacy is governed by the servers' storage structure. In classical graph-replicated PIR, the user retrieves a single message stored at the servers, while hiding the message index from each server. In our proposed privacy setting, the user is concerned with hiding the message index from a particular server, only if that server stores the message being retrieved, and privacy is not imposed otherwise. We coin this relaxed privacy requirement as local user privacy and the resulting PIR problem as local PIR on the graph. Our focus is on two-replicated PIR systems, where every message is replicated twice and stored on two distinct servers. Specifically, we study local PIR systems where the storage is represented by simple graphs, i.e., every pair of vertices is associated with at most one edge, and by their multigraph extension, i.e., $r$ parallel edges replace every edge. For these settings, we establish bounds on the local PIR capacity, defined as the maximum number of message symbols retrieved, per downloaded symbol. The local privacy requirement yields significant capacity gain over the classical PIR capacity under the same storage structure. For instance, in settings where the graph is a disjoint union of multiple identical sub-graphs, the gain in the local PIR capacity over classical PIR capacity is multiplicative in the number of sub-graphs. Further, for connected graphs, we derive capacity lower bounds for edge-transitive and bipartite graphs, which are greater than the best-known PIR capacity bounds. From these and by establishing matching upper bounds, we exactly characterize the capacity for star graphs, cyclic graphs, and path graphs with odd number of vertices. We introduce two local PIR schemes for general graphs.

cs.IT↗