Computing Research Repository

Li, Su; Gastpar, Michael

doi: 10.48550/arxiv.1809.09861pmid: N/A

Abstract:Multi-server single-message private information retrieval is studied in the presence of side information. In this problem, $K$ independent messages are replicatively stored at $N$ non-colluding servers. The user wants to privately download one message from the servers without revealing the index of the message to any of the servers, leveraging its $M$ side information messages. We assume that the servers only know the number of the side information messages available at the user but not their indices. We prove a converse bound on the maximum download rates, which coincides with the known achievability scheme proposed by Kadhe {\it et. al.}. Hence, we characterize the capacity for this problem, which is $(1+\frac{1}{N}+\frac{1}{N^2}+\dots+\frac{1}{N^{\left\lceil \frac{K}{M+1}\right\rceil-1}})^{-1}$. The proof leverages a novel concept that we call {\it virtual side information}, which, for a fixed query and any message, identifies the side information that would be needed in order to recover that message.

Kotiuga, P. Robert; Lahtinen, Valtteri

doi: 10.1007/s10444-024-10197-6pmid: N/A

Abstract:We look at computational physics from an electrical engineering perspective and suggest that several concepts of mathematics, not so well-established in computational physics literature, present themselves as opportunities in the field. We discuss elliptic complexes and highlight the category theoretical background and its role as a unifying language between algebraic topology, differential geometry, and modelling software design. In particular, the ubiquitous concept of naturality is central. Natural differential operators have functorial analogues on the cochains of triangulated manifolds. In order to establish this correspondence, we derive formulas involving simplices and barycentric coordinates, defining discrete vector fields and a discrete Lie derivative as a result of a discrete analogue of Cartan's magic formula. This theorem is the main mathematical result of the paper.