On the Asymptotic Capacity of X-Secure T-Private Information Retrieval With Graph-Based Replicated Storage | IEEE Journals & Magazine | IEEE Xplore

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


Abstract:

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 a...Show More

Abstract:

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 (ρmin-X -T )/N, where N is the total number of servers, and each message is replicated at least ρmin 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/ν2(G) where ν2(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 ∈ E between vertices u, v only if a subset of messages is replicated at both servers u and v.
Published in: IEEE Transactions on Information Theory ( Volume: 66, Issue: 10, October 2020)
Page(s): 6280 - 6296
Date of Publication: 21 July 2020

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