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The Asymptotic Capacity of Byzantine Symmetric Private Information Retrieval and Its Consequences

Mohamed Nomeir, Alptug Aytekin, Sennur Ulukus

TL;DR

This work defines a general Byzantine model for symmetric private information retrieval (SPIR) where up to $B$ servers can share storage, queries, masking variables, and coordination to disrupt privacy. It proves an asymptotic capacity bound $C_{BSPIR}(B,N,infty)=1-4B/N$ and provides an achievable scheme that attains this bound, along with a constellation of corollaries for eavesdroppers and pre-init collusions. The results show that Byzantine adversaries impose a rate loss of $2B$ per retrieved symbol, and in a variant with pre-init Byzantine behavior the loss becomes $3B+1$ over $N$, highlighting the robustness gains of the proposed construction. The paper also offers illustrative examples and rigorous privacy/security proofs to ensure symmetric privacy despite Byzantine cooperation.

Abstract

We consider the problem of finding the asymptotic capacity of symmetric private information retrieval (SPIR) with $B$ Byzantine servers. Prior to finding the capacity, a definition for the Byzantine servers is needed since in the literature there are two different definitions. In \cite{byzantine_tpir}, where it was first defined, the Byzantine servers can send any symbol from the storage, their received queries and some independent random symbols. In \cite{unresponsive_byzantine_1}, Byzantine servers send any random symbol independently of their storage and queries. It is clear that these definitions are not identical, especially when \emph{symmetric} privacy is required. To that end, we define Byzantine servers, inspired by \cite{byzantine_tpir}, as the servers that can share everything, before and after the scheme initiation. In this setting, we find an upper bound, for an infinite number of messages case, that should be satisfied for all schemes that protect against this setting and develop a scheme that achieves this upper bound. Hence, we identify the capacity of the problem.

The Asymptotic Capacity of Byzantine Symmetric Private Information Retrieval and Its Consequences

TL;DR

This work defines a general Byzantine model for symmetric private information retrieval (SPIR) where up to servers can share storage, queries, masking variables, and coordination to disrupt privacy. It proves an asymptotic capacity bound and provides an achievable scheme that attains this bound, along with a constellation of corollaries for eavesdroppers and pre-init collusions. The results show that Byzantine adversaries impose a rate loss of per retrieved symbol, and in a variant with pre-init Byzantine behavior the loss becomes over , highlighting the robustness gains of the proposed construction. The paper also offers illustrative examples and rigorous privacy/security proofs to ensure symmetric privacy despite Byzantine cooperation.

Abstract

We consider the problem of finding the asymptotic capacity of symmetric private information retrieval (SPIR) with Byzantine servers. Prior to finding the capacity, a definition for the Byzantine servers is needed since in the literature there are two different definitions. In \cite{byzantine_tpir}, where it was first defined, the Byzantine servers can send any symbol from the storage, their received queries and some independent random symbols. In \cite{unresponsive_byzantine_1}, Byzantine servers send any random symbol independently of their storage and queries. It is clear that these definitions are not identical, especially when \emph{symmetric} privacy is required. To that end, we define Byzantine servers, inspired by \cite{byzantine_tpir}, as the servers that can share everything, before and after the scheme initiation. In this setting, we find an upper bound, for an infinite number of messages case, that should be satisfied for all schemes that protect against this setting and develop a scheme that achieves this upper bound. Hence, we identify the capacity of the problem.

Paper Structure

This paper contains 9 sections, 6 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Main Result and Consequences
  4. Achievable Scheme
  5. Illustrative Example
  6. Wrong User Estimate for Byzantine Server Locations
  7. Correct User Estimate for Byzantine Server Locations
  8. Converse Proof
  9. Privacy and Security Proofs

Key Result

Theorem 1

Given $N$ servers with $B$ of them being Byzantine. The asymptotic capacity of the symmetric PIR is given by,

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Remark 5
  • Lemma 1
  • ...and 3 more