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Dual-Source SPIR over a noiseless MAC without Data Replication or Shared Randomness

Remi A. Chou

TL;DR

This work shows that symmetric private information retrieval can be achieved over a noiseless binary adder MAC with two non-colluding servers holding independent data, aided by a public noiseless channel and without data replication or pre-shared randomness. It fully characterizes the dual-source SPIR capacity region as (L1−1)R1+(L2−1)R2 ≤ 1/2 and provides constructive achievability for all L1,L2, including reductions to the base L1=L2=2 case and a general (L1−1)(L2−1) round scheme. A matching information-theoretic converse reduces to maximizing H(X1X2|Y) under Y=X1+X2, with the maximum equal to 1/2, establishing tightness. The results introduce a new path to SPIR that avoids data replication, shared randomness, and channel noise, expanding the design space for information-theoretic private retrieval.

Abstract

Information-theoretically secure Symmetric Private Information Retrieval (SPIR) is known to be infeasible over noiseless channels with a single server. Known solutions to overcome this infeasibility involve additional resources such as database replication, shared randomness, or noisy channels. In this paper, we propose an alternative approach for achieving SPIR with information-theoretic security guarantees, without relying on shared randomness, noisy channels, or data replication. Specifically, we demonstrate that it is sufficient to use a noiseless binary adder multiple-access channel, where inputs are controlled by two non-colluding servers and the output is observed by the client, alongside a public noiseless communication channel between the client and the servers. Furthermore, in this setting, we characterize the optimal file rates, i.e., the file lengths normalized by the number of channel uses, that can be transferred.

Dual-Source SPIR over a noiseless MAC without Data Replication or Shared Randomness

TL;DR

This work shows that symmetric private information retrieval can be achieved over a noiseless binary adder MAC with two non-colluding servers holding independent data, aided by a public noiseless channel and without data replication or pre-shared randomness. It fully characterizes the dual-source SPIR capacity region as (L1−1)R1+(L2−1)R2 ≤ 1/2 and provides constructive achievability for all L1,L2, including reductions to the base L1=L2=2 case and a general (L1−1)(L2−1) round scheme. A matching information-theoretic converse reduces to maximizing H(X1X2|Y) under Y=X1+X2, with the maximum equal to 1/2, establishing tightness. The results introduce a new path to SPIR that avoids data replication, shared randomness, and channel noise, expanding the design space for information-theoretic private retrieval.

Abstract

Information-theoretically secure Symmetric Private Information Retrieval (SPIR) is known to be infeasible over noiseless channels with a single server. Known solutions to overcome this infeasibility involve additional resources such as database replication, shared randomness, or noisy channels. In this paper, we propose an alternative approach for achieving SPIR with information-theoretic security guarantees, without relying on shared randomness, noisy channels, or data replication. Specifically, we demonstrate that it is sufficient to use a noiseless binary adder multiple-access channel, where inputs are controlled by two non-colluding servers and the output is observed by the client, alongside a public noiseless communication channel between the client and the servers. Furthermore, in this setting, we characterize the optimal file rates, i.e., the file lengths normalized by the number of channel uses, that can be transferred.

Paper Structure

This paper contains 22 sections, 12 theorems, 48 equations, 2 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Main Results
  4. Achievability part of Theorem \ref{['th2']}
  5. Special case $L_1 = L_2=2$
  6. Reliability
  7. Privacy of the receiver selection
  8. Privacy of the unselected strings for the senders
  9. Servers’ mutual privacy
  10. Achieved rate pair
  11. Case $L_1=L_2$
  12. General case
  13. Server's privacy with respect to Client
  14. Client privacy with respect to the servers
  15. Servers' mutual privacy
  16. ...and 7 more sections

Key Result

Theorem 1

The dual-source SPIR capacity region is Moreover, any rate pair in $\mathcal{C}_{\textup{SPIR}^2}(L_1,L_2)$ is achievable without time-sharing.

Figures (2)

  • Figure 1: Dual-source symmetric private information retrieval between one client and two servers who have access to a public channel and a binary adder MAC as described in \ref{['eqchannel']}. The file selection of the client is $(Z_1,Z_2)\in \mathcal{L}_1 \times \mathcal{L}_2$, i.e., the client wishes to obtain $(F_{1,Z_1}, F_{2,Z_2})$ from the servers with the constraints that the client must not learn information about $(F_{1,\mathcal{L}_1 \backslash\{{Z}_1\}}, F_{2,\mathcal{L}_2 \backslash\{{Z}_2\}})$, $(Z_1,Z_2)$ must remain private from both servers, and Server $1$ must not learn information about Server $2$'s content, and vice versa for Server $2$.
  • Figure 2: The x-coordinate (resp. y-coordinate) of $\mathcal{C}_{\textup{SPIR}}$ is defined as the SPIR capacity between Server $1$ (resp. Server $2$) and client in the absence of Server $2$ (resp. Server $1$), i.e., $\mathcal{C}_{\textup{SPIR}}=(0,0)$. $\mathcal{C}_{\textup{SPIR}^2}(L_1,L_2)$ is the dual-source SPIR capacity region characterized in Theorem \ref{['th1']}.

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Corollary 1
  • Example 1
  • Example 2
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 17 more