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Optimizing Leaky Private Information Retrieval Codes to Achieve ${O}(\log K)$ Leakage Ratio Exponent

Wenyuan Zhao, Yu-Shin Huang, Chao Tian, Alex Sprintson

TL;DR

The paper addresses leaky private information retrieval (L-PIR) under pure differential privacy by optimizing retrieval-pattern probabilities across a generalized TSC base code. It reveals a layered probability structure where keys with lower Hamming weight are more likely, and proves that focusing on a reduced TSC code with cyclic permutations suffices. The authors derive the optimal weight-based allocation $p_j=\frac{e^{(K-1-j)\epsilon}}{N\big(e^{\epsilon}+N-1\big)^{K-1}}$ and show the resulting leakage-exponent scales as $O(\log K)$ for fixed $D$ and $N$, a significant improvement over the prior $\Theta(K)$ bound. The results are established via reduction to the reduced scheme and KKT optimization, and they quantify the DP-PIR tradeoff with explicit expressions for download cost and leakage as functions of $\epsilon$, $K$, and $N$. This work substantially advances the practical design of privacy-leaky PIR systems by tightening the DP leakage-performance tradeoff.

Abstract

We study the problem of leaky private information retrieval (L-PIR), where the amount of privacy leakage is measured by the pure differential privacy parameter, referred to as the leakage ratio exponent. Unlike the previous L-PIR scheme proposed by Samy et al., which only adjusted the probability allocation to the clean (low-cost) retrieval pattern, we optimize the probabilities assigned to all the retrieval patterns jointly. It is demonstrated that the optimal retrieval pattern probability distribution is quite sophisticated and has a layered structure: the retrieval patterns associated with the random key values of lower Hamming weights should be assigned higher probabilities. This new scheme provides a significant improvement, leading to an ${O}(\log K)$ leakage ratio exponent with fixed download cost $D$ and number of servers $N$, in contrast to the previous art that only achieves a $Θ(K)$ exponent, where $K$ is the number of messages.

Optimizing Leaky Private Information Retrieval Codes to Achieve ${O}(\log K)$ Leakage Ratio Exponent

TL;DR

The paper addresses leaky private information retrieval (L-PIR) under pure differential privacy by optimizing retrieval-pattern probabilities across a generalized TSC base code. It reveals a layered probability structure where keys with lower Hamming weight are more likely, and proves that focusing on a reduced TSC code with cyclic permutations suffices. The authors derive the optimal weight-based allocation and show the resulting leakage-exponent scales as for fixed and , a significant improvement over the prior bound. The results are established via reduction to the reduced scheme and KKT optimization, and they quantify the DP-PIR tradeoff with explicit expressions for download cost and leakage as functions of , , and . This work substantially advances the practical design of privacy-leaky PIR systems by tightening the DP leakage-performance tradeoff.

Abstract

We study the problem of leaky private information retrieval (L-PIR), where the amount of privacy leakage is measured by the pure differential privacy parameter, referred to as the leakage ratio exponent. Unlike the previous L-PIR scheme proposed by Samy et al., which only adjusted the probability allocation to the clean (low-cost) retrieval pattern, we optimize the probabilities assigned to all the retrieval patterns jointly. It is demonstrated that the optimal retrieval pattern probability distribution is quite sophisticated and has a layered structure: the retrieval patterns associated with the random key values of lower Hamming weights should be assigned higher probabilities. This new scheme provides a significant improvement, leading to an leakage ratio exponent with fixed download cost and number of servers , in contrast to the previous art that only achieves a exponent, where is the number of messages.
Paper Structure (16 sections, 3 theorems, 46 equations, 2 figures, 3 tables)

This paper contains 16 sections, 3 theorems, 46 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Canonical PIR Coding Problem
  4. The Differential Privacy Requirement
  5. The Base PIR Code
  6. TSC Code with Permutation
  7. The Reduced TSC Code
  8. The L-PIR Code
  9. Main Result
  10. Proof of Theorem \ref{['thrm:epsbound']}
  11. Proof of Theorem \ref{['thrm:allocation-DP-PIR']}
  12. Optimality of the Reduced Scheme
  13. Optimal Allocation of the Reduced Code
  14. Conclusion
  15. The TSC Code When $K=2$ and $N=3$
  16. ...and 1 more sections

Key Result

Theorem 1

With a fixed number of servers and a fixed feasible download cost, the leakage ratio exponent $\epsilon$'s are bounded as where $\alpha := (D-1)(N-1)$. As a direct consequence, we have $\epsilon^{\text{UB}}=\Theta\left(K\right)$, while $\epsilon^{\text{TSC}}=O(\log K)$.

Figures (2)

  • Figure 1: The effect of $(N,K)$ on the $(D, \epsilon)$ tradeoff. TSC is the proposed scheme; UB is the scheme derived in Samy2021; LB denotes the lower bound in Samy2021.
  • Figure 2: The download cost $D$ and multiplicative gap between TSC, UB, and LB when user privacy leakage $\epsilon$ is fixed.

Theorems & Definitions (6)

  • Definition 1: Differential Privacy
  • Theorem 1
  • Theorem 2
  • Proposition 1
  • proof
  • proof