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An Improved Lower Bound on Oblivious Transfer Capacity via Interactive Erasure Emulation

So Suda, Shun Watanabe, Haruya Yamaguchi

TL;DR

This work addresses the OT capacity of noisy channels under a passive adversary, where prior results were limited to generalized erasure channels. It introduces a recursive, interactive erasure-emulation protocol that extends beyond one-shot GEC-based constructions to improve the lower bound on $C_{ ext{OT}}(W)$ for the BSC and BSEC. The central result provides a multi-round lower bound, $C_{ ext{OT}}(W_{ ext{BSEC}(p_1,q_1)}) \ge \sum_{t=1}^T (\prod_{j=1}^{t-1} \frac{(1-2p_j)}{2}) p_t (1-H(q_t))$, with $p_{t+1}=2 q_t (1-q_t)$ and $q_{t+1}= \frac{q_t^2}{(1-q_t)^2+q_t^2}$, and demonstrates that the bound nearly matches known upper bounds for certain parameter ranges. The work also discusses potential necessity of multi-round interaction for attaining OT capacity and provides an illustrative example suggesting two rounds may be required. Overall, the approach advances understanding of OT realization from noisy channels and highlights the value of interactive protocols in information-theoretic cryptography.

Abstract

We revisit the oblivious transfer (OT) capacities of noisy channels against the passive adversary, which have been identified only for a limited class of channels. In the literature, the general construction of oblivious transfer has been known only for generalized erasure channels (GECs); for other channels, we first convert a given channel to a GEC via alphabet extension and erasure emulation, and then apply the general construction for GEC. In this paper, we derive an improved lower bound on the OT capacity of the binary symmetric channel (BSC) and binary symmetric erasure channel (BSEC) by proposing a new protocol; by using interactive communication between the sender and the receiver, our protocol emulates erasure events recursively in multiple rounds. We also discuss a potential necessity of multiple rounds interactive communication to attain the OT capacity.

An Improved Lower Bound on Oblivious Transfer Capacity via Interactive Erasure Emulation

TL;DR

This work addresses the OT capacity of noisy channels under a passive adversary, where prior results were limited to generalized erasure channels. It introduces a recursive, interactive erasure-emulation protocol that extends beyond one-shot GEC-based constructions to improve the lower bound on for the BSC and BSEC. The central result provides a multi-round lower bound, , with and , and demonstrates that the bound nearly matches known upper bounds for certain parameter ranges. The work also discusses potential necessity of multi-round interaction for attaining OT capacity and provides an illustrative example suggesting two rounds may be required. Overall, the approach advances understanding of OT realization from noisy channels and highlights the value of interactive protocols in information-theoretic cryptography.

Abstract

We revisit the oblivious transfer (OT) capacities of noisy channels against the passive adversary, which have been identified only for a limited class of channels. In the literature, the general construction of oblivious transfer has been known only for generalized erasure channels (GECs); for other channels, we first convert a given channel to a GEC via alphabet extension and erasure emulation, and then apply the general construction for GEC. In this paper, we derive an improved lower bound on the OT capacity of the binary symmetric channel (BSC) and binary symmetric erasure channel (BSEC) by proposing a new protocol; by using interactive communication between the sender and the receiver, our protocol emulates erasure events recursively in multiple rounds. We also discuss a potential necessity of multiple rounds interactive communication to attain the OT capacity.
Paper Structure (5 sections, 1 theorem, 14 equations, 2 figures, 3 algorithms)

This paper contains 5 sections, 1 theorem, 14 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Review of Standard Protocol for BSEC
  4. Recursive Protocol
  5. Discussions

Key Result

Theorem 1

By using Protocol protocol-recursive with $T$ rounds, we can derive the following lower bound on the OT capacity of BSEC $W_{\mathrm{BSEC}(p_1,q_1)}$:For $t=1$, $\prod_{j=1}^{t-1} \frac{(1-2p_j)}{2}$ is regarded as $1$.

Figures (2)

  • Figure 1: A description emulated BSEC induced from the BSC; the erasure probability $p_2$ and the crossover probability $q_2$ are computed by \ref{['eq:erasure-crossover-probability-1']}.
  • Figure 2: A comparison of the lower bound on the OT capacity in Theorem \ref{['theorem:recursive']} for $T=1,2,3$ and the upper bound on the OT capacity in \ref{['eq:upper-1']}, where the horizontal axis is $0 \le q_1 \le 1$ and $p_1=0.1$ is fixed.

Theorems & Definitions (2)

  • Theorem 1
  • Example 2