Commitment over Gaussian Unfair Noisy Channels

TL;DR

This paper investigates the limits of cryptographic commitment over unreliable Gaussian noisy channels, revealing that unlike classical Gaussian channels, their commitment capacity can be finite or zero, depending on channel properties.

Contribution

It provides the first analysis of commitment capacity for Gaussian UNCs, establishing bounds, conditions for infinite capacity, and the impact of channel elasticity.

Findings

Gaussian UNCs can have finite or zero commitment capacity.

Infinite capacity occurs when channel elasticity is zero.

Threshold for positive capacity depends on channel elasticity at high power.

Abstract

Commitment is a key primitive which resides at the heart of several cryptographic protocols. Noisy channels can help realize information-theoretically secure commitment schemes, however, their imprecise statistical characterization can severely impair such schemes, especially their security guarantees. Keeping our focus on channel unreliability in this work, we study commitment over unreliable continuous alphabet channels called the Gaussian unfair noisy channels or Gaussian UNCs. We present the first results on the optimal throughput or commitment capacity of Gaussian UNCs. It is known that classical Gaussian channels have infinite commitment capacity, even under finite transmit power constraints. For unreliable Gaussian UNCs, we prove the surprising result that their commitment capacity may be finite, and in some cases, zero. When commitment is possible, we present achievable rate lower bounds by constructing positive - throughput protocols under given input power constraint, and (two-sided) channel elasticity at committer Alice and receiver Bob. Our achievability results establish an interesting fact - Gaussian UNCs with zero elasticity have infinite commitment capacity - which brings a completely new perspective to why classic Gaussian channels, i.e., Gaussian UNCs with zero elasticity, have infinite capacity. Finally, we precisely characterize the positive commitment capacity threshold for a Gaussian UNC in terms of the channel elasticity, when the transmit power tends to infinity.