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IEEE International Symposium on Information Theory (ISIT)

Bit Commitment over Gaussian Channels

João Barros, Hideki Imai, Anderson C. A. Nascimento, Stefan Skludarek

Published Jul 1, 2006• 7 citations
Publisher

Abstract

We consider bit commitment over additive white Gaussian noise channels. Our main result is that the maximum rate at which this class of channels can be used for implementing commitment protocols (the commitment capacity) is provably infinite, even under an average power constraint.
DOI
10.1109/isit.2006.262084

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Bit commitment is an important tool for constructing zero-knowledge proofs and multi-party computation. Unconditionally secure bit commitment can be based, in particular, on noisy channel or correlation where noise considered a valuable resource. Recently, Winter, Nascimento and Imai introduced the concept of commitment capacity, the maximal ratio between the length of a string which the sender commits to and the number of times the noisy channel/correlation is used. They also proved that for any discrete memoryless channel there exists a secure protocol achieving its commitment capacity however, no particular construction was given. Solving their open question, we provide an efficient protocol for achieving the commitment capacity of discrete memoryless systems (noisy channels and correlations).

IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2006•27 citations
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. This result brings a completely new perspective as 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.
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.
We establish a previously unnoticed connection between commitment (BC), a fundamental cryptographic primitive, and deterministic identification (DI), a post-Shannon communication setting. Specifically, for additive white Gaussian noise channels G\mathcal{G}G we show that a reliable commitment protocol can be obtained from deterministic identification codes in both the honest but curious and fully dishonest security settings. This viewpoint allows BC schemes to inherit the asymptotic performance of DI codes. Indeed, we show that commitment is naturally achievable in the linearithmic regime, i.e., with message sets of size Nn=exp⁡[Θ(nlog⁡n)]N_n=\exp [Θ(n\log n)]Nn​=exp[Θ(nlogn)], and we establish a lower bound on the linearithmic-scale BC capacity of C˙hBC(G)≥12\dot C_{\text{hBC}}(\mathcal G)\geq\frac12C˙hBC​(G)≥21​ in the honest but curious case and C˙BC(G)≥14\dot C_{\text{BC}}(\mathcal G)\geq\frac14C˙BC​(G)≥41​ in the fully dishonest picture. This refines the best previously known characterisation of the commitment capacity over G\mathcal{G}G, which established only an infinite linear-scale capacity.
Bit commitment is an important tool for constructing zero-knowledge proofs and multi-party computation. Unconditionally secure bit commitment can be based, in particular, on noisy channel or correlation where noise considered a valuable resource. Recently, Winter, Nascimento and Imai introduced the concept of commitment capacity, the maximal ratio between the length of a string which the sender commits to and the number of times the noisy channel/correlation is used. They also proved that for any discrete memoryless channel there exists a secure protocol achieving its commitment capacity however, no particular construction was given. Solving their open question, we provide an efficient protocol for achieving the commitment capacity of discrete memoryless systems (noisy channels and correlations).