We study commitments over unreliable compound noisy channels, where parties may know the compound channel state but lack control over that state. Commitment is one of the common building blocks for many cryptographic protocols which realize multi-party computational functionalities in a secure manner. A noisy channel, among others, is widely acknowledged as a promising resource for realizing information-theoretically secure cryptographic primitives, including commitment. However, unreliable noisy channels with poorly or imprecisely characterized statistical behaviour can severely degrade commitment guarantees. Our focus is unreliability on account of a compound channel state, albeit under public awareness of that state.Building on prior work, we seek to characterize the optimal commitment throughput or commitment capacity of compound binary symmetric channels when parties may be state-aware. State-awareness implies a passive and publicly known capability of precise channel knowledge (whether said party is honest or dishonest); however, state-awareness precludes any active and private channel state control as in, for instance, the classic unfair noisy channels (UNCs). We present new results on the commitment capacity under all possible configurations where individual parties may (or may not) be state-aware. An important takeaway of our work is the following: even a fairly weak capability of state-awareness (albeit when asymmetric and only at the committer-side) can degrade the commitment throughput to the same extent as under strongly capable parties that can privately control the compound state.
We study commitments over unreliable compound noisy channels, where parties may know the compound channel state but lack control over that state. Commitment is one of the common building blocks for many cryptographic protocols which realize multi-party computational functionalities in a secure manner. A noisy channel, among others, is widely acknowledged as a promising resource for realizing information-theoretically secure cryptographic primitives, including commitment. However, unreliable noisy channels with poorly or imprecisely characterized statistical behaviour can severely degrade commitment guarantees. Our focus is unreliability on account of a compound channel state, albeit under public awareness of that state.Building on prior work, we seek to characterize the optimal commitment throughput or commitment capacity of compound binary symmetric channels when parties may be state-aware. State-awareness implies a passive and publicly known capability of precise channel knowledge (whether said party is honest or dishonest); however, state-awareness precludes any active and private channel state control as in, for instance, the classic unfair noisy channels (UNCs). We present new results on the commitment capacity under all possible configurations where individual parties may (or may not) be state-aware. An important takeaway of our work is the following: even a fairly weak capability of state-awareness (albeit when asymmetric and only at the committer-side) can degrade the commitment throughput to the same extent as under strongly capable parties that can privately control the compound state.
As computation and communication become increasingly ubiquitous, data security remains a key consideration. Hence, cryptographic primitives lie at the heart of several modern digital technologies. In this work, we study one such important cryptographic primitive called commitment. Here two parties, say Alice and Bob, aim to realize a protocol with the following functionality: Alice seeks to share or commit to a bit string securely with Bob with the requirement that the string is hidden from Bob until she chooses to reveal it to him. Bob wants to ensure that the protocol binds Alice to her original choice of bit string by allowing him to successfully detect when she cheats at the time of revealing her string. This seemingly elementary cryptographic functionality is extremely powerful and finds wide use in technologies like blockchains, digital contracts, cryptocurrencies, etc.. More generally, commitment has also been widely studied for applications like sealed-bid auctions, distributed coin tossing, etc.. Building on Wyner's classic work on wiretap channels, unconditionally-secure or information-theoretically secure protocols have been realized for a wide variety of problems, including commitment, using noisy channels as a resource. Crucially, however, the security guarantees are intimately tied to the knowledge of the noise statistics; these could be compromised when noise statistics are imprecisely known. In this work, we seek to study information-theoretically secure commitment over a class of unreliable noisy channels called compound channels. In particular, we study commitment over compound discrete memoryless channels (compound-DMCs) where the instantiated channel is known to be a DMC but its realization or channel state is not known exactly and can only be localized to a potential set of possible states. We precisely characterize the optimal throughput or commitment capacity over general compound DMCs. We also study the effect of state-awareness at the two parties, when the parties are honest and when they are malicious, on the commitment capacity of such compound DMCs. Interestingly, we show that committer-side state-awareness can non-trivially increase the commitment throughput over the compound DMC which is not the case when one has only receiver-side state awareness. In particular, we completely characterize the commitment capacity for all state-aware configurations for compound DMCs.
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.
Noisy channels are a valuable resource from a cryptographic point of view. They can be used for exchanging secret-keys as well as realizing other cryptographic primitives such as commitment and oblivious transfer. To be really useful, noisy channels have to be considered in the scenario where a cheating party has some degree of control over the channel characteristics. Damgård et al. (EUROCRYPT 1999) proposed a more realistic model where such level of control is permitted to an adversary, the so called unfair noisy channels, and proved that they can be used to obtain commitment and oblivious transfer protocols. Given that noisy channels are a precious resource for cryptographic purposes, one important question is determining the optimal rate in which they can be used. The commitment capacity has already been determined for the cases of discrete memoryless channels and Gaussian channels. In this work we address the problem of determining the commitment capacity of unfair noisy channels. We compute a single-letter characterization of the commitment capacity of unfair noisy channels. In the case where an adversary has no control over the channel (the fair case) our capacity reduces to the well-known capacity of a discrete memoryless binary symmetric channel.
In the commitment problem, two mutually distrustful parties Alice and Bob interact in a two-phase protocol, viz., commit and reveal phase, to achieve commitment over a bit string that Alice possesses. The protocol successfully achieves commitment if, firstly, Alice can commit to sharing a string with Bob, with the guarantee that this string remains hidden from Bob until she chooses to reveal it to him. Secondly, when Alice does reveal a string, Bob is able to detect precisely whether the revealed string is different from the one Alice committed to sharing. Information-theoretically secure commitment is impossible if Alice and Bob communicate only noiselessly; however, communication using a noisy channel can be a resource to realize commitment. Even though a noisy channel may be available, it is possible that the corresponding channel law is imprecisely known or poorly characterized. We define and study a compound-binary symmetric channel (compound-BSC) which models such a scenario. A compound-BSC is a BSC whose transition probability is fixed but unknown to either party; the set of potential values which this transition probability can take, though, is known to both parties a priori. In this work, we completely characterize the maximum commitment throughput or commitment capacity of a compound-BSC. We provide an optimal, computationally-efficient scheme for our achievability, and we derive a converse for general alphabet compound DMCs, which is then specialized for compound-BSCs.
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.
In this work, we study commitment over a class of channels called reverse elastic channels (RECs). In the commitment problem, two mutually distrustful parties, say Alice and Bob, seek to commit on a bit string available to Alice. The parties interact via a commitment protocol comprising two phases, viz., commit phase followed by reveal phase. Alice commits to a string, and transmits it to Bob securely in a manner Bob cannot learn it until Alice chooses to reveal it; at the time of reveal, however, Bob can successfully detect if Alice cheats. It is well known that noisy channels are a promising resource to realize information-theoretically secure commitment; however, oftentimes, channel behaviour may be poorly characterized thereby limiting the commitment throughput and/or degrading the security guarantees. Particularly problematic is a scenario where dishonest parties can actively alter the channel characteristics. RECs are an interesting class of such unreliable channels, where essentially only a dishonest committer Alice can meaningfully alter the channel; RECs have attracted active recent interest. Our principal contribution is the REC commitment capacity characterization for all parameters; this proves a recent related conjecture. Apart from presenting an achievable scheme, a key result in our work is a tight converse which analyses a specific cheating strategy by Alice. The significance of RECs stems from the fact that along with elastic channels (ECs), where only a dishonest receiver Bob can alter the channel, these two channel models represent special cases of the more widely studied unfair noisy channels (UNCs). Interestingly, for a given set of parameters, our result shows that the REC commitment capacity is no larger than that for the ECs. Furthermore, similar to the ECs, RECs offer non-trivial commitment throughput for all meaningful parameters; this is in stark contrast to UNCs where the throughput may possibly be zero.