On Commitment over General Compound Channels

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.