Commitment over Compound Binary Symmetric Channels

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.