An important problem in secure multi-party computation is the design of protocols that can tolerate adversaries that are capable of corrupting parties dynamically and learning their internal states. In this paper, we make significant progress in this area in the context of password-authenticated key exchange (PAKE) and oblivious transfer (OT) protocols. More precisely, we first revisit the notion of projective hash proofs and introduce a new feature that allows us to explain any message sent by the simulator in case of corruption, hence the notion of Explainable Projective Hashing. Next, we demonstrate that this new tool generically leads to efficient PAKE and OT protocols that are secure against semi-adaptive adversaries without erasures in the Universal Composability (UC) framework. We then show how to make these protocols secure even against adaptive adversaries, using non-committing encryption, in a much more efficient way than generic conversions from semi-adaptive to adaptive security. Finally, we provide concrete instantiations of explainable projective hash functions that lead to the most efficient PAKE and OT protocols known so far, with UC-security against adaptive adversaries, with or without erasures, in the single global CRS setting.
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