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When analyzing the round complexity of multi-party protocols, one often overlooks the fact that underlying resources, such as a broadcast channel, can by themselves be expensive to implement. For example, it is well known that it is impossible to implement a broadcast channel by a (deterministic) protocol in a sublinear (in the number of corrupted parties) number of rounds. The seminal works of Rabin and Ben-Or from the early 1980s demonstrated that limitations as the above can be overcome by using randomization and allowing parties to terminate at different rounds, igniting the study of protocols over point-to-point channels with probabilistic termination and expected constant round complexity. However, absent a rigorous simulation-based definition, the suggested protocols are proven secure in a property-based manner or via ad hoc simulation-based frameworks, therefore guaranteeing limited, if any, composability. In this work, we put forth the first simulation-based treatment of multi-party cryptographic protocols with probabilistic termination. We define secure multi-party computation (MPC) with probabilistic termination in the UC framework and prove a universal composition theorem for probabilistic termination protocols. Our theorem allows to compile a protocol using deterministic termination hybrids into a protocol that uses expected constant round protocols for emulating these hybrids, preserving the expected round complexity of the calling protocol. We showcase our definitions and compiler by providing the first composable protocols (with simulation-based security proofs) for the following primitives, relying on point-to-point channels: (1) expected constant round perfect Byzantine agreement, (2) expected constant round perfect parallel broadcast, and (3) perfectly secure MPC with round complexity independent of the number of parties.

We devise a notion of polynomial runtime suitable for the simulation-based security analysis of multi-party cryptographic protocols. Somewhat surprisingly, straightforward notions of polynomial runtime lack expressivity for reactive tasks and/or lead to an unnatural simulation-based security notion. Indeed, the problem has been recognized in previous works, and several notions of polynomial runtime have already been proposed. However, our new notion, dubbed reactive polynomial time, is the first to combine the following properties: We work in the Universal Composability (UC) protocol framework. We remark that while the UC framework already features a universal composition theorem, we develop new techniques to prove secure composition in the case of reactively polynomial-time protocols and attacks.

We introduce the notion of resource-fair protocols. Informally, this property states that if one party learns the output of the protocol, then so can all other parties, as long as they expend roughly the same amount of resources. As opposed to previously proposed definitions related to fairness, our definition follows the standard simulation paradigm and enjoys strong composability properties. In particular, our definition is similar to the security definition in the universal composability (UC) framework, but works in a model that allows any party to request additional resources from the environment to deal with dishonest parties that may prematurely abort. In this model we specify the ideally fair functionality as allowing parties to “invest resources” in return for outputs, but in such an event offering all other parties a fair deal. (The formulation of fair dealings is kept independent of any particular functionality, by defining it using a “wrapper.”) Thus, by relaxing the notion of fairness, we avoid a well-known impossibility result for fair multi-party computation with corrupted majority; in particular, our definition admits constructions that tolerate arbitrary number of corruptions. We also show that, as in the UC framework, protocols in our framework may be arbitrarily and concurrently composed. Turning to constructions, we define a “commit-prove-fair-open” functionality and design an efficient resource-fair protocol that securely realizes it, using a new variant of a cryptographic primitive known as “time-lines.” With (the fairly wrapped version of) this functionality we show that some of the existing secure multi-party computation protocols can be easily transformed into resource-fair protocols while preserving their security.