International Association
for Cryptologic Research

An important benchmark for multi-party computation protocols (MPC) is their round complexity . For several important MPC tasks, such as broadcast, (tight) lower bounds on the round complexity are known. However, some of these lower bounds can be circumvented when the termination round of every party is not a priori known, and simultaneous termination is not guaranteed. Protocols with this property are called probabilistic-termination ( PT ) protocols. Running PT protocols in parallel affects the round complexity of the resulting protocol in somewhat unexpected ways. For instance, an execution of m protocols with constant expected round complexity might take $$O(\log m)$$ O ( log m ) rounds to complete. In a seminal work, Ben-Or and El-Yaniv (Distributed Computing ‘03) developed a technique for a parallel execution of arbitrarily many broadcast protocols, while preserving expected round complexity. More recently, Cohen et al.  (CRYPTO ‘16) devised a framework for universal composition of PT protocols, and provided the first composable parallel-broadcast protocol with a simulation-based proof. These constructions crucially rely on the fact that broadcast is “privacy-free,” and do not generalize to arbitrary protocols in a straightforward way. This raises the question of whether it is possible to execute arbitrary PT protocols in parallel, without increasing the round complexity. In this paper we tackle this question and provide both feasibility and infeasibility results. We construct a round-preserving protocol compiler, tolerating any dishonest minority of actively corrupted parties, that compiles arbitrary protocols into a protocol realizing their parallel composition, while having a black-box access to the underlying protocols . Furthermore, we prove that the same cannot be achieved, using known techniques, given only black-box access to the functionalities realized by the protocols, unless merely security against semi-honest corruptions is required, for which case we provide a protocol. To prove our results, we utilize the language and results by Cohen et al. , which we extend to capture parallel composition and reactive functionalities, and to handle the case of an honest majority.

Traditional bounds on synchronous Byzantine agreement (BA) and secure multi-party computation (MPC) establish that in absence of a private correlated-randomness setup, such as a PKI, protocols can tolerate up to $t<n/3$ of the parties being malicious. The introduction of ``Nakamoto style'' consensus, based on Proof-of-Work (PoW) blockchains, put forth a somewhat different flavor of BA, showing that even a majority of corrupted parties can be tolerated as long as the majority of the computation resources remain at honest hands. This assumption on honest majority of some resource was also extended to other resources such as stake, space, etc., upon which blockchains achieving Nakamoto-style consensus were built that violated the $t<n/3$ bound in terms of number of party corruptions. The above state of affairs begs the question of whether the seeming mismatch is due to different goals and models, or whether the resource-restricting paradigm can be generically used to circumvent the $n/3$ lower bound. In this work we study this question and formally demonstrate how the above paradigm changes the rules of the game in cryptographic definitions. First, we abstract the core properties that the resource-restricting paradigm offers by means of a functionality {\em wrapper}, in the UC framework, which when applied to a standard point-to-point network restricts the ability (of the adversary) to send new messages. We show that such a wrapped network can be implemented using the resource-restricting paradigm---concretely, using PoWs and honest majority of computing power---and that the traditional $t<n/3$ impossibility results fail when the parties have access to such a network. Our construction is in the {\em fresh} Common Reference String (CRS) model---i.e., it assumes a CRS which becomes available to the parties at the same time as to the adversary. We then present constructions for BA and MPC, which given access to such a network tolerate $t<n/2$ corruptions without assuming a private correlated randomness setup. We also show how to remove the freshness assumption from the CRS by leveraging the power of a random oracle. Our MPC protocol achieves the standard notion of MPC security, where parties might have dedicated roles, as is for example the case in Oblivious Transfer protocols. This is in contrast to existing solutions basing MPC on PoWs, which associate roles to pseudonyms but do not link these pseudonyms with the actual parties.

An intensive effort by the cryptographic community to minimize the round complexity of secure multi-party computation (MPC) has recently led to optimal two-round protocols from minimal assumptions. Most of the proposed solutions, however, make use of a broadcast channel in every round, and it is unclear if the broadcast channel can be replaced by standard point-to-point communication in a round-preserving manner, and if so, at what cost on the resulting security. In this work, we provide a complete characterization of the trade-off between number of broadcast rounds and achievable security level for two-round MPC tolerating arbitrarily many active corruptions. Specifically, we consider all possible combinations of broadcast and point-to-point rounds against the three standard levels of security for maliciously se- cure MPC protocols, namely, security with identifiable, unanimous, and selective abort. For each of these notions and each combination of broadcast and point-to-point rounds, we provide either a tight feasibility or an infeasibility result of two-round MPC. Our feasibility results hold assuming two-round OT in the CRS model, whereas our impossibility results hold given any correlated randomness.

The formalization of concrete, non-idealized hash function properties sufficient to prove the security of Bitcoin and related protocols has been elusive, as all previous security analyses of blockchain protocols have been performed in the random oracle model. In this paper we identify three such properties, and then construct a blockchain protocol whose security can be reduced to them in the standard model assuming a common reference string (CRS). The three properties are: {\em collision resistance}, {\em computational randomness extraction} and {\em iterated hardness}. While the first two properties have been extensively studied, iterated hardness has been empirically stress-tested since the rise of Bitcoin; in fact, as we demonstrate in this paper, any attack against it (assuming the other two properties hold) results in an attack against Bitcoin. In addition, iterated hardness puts forth a new class of search problems which we term {\em iterated search problems} (ISP). ISPs enable the concise and modular specification of blockchain protocols, and may be of independent interest.

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.

The Bitcoin backbone protocol (Eurocrypt 2015) extracts basic properties of Bitcoin’s underlying blockchain data structure, such as “common prefix” and “chain quality,” and shows how fundamental applications including consensus and a robust public transaction ledger can be built on top of them. The underlying assumptions are “proofs of work” (POWs), adversarial hashing power strictly less than 1/2 and no adversarial pre-computation—or, alternatively, the existence of an unpredictable “genesis” block.In this paper we first show how to remove the latter assumption, presenting a “bootstrapped” Bitcoin-like blockchain protocol relying on POWs that builds genesis blocks “from scratch” in the presence of adversarial pre-computation. Importantly, the round complexity of the genesis block generation process is independent of the number of participants.Next, we consider applications of our construction, including a PKI generation protocol and a consensus protocol without trusted setup assuming an honest majority (in terms of computational power). Previous results in the same setting (unauthenticated parties, no trusted setup, POWs) required a round complexity linear in the number of participants.