On the Round Complexity of Randomized Byzantine Agreement

We prove lower bounds on the round complexity of randomized Byzantine agreement (BA) protocols, bounding the halting probability of such protocols after one and two rounds. In particular, we prove that: 1) BA protocols resilient against n/3 [resp., n/4] corruptions terminate (under attack) at the end of the first round with probability at most o(1) [resp., 1/2+ o(1)]. 2) BA protocols resilient against n/4 corruptions terminate at the end of the second round with probability at most 1-Theta(1). 3) For a large class of protocols (including all BA protocols used in practice) and under a plausible combinatorial conjecture, BA protocols resilient against n/3 [resp., n/4] corruptions terminate at the end of the second round with probability at most o(1) [resp., 1/2 + o(1)]. The above bounds hold even when the parties use a trusted setup phase, e.g., a public-key infrastructure (PKI). The third bound essentially matches the recent protocol of Micali (ITCS\u2717) that tolerates up to n/3 corruptions and terminates at the end of the third round with constant probability

Peer-to-Peer Secure Multi-Party Numerical Computation Facing Malicious Adversaries

We propose an efficient framework for enabling secure multi-party numerical computations in a Peer-to-Peer network. This problem arises in a range of applications such as collaborative filtering, distributed computation of trust and reputation, monitoring and other tasks, where the computing nodes is expected to preserve the privacy of their inputs while performing a joint computation of a certain function. Although there is a rich literature in the field of distributed systems security concerning secure multi-party computation, in practice it is hard to deploy those methods in very large scale Peer-to-Peer networks. In this work, we try to bridge the gap between theoretical algorithms in the security domain, and a practical Peer-to-Peer deployment. We consider two security models. The first is the semi-honest model where peers correctly follow the protocol, but try to reveal private information. We provide three possible schemes for secure multi-party numerical computation for this model and identify a single light-weight scheme which outperforms the others. Using extensive simulation results over real Internet topologies, we demonstrate that our scheme is scalable to very large networks, with up to millions of nodes. The second model we consider is the malicious peers model, where peers can behave arbitrarily, deliberately trying to affect the results of the computation as well as compromising the privacy of other peers. For this model we provide a fourth scheme to defend the execution of the computation against the malicious peers. The proposed scheme has a higher complexity relative to the semi-honest model. Overall, we provide the Peer-to-Peer network designer a set of tools to choose from, based on the desired level of security.Comment: Submitted to Peer-to-Peer Networking and Applications Journal (PPNA) 200

On Broadcast in Generalized Network and Adversarial Models

Broadcast is a primitive which allows a specific party to distribute a message consistently among n parties, even if up to t parties exhibit malicious behaviour. In the classical model with a complete network of bilateral authenticated channels, the seminal result of Pease et al. [Pease et al., 1980] shows that broadcast is achievable if and only if t < n/3. There are two generalizations suggested for the broadcast problem - with respect to the adversarial model and the communication model. Fitzi and Maurer [Fitzi and Maurer, 1998] consider a (non-threshold) general adversary that is characterized by the subsets of parties that could be corrupted, and show that broadcast can be realized from bilateral channels if and only if the union of no three possible corrupted sets equals the entire set of n parties. On the other hand, Considine et al. [Considine et al., 2005] extend the standard model of bilateral channels with the existence of b-minicast channels that allow to locally broadcast among any subset of b parties; the authors show that in this enhanced model of communication, secure broadcast tolerating up to t corrupted parties is possible if and only if t < (b-1)/(b+1)n. These generalizations are unified in the work by Raykov [Raykov P., 2015], where a tight condition on the possible corrupted sets is presented such that broadcast is achievable from a complete set of b-minicasts. This paper investigates the achievability of broadcast in general networks, i.e., networks where only some subsets of minicast channels may be available, thereby addressing open problems posed in [Jaffe et al., 2012; Raykov P., 2015]. To that end, we propose a hierarchy over all possible general adversaries, and identify for each class of general adversaries 1) a set of minicast channels that are necessary to achieve broadcast and 2) a set of minicast channels that are sufficient to achieve broadcast. In particular, this allows us to derive bounds on the amount of b-minicasts that are necessary and that suffice towards constructing broadcast in general b-minicast networks

Betrayal, Distrust, and Rationality: Smart Counter-Collusion Contracts for Verifiable Cloud Computing

Cloud computing has become an irreversible trend. Together comes the pressing need for verifiability, to assure the client the correctness of computation outsourced to the cloud. Existing verifiable computation techniques all have a high overhead, thus if being deployed in the clouds, would render cloud computing more expensive than the on-premises counterpart. To achieve verifiability at a reasonable cost, we leverage game theory and propose a smart contract based solution. In a nutshell, a client lets two clouds compute the same task, and uses smart contracts to stimulate tension, betrayal and distrust between the clouds, so that rational clouds will not collude and cheat. In the absence of collusion, verification of correctness can be done easily by crosschecking the results from the two clouds. We provide a formal analysis of the games induced by the contracts, and prove that the contracts will be effective under certain reasonable assumptions. By resorting to game theory and smart contracts, we are able to avoid heavy cryptographic protocols. The client only needs to pay two clouds to compute in the clear, and a small transaction fee to use the smart contracts. We also conducted a feasibility study that involves implementing the contracts in Solidity and running them on the official Ethereum network.Comment: Published in ACM CCS 2017, this is the full version with all appendice