The Round Complexity of General VSS

The round complexity of verifiable secret sharing (VSS) schemes has been studied extensively for threshold adversaries. In particular, Fitzi et al. showed an efficient 3-round VSS for $n \geq 3t+1$ \cite{FitziVSSTCC06}, where an infinitely powerful adversary can corrupt t (or less) parties out of $n$ parties. This paper shows that for non-threshold adversaries, -Two round VSS is possible iff the underlying adversary structure satisfies ${\cal Q}^4$ condition; -Three round VSS is possible iff the underlying adversary structure satisfies ${\cal Q}^3$ condition. Further as a special case of our three round protocol, we can obtain a more efficient 3-round VSS than the VSS of Fitzi et al. for $n = 3t+1$. More precisely, the communication complexity of the reconstruction phase is reduced from ${\cal O}(n^3)$ to ${\cal O}(n^2)$. We finally point out a flaw in the reconstruction phase of VSS of Fitzi et al., and show how to fix it

Improving the round complexity of VSS in point-topoint networks

We revisit the following question: what is the optimal round complexity of verifiable secret sharing (VSS)? We focus here on the case of perfect VSS where the number of corrupted parties t satisfies t < n/3, with n the total number of parties. Work of Gennaro et al. (STOC 2001) and Fitzi et al. (TCC 2006) shows that, assuming a broadcast channel, 3 rounds are necessary and sufficient for efficient VSS. Existing protocols, however, treat the broadcast channel as being available “for free ” and do not attempt to minimize its usage. This approach leads to relatively poor round complexity when such protocols are compiled to run over a point-to-point network. We show here a VSS protocol that is simultaneously optimal in terms of both the number of rounds and the number of invocations of broadcast. Our protocol also satisfies a certain “2-level sharing ” property that makes it useful for constructing protocols for general secure computation.

Studies on Verifiable Secret Sharing, Byzantine Agreement and Multiparty Computation

This dissertation deals with three most important as well as fundamental problems in secure distributed computing, namely Verifiable Secret Sharing (VSS), Byzantine Agreement (BA) and Multiparty Computation (MPC). VSS is a two phase protocol (Sharing and Reconstruction) carried out among $n$ parties in the presence of a centralized adversary who can corrupt up to $t$ parties. Informally, the goal of the VSS protocol is to share a secret $s$, among the $n$ parties during the sharing phase in a way that would later allow for a unique reconstruction of this secret in the reconstruction phase, while preserving the secrecy of $s$ until the reconstruction phase. VSS is used as a key tool in MPC, BA and many other secure distributed computing problems. It can take many different forms, depending on the underlying network (synchronous or asynchronous), the nature (passive or active) and computing power (bounded or unbounded) of the adversary, type of security (cryptographic or information theoretic) etc. We study VSS in information theoretic setting over both synchronous as well as asynchronous network, considering an active unbounded powerful adversary. Our main contributions for VSS are: \begin{itemize} \item In synchronous network, we carry out in-depth investigation on the round complexity of VSS by allowing a probability of error in computation and show that existing lower bounds for the round complexity of error-free VSS can be circumvented by introducing a negligible probability of error. \item We study the communication and round efficiency of VSS in synchronous network and present a robust VSS protocol that is simultaneously communication efficient and round efficient. In addition, our protocol is the best known communication and round efficient protocol in the literature. \item In asynchronous network, we study the communication complexity of VSS and propose a number of VSS protocols. Our protocols are highly communication efficient and show significant improvement over the existing protocols in terms of communication complexity. \end{itemize} The next problem that we deal with is Byzantine Agreement (BA). BA is considered as one of the most fundamental primitives for fault tolerant distributed computing and cryptographic protocols. BA among a set of $n$ parties, each having a private input value, allows them to reach agreement on a common value even if some of the malicious parties (at most $t$) try to prevent agreement among the parties. Similar to the case of VSS, several models for BA have been proposed during the last three decades, considering various aspects like the underlying network, the nature and computing power of adversary, type of security. One of these models is BA over asynchronous network which is considered to be more realistic network than synchronous in many occasions. Though important, research in BA in asynchronous network has received much less attention in comparison to the BA protocols in synchronous network. Even the existing protocols for asynchronous BA involve high communication complexity and in general are very inefficient in comparison to their synchronous counterparts. We focus on BA in information theoretic setting over asynchronous network tolerating an active adversary having unbounded computing power and mainly work towards the communication efficiency of the problem. Our contributions for BA are as follows: \begin{itemize} \item We propose communication efficient asynchronous BA protocols that show huge improvement over the existing protocols in the same setting. Our protocols for asynchronous BA use our VSS protocols in asynchronous network as their vital building blocks. \item We also construct a communication optimal asynchronous BA protocol for sufficiently long message size. Precisely, our asynchronous BA communicates O(\ell n) bits for \ell bit message, for sufficiently large \ell. \end{itemize} The studies on VSS and BA naturally lead one towards MPC problems. The MPC can model almost any known cryptographic application and uses VSS as well as BA as building blocks. MPC enables a set of $n$ mutually distrusting parties to compute some function of their private inputs, such that the privacy of the inputs of the honest parties is guaranteed (except for what can be derived from the function output) even in the presence of an adversary corrupting up to $t$ of the parties and making them misbehave arbitrarily. Much like VSS and BA, MPC can also be studied in various models. Here, we attempt to solve MPC in information theoretic setting over synchronous as well as asynchronous network, tolerating an active unbounded powerful adversary. As for MPC, our main contributions are: \begin{itemize} \item Using one of our synchronous VSS protocol, we design a synchronous MPC that minimizes the communication and round complexity simultaneously, where existing MPC protocols try to minimize one complexity measure at a time (i.e the existing protocols minimize either communication complexity or round complexity). \item We study the communication complexity of asynchronous MPC protocols and design a number of protocols for the same that show significant gain in communication complexity in comparison to the existing asynchronous MPC protocols. \item We also study a specific instance of MPC problem called Multiparty Set Intersection (MPSI) and provide protocols for the same. \end{itemize} In brief, our work in this thesis has made significant advancement in the state-of-the-art research on VSS, BA and MPC by presenting several inherent lower bounds and efficient/optimal solutions for the problems in terms of their key parameters such as communication complexity and time/round complexity. Thus our work has made a significant contribution to the field of secure distributed computing by carrying out a foundation research on the three most important problems of this field

A High Reliability Asymptotic Approach for Packet Inter-Delivery Time Optimization in Cyber-Physical Systems

In cyber-physical systems such as automobiles, measurement data from sensor nodes should be delivered to other consumer nodes such as actuators in a regular fashion. But, in practical systems over unreliable media such as wireless, it is a significant challenge to guarantee small enough inter-delivery times for different clients with heterogeneous channel conditions and inter-delivery requirements. In this paper, we design scheduling policies aiming at satisfying the inter-delivery requirements of such clients. We formulate the problem as a risk-sensitive Markov Decision Process (MDP). Although the resulting problem involves an infinite state space, we first prove that there is an equivalent MDP involving only a finite number of states. Then we prove the existence of a stationary optimal policy and establish an algorithm to compute it in a finite number of steps. However, the bane of this and many similar problems is the resulting complexity, and, in an attempt to make fundamental progress, we further propose a new high reliability asymptotic approach. In essence, this approach considers the scenario when the channel failure probabilities for different clients are of the same order, and asymptotically approach zero. We thus proceed to determine the asymptotically optimal policy: in a two-client scenario, we show that the asymptotically optimal policy is a "modified least time-to-go" policy, which is intuitively appealing and easily implementable; in the general multi-client scenario, we are led to an SN policy, and we develop an algorithm of low computational complexity to obtain it. Simulation results show that the resulting policies perform well even in the pre-asymptotic regime with moderate failure probabilities

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

Multi-party Quantum Computation

We investigate definitions of and protocols for multi-party quantum computing in the scenario where the secret data are quantum systems. We work in the quantum information-theoretic model, where no assumptions are made on the computational power of the adversary. For the slightly weaker task of verifiable quantum secret sharing, we give a protocol which tolerates any t < n/4 cheating parties (out of n). This is shown to be optimal. We use this new tool to establish that any multi-party quantum computation can be securely performed as long as the number of dishonest players is less than n/6.Comment: Masters Thesis. Based on Joint work with Claude Crepeau and Daniel Gottesman. Full version is in preparatio