We present efficient and practical algorithms for a large, distributed system of processors to achieve reliable computations in a secure manner. Specifically, we address the problem of computing a general function of several private inputs distributed among the processors of a network, while ensuring the correctness of the results and the privacy of the inputs, despite accidental or malicious faults in the system. [...] Our algorithms maintain a low cost in local processing time, are the first to achieve optimal levels of fault-tolerance, and most importantly, have low communication complexity. In contrast to the best known previous methods, which require large numbers of rounds even for fairly simple computations, we devise protocols that use small messages and a constant number of rounds regardless of the complexity of the function to be computed. Through direct algebraic approaches, we separate the communication complexity of secure computing from the computational complexity of the function to be computed. We examine security under both the modern approach of computational complexity-based cryptography and the classical approach of unconditional, information-theoretic security. We [...] support formal proofs of claims to security, addressing an important deficiency in the literature. Our protocols are provably secure. In the realm of information-theoretic security, we characterize those functions which two parties can compute jointly with absolute privacy. We also characterize those functions which a weak processor can compute using the aid of powerful processors without having to reveal the instances of the problem it would like to solve. Our methods include a promising new technique called a locally random reduction, which has given rise not only to efficient solutions for many of the problems considered in this work but to several powerful new results in complexity theory.
We present efficient and practical algorithms for a large, distributed system of processors to achieve reliable computations in a secure manner. Specifically, we address the problem of computing a general function of several private inputs distributed among the processors of a network, while ensuring the correctness of the results and the privacy of the inputs, despite accidental or malicious faults in the system. [...] Our algorithms maintain a low cost in local processing time, are the first to achieve optimal levels of fault-tolerance, and most importantly, have low communication complexity. In contrast to the best known previous methods, which require large numbers of rounds even for fairly simple computations, we devise protocols that use small messages and a constant number of rounds regardless of the complexity of the function to be computed. Through direct algebraic approaches, we separate the communication complexity of secure computing from the computational complexity of the function to be computed. We examine security under both the modern approach of computational complexity-based cryptography and the classical approach of unconditional, information-theoretic security. We [...] support formal proofs of claims to security, addressing an important deficiency in the literature. Our protocols are provably secure. In the realm of information-theoretic security, we characterize those functions which two parties can compute jointly with absolute privacy. We also characterize those functions which a weak processor can compute using the aid of powerful processors without having to reveal the instances of the problem it would like to solve. Our methods include a promising new technique called a locally random reduction, which has given rise not only to efficient solutions for many of the problems considered in this work but to several powerful new results in complexity theory.
We present efficient and practical algorithms for a large, distributed system of processors to achieve reliable computations in a secure manner. Specifically, we address the problem of computing a general function of several private inputs distributed among the processors of a network, while ensuring the correctness of the results and the privacy of the inputs, despite accidental or malicious faults in the system.
Communication is often the most significant bottleneck in distributed computing. Our algorithms maintain a low cost in local processing time, are the first to achieve optimal levels of fault-tolerance, and most importantly, have low communication complexity. In contrast to the best known previous methods, which require large numbers of rounds even for fairly simple computations, we devise protocols that use small messages and a constant number of rounds regardless of the complexity of the function to be computed. Through direct algebraic approaches, we separate the communication complexity of secure computing from the computational complexity of the function to be computed.
We examine security under both the modern approach of computational complexity-based cryptography and the classical approach of unconditional, information-theoretic security. We develop a clear and concise set of definitions that support formal proofs of claims to security, addressing an important deficiency in the literature. Our protocols are provably secure.
In the realm of information-theoretic security, we characterize those functions which two parties can compute jointly with absolute privacy. We also characterize those functions which a weak processor can compute using the aid of powerful processors without having to reveal the instances of the problem it would like to solve. Our methods include a promising new technique called a locally random reduction, which has given rise not only to efficient solutions for many of the problems considered in this work but to several powerful new results in complexity theory.
In the problem of Secure Multi-party Computation, the objective is to design a protocol that allows a group of parties to securely compute functions of their collective private data, while ensuring that no party reveals any more information about its private data other than what must be inherently revealed by the computation results (privacy) and that no parties can disrupt or influence the computation results beyond the effects of changing their input data (correctness and consistency). Information theoretic approaches toward this problem, that provide provable (unconditional) security guarantees even against adversarial parties that have unbounded computational power, have established that general computation is possible in a variety of scenarios. However, these general solutions are not always efficient or finely tuned to the requirements of specific problems and applications.
The focus of this thesis is on the development of information theoretic approaches for multi-party computation applications with the common theme of secure and efficient computation and inference over a distributed data network. Specific applications include: 1) private information retrieval, where the objective is to privately retrieve data without revealing what was selected; 2) secure statistical analysis, the problem of extracting statistics without revealing anything else about the underlying data; 3) secure sampling, which is the secure generation of new data with a given distribution; and 4) secure authentication, where the identity of a party needs to authenticated via inference on his/her credentials and stored registration data.
The contributions of this thesis toward addressing these applications include the following: The development of an Oblivious Transfer (OT) protocol, applicable to private information retrieval, that trades off a small amount of privacy while being several times more efficient than other OT protocols. The efficiency of general se- cure two-party computation via OT realized from noisy channels is also analyzed. A technique for approximate secure multi-party computation applicable to secure statistical analysis of large scale distributed databases is developed. By exploiting a dimensionality reduction and the structure of a broad class of functions based on the computed statistics, the feasibility of simultaneously achieving both arbitrarily high accuracy and arbitrarily high communication efficiency is demonstrated. The region of distributions that can be securely sampled in the two-party scenario is characterized. It is further established that for those distributions, it can be accomplished with a protocol that only sends one message. Inner and outer bounds on the region of distributions that can be securely sampled in the general multi-party scenario are also developed. A new three-party protocol, applicable to the problem of secure biometric authentication, that securely computes the Hamming distance even when one of the parties arbitrarily deviates from the protocol is proposed and analyzed. A two-factor secure biometric authentication system that is robust against the compromise of registered biometric data, allowing for revocability and resistance against cross-enrollment attacks is also proposed.
In this dissertation, we study the design of secure and efficient protocols for distributed linear algebra problems. All existing solutions to the problem are either inefficient in terms of round and communication complexity and/or provide security against very weak adversaries.
We design protocols for two different adversarial settings. (i) Computational: where the adversary's computational resources are bounded, but the adversary is allowed to corrupt all but one of the participants. (ii) Information theoretic: where the adversary is unconditionally powerful, but is only allowed to corrupt a third of the participants. The security in both cases is defined in the real world/ideal world simulation paradigm, where we assume that the adversary can behave arbitrarily malicious.
Our main constructions are O(s) round protocols with O(sn2+1/s) communication for testing singularity of a shared n x n matrix, for any positive integer s, in both adversarial settings. We then use and extend these constructions to design secure protocols for computing the rank of a shared matrix and solving shared linear systems with similar efficiency. Our protocols are almost optimal in terms of round and communication complexity, as they roughly match the efficiency of the best possible protocols for the case where security is not a concern.
We use a wide range of techniques from computer algebra to reduce the problem of securely deciding singularity to the problem of secure matrix multiplication. We then design new and efficient protocols for secure matrix multiplication in both adversarial environments. In the computational setting, and when there are only two parties, we apply cut-and-choose techniques to encrypted algebraic encodings of the inputs, outputs, and intermediate values in order to prevent malicious behavior. We achieve even better amortized efficiency when running multiple instances of the protocol, via use of share-packing techniques. When the number of parties is more that two, our construction relies on efficient instantiations of the inner and outer protocols in a recently introduced compiler by Ishai et al. [35]. In the information theoretic setting, our protocol takes advantage of a homomorphic commitment scheme to achieve the improved efficiency.
In the setting of secure multiparty computation, several distrustful parties wish to carry out a distributed computing task on their local private data while satisfying several security properties such as correctness, privacy, independence of inputs and fairness. The aim of secure multiparty computation (MPC) is to enable the parties to carry out the computation in a secure manner, eliminating any attempt of an adversarial entity to harm the execution. The concept of secure computation is very general and fundamental in cryptography, and models any distributed computing task, including simple computations as coin-tossing and broadcast, as well as more complex tasks such as electronic auctions and anonymous transactions. In this thesis, we study two foundational aspects of secure computation: perfect security and fairness. In the first part of this thesis, we study perfect security in secure computation. A protocol that is perfectly secure cannot be broken even when the adversary has unlimited computational power and its security holds unconditionally. One of the most fundamental results of secure computation was presented by Ben-Or, Goldwasser and Wigderson (BGW) in 1988. They demonstrated that any n-party functionality can be computed with perfect security, in the private channels model. When the adversary is semi-honest this holds as long as t < n/2 parties are corrupted, and when the adversary is malicious this holds as long as t < n/3 parties are corrupted. Unfortunately, a full proof of these results was never published. In this thesis, we remedy this situation and provide a full proof of security of the BGW protocol. This also includes a full description of the protocol for the malicious setting. In addition to the above, we observe that by some simple and natural modifications, the BGW protocol can be significantly simplified and one of its expensive (and perhaps most complicated) subprotocols can be saved. We present a new multiplication protocol that is based on the original construction, but is simpler and achieves higher efficiency. In the second part of this thesis, we study fairness in secure two party computation. Informally, the fairness property guarantees that if one party receives its output, then the other party does too. The well-known impossibility result of Cleve (STOC 1986) implies that, in general, it is impossible to securely compute a function with complete fairness without an honest majority. Until recently, the accepted belief has been that nothing non-trivial can be computed with complete fairness in the two party setting. The surprising work of Gordon, Hazay, Katz and Lindell (STOC 2008) shows that this belief is false, and that there exist some non-trivial (deterministic, finite-domain) Boolean functions that can be computed fairly. This raises the fundamental question of characterizing complete fairness in secure two-party computation. We focus on this question and give both positive and negative results. We first characterize
In this dissertation, we consider the design of broadcast and secure multi-party computation (MPC) protocols in the presence of adversarial faults.
Secure multi-party computation is the most generic problem in fault-tolerant distributed computing. In principle, a multi-party computation protocol can be used to solve any distributed cryptographic problem. Informally, the problem of multi-party computation is the following: suppose we have n parties P1, P2,…, Pn where each party Pi has a private input xi. Together, the parties want to compute a function of their inputs (y1, y2,…, yn) = f(x1,x2,…, xn). However, some parties can be corrupted and do not execute a prescribed protocol faithfully. Even worse, they may be controlled by an adversary and attack the protocol in a coordinated manner. Despite the presence of such an adversary, a secure MPC protocol should ensure that each (corrupted) party Pi learn only its output yi but nothing more.
Broadcast in the presence of adversarial faults is one of the simplest special cases of multi-party computation and important component of larger protocols. In short, broadcast allows a party to send the same message to all parties, and all parties to be assured they have received identical messages.
The contribution of this dissertation is twofold. First, we construct broadcast and secure multi-party computation protocols for honest majority in a point-to-point network whose round complexities improve significantly upon prior work. In particular, we give the first expected constant-round authenticated broadcast protocol for honest majority assuming only public-key infrastructure and signatures. Second, we initiate the study of broadcast in radio networks in the presence of adversarial faults. In radio networks, parties communicate through multicasting messages; a message can only be received by the parties within some radius from the sender. Feasibility and impossibility results are given, and our bounds are tight.
A common task in parallel processing is the distributed computation of a function by a number of processors each of which possesses partial information relevant to the value of that function. In this thesis we develop protocols which allow for such computation to take place while maintaining the value of the function secret to an eavesdropper. Of interest is the communication complexity of such protocols. We begin by considering two processors and two channels, one secret and one public, and present a protocol which minimizes the number of bits exchanged over the secret channel, while maintaining ϵ-uncertainty about the value of the function for the eavesdropper. We show that all binary functions can be kept ϵ-secret using a constant number of bits independent of the size of their domain. We then generalize our results to N processors communicating over a network of arbitrary topology.
In a related problem, also of interest in parallel processing, we consider processors communicating over a mesh network with the objective of broadcasting information amongst each other. One instance of the problem involves a number of nodes all with the same message to be broadcasted. For that problem a lower-bound on the time to complete the broadcast, and an algorithm which achieves this bound are presented. In another instance, every node in the mesh has packets to be broadcast arriving independently, according to a Poisson random process. The stability region for performing such broadcasts is characterized, and broadcast algorithms which operate efficiently within that region are presented. These algorithms involve interacting queues whose analysis is known to be very difficult. Toward that end we develop an approximate model for analyzing interacting queues. This new approximation models an N-dimensional infinite Markov chain by means of two Markov chains, one being one-dimensional and infinite and the other being N-dimensional and finite. The transition probabilities of each chain are expressed in terms of statistics of the other chain. The two chains can be solved together using standard iterative methods to yield an approximation to the original N-dimensional infinite chain. The results of the approximation compare very well with simulation, especially when arrival rates are low. We show how this model can also be used to analyze the ALOHA multiple access protocol.