We consider the problem of Non-Interactive Two-Party Secure Computation (NISC), where Rachel wishes to publish an encryption of her input x, in such a way that any other party, who holds an input y, can send her a single message which conveys to her the value f(x, y), and nothing more. We demand security against malicious parties. While such protocols are easy to construct using garbled circuits and general non-interactive zero-knowledge proofs, this approach inherently makes a non-black-box use of the underlying cryptographic primitives and is infeasible in practice.
We consider the problem of Non-Interactive Two-Party Secure Computation (NISC), where Rachel wishes to publish an encryption of her input x, in such a way that any other party, who holds an input y, can send her a single message which conveys to her the value f(x, y), and nothing more. We demand security against malicious parties. While such protocols are easy to construct using garbled circuits and general non-interactive zero-knowledge proofs, this approach inherently makes a non-black-box use of the underlying cryptographic primitives and is infeasible in practice.
In this paper, we study the question of whether or not it is possible to construct protocols for general secure computation in the setting of malicious adversaries and no honest majority that use the underlying primitive (e.g., enhanced trapdoor permutation) in a black-box way only. Until now, all known general constructions for this setting were inherently non-blackbox since they required the parties to prove zero-knowledge statements that are related to the computation of the underlying primitive. Our main technical result is a fully black-box reduction from oblivious transfer with security against malicious parties to oblivious transfer with security against semi-honest parties. As a corollary, we obtain the first constructions of general multiparty protocols (with security against malicious adversaries and without an honest majority) which only make a black-box use of semi-honest oblivious transfer, or alternatively a black-box use of lower-level primitives such as enhanced trapdoor permutations or homomorphic encryption. In order to construct this reduction we introduce a new notion of security called privacy in the presence of defensible adversaries. This notion states that if an adversary can produce (retroactively, after the protocol terminates) an input and random tape that make
Motivated by theoretical and practical interest, the challenging task of designing cryptographic protocols having only black-box access to primitives has generated various breakthroughs in the last decade. Despite such positive results, even though nowadays we know black-box constructions for secure two-party and multi-party computation even in constant rounds, there still are in Cryptography several constructions that critically require non-black-box use of primitives in order to securely realize some fundamental tasks. As such, the study of the gap between black-box and nonblack-box constructions still includes major open questions.
It seems to be a generally acknowledged fact that you should never trust a computer and that you should trust the person operating the computer even less. This in particular becomes a problem when the party that you do not trust is one which is separated from you and is one on which you depend, e.g. because that party is the holder of some data or some capability that you need in order to operate correctly. How do you perform a given task involving interaction with other parties while allowing these parties a minimal influence on you and, if privacy is an issue, allowing them to learn as little about you as possible. This is the general problem of secure multiparty computation. The usual way of formalizing the problem is to say that a number of parties who do not trust each other wish to compute some function of their local inputs, while keeping their inputs as secret as possible and guaranteeing the correctness of the output. Both goals should be obtained even if some parties stop participating or some malevolent coalition of the parties start deviating arbitrarily from the agreed protocol. The task is further complicated by the fact that besides their mutual distrust, nor do the parties trust the channels by which they communicate. A general solution to the secure multiparty computation problem is a compiler which given any feasible function describes an efficient protocol which allows the parties to compute the function securely on their local inputs over an open network. Over the past twenty years the secure multiparty computation problem has been the subject of a large body of research, both research into the models of multiparty computation and research aimed at realizing general secure multiparty computation. The main approach to realizing secure multiparty computation has been based on verifiable secret sharing as computation, where each party holds a secret share of each input and during the execution computes secret shares of all intermediate values. This approach allows the parties to keep all inputs and intermediate values secret and to pool the shares of the output values to learn exactly these values. More recently an approach based on joint homomorphic encryption was introduced, allowing for an efficient realization of general multiparty computation secure against an eavesdropper. A joint encryption scheme is an encryption scheme where all parties can encrypt, but where it takes the participation of all parties to open an encryption. The computation then starts by all parties broadcasting encryptions of their inputs and progresses through computing encryptions of the intermediary values using the homomorphic properties of the encryption scheme. The encryptions of the outputs can then be jointly decrypted. In this dissertation we extend this approach by using threshold homomorphic encryption to provide full-fledged security against an active attacker which might control some of the parties and which might take over parties during the course of the protocol execution. As opposed to a joint encryption scheme a threshold encryption scheme only requires that a given number of the parties are operating correctly for decryption to be possible. In this dissertation we show that threshold homomorphic encryption allows to solve the secure general multiparty computation problem more efficiently than previous approaches to the problem. Starting from an open point-to-point network there is a long way to general secure multiparty computation. The dissertation contains contributions at several points along the way. In particular we investigate how to realize secure channels. We also show how threshold signature schemes can be used to efficiently realize a broadcast channel and how threshold cryptography can be used to provide the parties of the network with a source of shared randomness. All three tools are well-known to be extremely powerful resources in a network, and our principle contribution is to provide efficient realizations. The dissertation also contains contributions to the definitional part of the field of multiparty computation. Most significantly we augment the recent model of universally composable security with a formal notion of what it means for a protocol to only realize a given problem under a given restricted input-output behavior of the environment. This e.g. allows us to give the first specification of a universally composable zero-knowledge proof of membership which does not at the same time require the proof system to be a proof of knowledge.
We consider a minimal scenario for secure computation: Parties A and B have private inputs x and y and a shared random string r. A and B are each allowed to send a single message to a third party C, from which C is to learn the value of f(x, y) for some function f, but nothing else. We show that this model is surprisingly powerful: every function f can be securely computed in this fashion. If the messages are required to be of polynomial size, then we exhibit an efficient protocol for any function f computable in nondeterministic logspace. Using a computational notion of security, we exhibit efficient protocols for any polynomial-time computable function f, assuming the existence of one-way functions. The above results generalize to the case where there are more than two parties with private inputs. The minimalistic nature of our model makes it easy to transform positive results achieved in our model to other more general models of secure computation. It also gives hope for lowerbound proofs. We give an alternative characterization of our model in terms of graph embeddings, and use this to show that for most Boolean functions on {0, 1} n × {0, 1} n, the need to hide just one of the input bits from C requires a communication overhead of n bits. 1
We study the complexity of securely evaluating arithmetic circuits over finite rings. This question is motivated by natural secure computation tasks. Focusing mainly on the case of two-party protocols with security against malicious parties, our main goals are to: (1) only make black-box calls to the ring operations and standard cryptographic primitives, and (2) minimize the number of such black-box calls as well as the communication overhead. We present several solutions which differ in their efficiency, generality, and underlying intractability assumptions. These include: 1. An unconditionally secure protocol in the OT-hybrid model which makes a black-box use of an arbitrary ring R, but where the number of ring operations grows linearly with (an upper bound on) log∣R∣. 2. Computationally secure protocols in the OT-hybrid model which make a black-box use of an underlying ring, and in which the number of ring operations does not grow with the ring size. These results extend a previous approach of Naor and Pinkas for secure polynomial evaluation (SIAM J. Comput., 35(5), 2006). 3. A protocol for the rings Zm=Z/mZ which only makes a black-box use of a homomorphic encryption scheme. When m is prime, the (amortized) number of calls to the encryption scheme for each gate of the circuit is constant. All of our protocols are in fact UC-secure in the OT-hybrid model and can be generalized to multiparty computation with an arbitrary number of malicious parties.
The notion of secure computation is central to cryptography. Introduced in the seminal works of Yao [FOCS'82, FOCS'86] and Goldreich, Micali and Wigderson [STOC'87], secure multiparty computation allows a group of (mutually) distrustful parties to jointly compute any functionality over their individual private inputs in such a manner that the honest parties obtain the correct outputs and no group of malicious parties learn anything beyond their inputs and the prescribed outputs. General feasibility results for secure computation were given by Yao [FOCS'86] and Goldreich et al. [STOC'87] more than two decades ago. Subsequent to these works, designing secure computation protocols that can tolerate more powerful adversaries and satisfy stronger notions of security has been a very active area of research. In this dissertation, we study two such new frontiers in the area of secure computation.
In the first part of this dissertation, we initiate a study of designing leakage-resilient interactive protocols. Specifically, we consider the scenario where an adversary, in addition to corrupting a subset of parties, can leak (potentially via physical attacks) arbitrary information from the secret state of any honest party. This is in contrast to the standard notion of secure computation where it is assumed that the adversary only has black-box'' access to the honest parties. In particular, we formalize a meaningful definition of leakage-resilient zero knowledge proof systems and provide constructions that satisfy our notion. We also discuss various applications of our results.
The second part of this dissertation concerns with the general question of whether it is possible to securely run cryptographic protocols over an insecure network environment such as the Internet. It is well-known that the standard notion of secure computation is only relevant to the stand-alone setting where a single protocol is being executed in isolation; as such it does not guarantee security when multiple protocol sessions may be executed concurrently under the control of an adversary who is present across all sessions. We consider the open problem of constructing secure password-based authenticated key exchange protocols in such a setting in the plain model (i.e., without assuming any trusted infrastructure or random oracles). We give the first construction of such a protocol based on standard cryptographic assumptions. Our results are in fact much more general, and extend to other functionalities w.r.t. a (necessarily) weakened notion of concurrently secure computation.
The results presented in this dissertation stem from two papers which are joint works with Sanjam Garg and Amit Sahai, and with Vipul Goyal and Rafail Ostrovsky, respectively.