It is well known that it is impossible for two parties to toss a coin fairly (Cleve, STOC 1986). This result implies that it is impossible to securely compute with fairness any function that can be used to toss a coin fairly. In this paper, we focus on the class of deterministic Boolean functions with finite domain, and we ask for which functions in this class is it possible to information-theoretically toss an unbiased coin, given a protocol for securely computing the function with fairness. We provide a complete characterization of the functions in this class that imply and do not imply fair coin tossing. This characterization extends our knowledge of which functions cannot be securely computed with fairness. In addition, it provides a focus as to which functions may potentially be securely computed with fairness, since a function that cannot be used to fairly toss a coin is not ruled out by the impossibility result of Cleve (which is the only known impossibility result for fairness). In addition to the above, we draw corollaries to the feasibility of achieving fairness in two possible fail-stop models. Keywords: Fairness, coin tossing, secure computation, malicious and fail-stop adversaries
It is well known that it is impossible for two parties to toss a coin fairly (Cleve, STOC 1986). This result implies that it is impossible to securely compute with fairness any function that can be used to toss a coin fairly. In this paper, we focus on the class of deterministic Boolean functions with finite domain, and we ask for which functions in this class is it possible to information-theoretically toss an unbiased coin, given a protocol for securely computing the function with fairness. We provide a complete characterization of the functions in this class that imply and do not imply fair coin tossing. This characterization extends our knowledge of which functions cannot be securely computed with fairness. In addition, it provides a focus as to which functions may potentially be securely computed with fairness, since a function that cannot be used to fairly toss a coin is not ruled out by the impossibility result of Cleve (which is the only known impossibility result for fairness). In addition to the above, we draw corollaries to the feasibility of achieving fairness in two possible fail-stop models. Keywords: Fairness, coin tossing, secure computation, malicious and fail-stop adversaries
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. Since then, 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, nite-domain) boolean functions that can be computed fairly. This raises the fundamental question of characterizing complete fairness in secure two-party computation. In this work we show that not only that some or few functions can be computed fairly, but rather an enormous number of functions can be computed fairly. In fact, almost all boolean functions with distinct domain sizes can be computed with complete fairness (for instance, more than 99:999 % of the boolean functions with domain sizes 31 30). The class of functions that is shown to be possible includes also rather involved and highly non-trivial tasks, such as set-membership, evaluation of a private (boolean) function, private matchmaking and set-disjointness.
Abstract. Fairness is a desirable property in secure computation; informally it means that if one party gets the output of the function, then all parties get the output. Alas, an implication of Cleve’s result [18th ACM Symposium on the Theory of Computing, 1986] is that when there is no honest majority, in particular in the important case of the two-party setting, there exist functions that cannot be computed with fairness. In a surprising result, Gordon et al. [18th ACM Symposium on the Theory of Computing, 2008; J. ACM, 58 (2011), 24] showed that some interesting functions can be computed with fairness in the two-party setting and reopened the question of understanding which Boolean functions can be computed with fairness, and which cannot. Our main result in this work is a complete characterization of the (symmetric) Boolean functions that can be computed with fairness in the two-party setting; this settles an open problem of Gordon et al. The statement of the characterization is quite simple: A function can be computed with fairness if and only if the all-one vector or the all-zero vector are in the affine span of either the rows or the columns of the matrix describing the function. This is true for both deterministic and randomized functions. To prove the possibility result, we modify the protocol of Gordon et al.; the resulting protocol computes with full security (and in particular with fairness) all functions that are computable with fairness. Complementing this result, we also show that any function that does not satisfy the aforementioned condition can be reduced to a fair sampling protocol, which, by Agrawal and Prabhakaran [ Advances in Cryptology – CRYPTO 2013, 2013], cannot be computed with fairness.
In the setting of secure two-party computation, two mutually distrusting parties wish to compute some function of their inputs while preserving, to the extent possible, various security properties such as privacy, correctness, and more. One desirable property is fairness which guarantees, informally, that if one party receives its output, then the other party does too. Cleve (STOC 1986) showed that complete fairness cannot be achieved in general without an honest majority. Since then, the accepted folklore has been that nothing non-trivial can be computed with complete fairness in the two-party setting, and the problem has been treated as closed since the late ’80s. In this paper, we demonstrate that this folklore belief is false by showing completely-fair protocols for various non-trivial functions in the two-party setting based on standard cryptographic assumptions. We first show feasibility of obtaining complete fairness when computing any function over polynomial-size domains that does not contain an “embedded XOR”; this class of functions includes boolean AND/OR as well as Yao’s “millionaires ’ problem”. We also demonstrate feasibility for certain functions that do contain an embedded XOR, and prove a lower bound showing that any completely-fair protocol for such functions must have round complexity super-logarithmic in the security parameter. Our results demonstrate that the question of completely-fair secure computation without an honest majority is far from closed.
We address one of the foundational problems in cryptography: the bias of coin-\nipping pro-tocols. Coin-\nipping protocols allow mutually distrustful parties to generate a common unbiased random bit, guaranteeing that even if one of the parties is malicious, it cannot signicantly bias the output of the honest party. A classical result by Cleve [STOC '86] showed that for any two-party r-round coin-\nipping protocol there exists an efficient adversary that can bias the output of the honest party by Ω(1=r). However, the best previously known protocol only guarantees O(1= p r) bias, and the question of whether Cleve's bound is tight has remained open for more than twenty years. In this paper we establish the optimal trade-off between the round complexity and the bias of two-party coin-\nipping protocols. Under standard assumptions (the existence of oblivious transfer), we show that Cleve's lower bound is tight: we construct an r-round protocol with bias O(1=r).
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
A seminal result of Cleve (STOC ’86) is that, in general, complete fairness is impossible to achieve in two-party computation. In light of this, various techniques for obtaining partial fairness have been suggested in the literature. We propose a definition of partial fairness within the standard real-/ideal-world paradigm that addresses deficiencies of prior definitions. also show broad feasibility results with respect to our definition: partial fairness is possible for any (randomized) functionality f: X × Y → Z 1 × Z 2 at least one of whose domains or ranges is polynomial in size. Our protocols are always private, and when one of the domains has polynomial size our protocols also simultaneously achieve the usual notion of security with abort. In contrast to some prior work, we rely on standard assumptions only. We also show that, as far as general feasibility is concerned, our results are optimal. Specifically, there exist functions with super-polynomial domains and ranges for which it is impossible to achieve our definition.