Reducing the minimum assumptions needed to construct various cryptographic primitives is an important andinteresting task in theoretical cryptography. Oblivious Transfer, one of the most basic cryptographic building blocks, is also studied under this scenario. Reducing the minimum assumptions for Oblivious Transfer seems not an easy task, as there are a few impossibility results under black-box reductions. Until recently, it is widely believed that Oblivious Transfer can be constructed with trapdoor permutations but not trapdoor functions in general. In this paper, we enhance previous results and show one Oblivious Transfer protocol based on a collection of trapdoor functions with some extra properties. We also provide reasons for adding the extra properties and argue that the assumptions in the protocol are nearly minimum.
Reducing the minimum assumptions needed to construct various cryptographic primitives is an important andinteresting task in theoretical cryptography. Oblivious Transfer, one of the most basic cryptographic building blocks, is also studied under this scenario. Reducing the minimum assumptions for Oblivious Transfer seems not an easy task, as there are a few impossibility results under black-box reductions. Until recently, it is widely believed that Oblivious Transfer can be constructed with trapdoor permutations but not trapdoor functions in general. In this paper, we enhance previous results and show one Oblivious Transfer protocol based on a collection of trapdoor functions with some extra properties. We also provide reasons for adding the extra properties and argue that the assumptions in the protocol are nearly minimum.
In cryptographic protocols, honest parties would prefer that their security is assured even in presence of adversarial parties who have unbounded computational power. Information theoretic secure realization of cryptographic primitives provides such guarantees; but for most tasks such strong security guarantees cannot be provided for any reasonable notion of security. The standard technique used in cryptography is to assume the existence of some puzzle, whose hard instances are easy to generate but no efficient algorithm can solve them. Such assumptions, which define intractable problems for efficient algorithms, are called computational intractability assumptions.
In this work, we motivate a study of computational intractability assumptions which is goal driven and lends support to the fundamental nature of some of the traditional assumptions beyond being historical accidents. Secure multi-party computation deals with the study of constructing secure protocols for general cryptographic tasks which conform to various notions of security. Inspired by complexity theory, we use the notion of reduction to further our understanding of computational intractability assumptions. Our framework explores the hardness of natural assumptions of the form: “Task F can be securely computed given an ideally secure facility for computing G ”.
Formally, we characterize the minimal computational intractability assumption which is sufficient to securely realize a functionality using trusted copies of some other functionality. A functionality F reduces to G , similar to the complexity-theoretic notion of reduction, if F can be securely realized when parties can access trusted copies of G . Thus, if F does not reduce to G information-theoretically, we are interested in characterizing the minimal assumption, which is sufficient, to securely realize the reduction of F to G .
In our framework, we explore the relative strengths of these minimal computational assumptions. The fundamental problem, then, is to establish relations, like implication, equivalence and separation, among the assumptions which correspond to various reductions. In this work, we further our understanding of the minimal computational assumptions corresponding to such reductions by showing the following results:
1. We identify a spectrum for the set of assumptions that correspond to reduction between a pair of tasks under different security notions. These reductions are implied by the computational intractability assumption: “semi-honest secure protocol for oblivious transfer exists” (sh-OT assumption). Additionally, the information theoretically impossible reductions considered in this thesis imply the assumption: “one-way functions exist” (OWF assumption). We conjecture that OWF assumption is necessary for these reductions. In fact, the reductions considered in this thesis are either information theoretically true, false, equivalent to OWF assumption or equivalent to sh-OT assumption [MPR10a].
We expand our study to encompass reductions involving randomized functionalities and study the consequences of providing parties with a trusted source of common unbiased coins. For a strong notion of security they turn out to be useless and unless sh-OT assumption holds these randomness sources are not useful [MOPR11, MP11].
2. We [MPS10] seek to lend additional support to the widely believed premise that “non-trivial cryptography entails the existence of one-way functions”. We show that (constant round) weak coin-tossing protocols, imply OWF assumption. For the general problem, i.e. secure weak coin-tossing protocols with polynomial round complexity, we show a slightly weaker implication NP n BPP.
3. Finally, we indicate the possibility of a wide range of intractability assumptions in our spectrum, intermediate to OWF assumption and sh-OT assumption. We show that a class of natural assumptions corresponding to reductions that are relativistically separated from OWF assumption. Similar techniques have been used to show their separation from the assumption that “public-key encryption exists” (PKE assumption) [MMP11]; but these assumptions are not known to imply sh-OT assumption. We conjecture that, in fact, sh-OT assumption is relativistically separated from these assumptions.
In this paper we study the relationships among some of the most fundamental primitives and protocols in cryptography: public-key encryption (i.e. trapdoor predicates), oblivious transfer (which is equivalent to general secure multi-party computation), key agreement and trapdoor permutations. Our main results show that public-key encryption and oblivious transfer are incomparable under black-box reductions. These separations are tightly matched by our positive results where a restricted (strong) version of one primitive does imply the other primitive. We also show separations between oblivious transfer and key agreement. Finally, we conclude that neither oblivious transfer nor trapdoor predicates imply trapdoor permutations. Our techniques for showing negative results follow the oracle separations of R. Impagliazzo and S. Rudich (1989).
Oblivious transfer is one of the most important cryptographic primitives, both for theoretical and practical reasons and several protocols were proposed during the years. We provide the first oblivious transfer protocol which is simultaneously optimal on the following list of parameters: Security: it has universal composition. Trust in setup assumptions: only one of the parties needs to trust the setup (and some setup is needed for UC security). Trust in computational assumptions: only one of the parties needs to trust a computational assumption. Round complexity: it uses only two rounds. Communication complexity: it communicates O(1) group elements to transfer one out of two group elements. The Big-O notation hides 32, meaning that the communication is probably not optimal, but is essentially optimal in that the overhead is at least constant. Our construction is based on pairings, and we assume the presence of a key registration authority.
Oblivious transfer is one of the most basic and important building blocks in cryptography. As such, understanding its cost is of prime importance. Beaver (STOC 1996) showed that it is possible to obtain poly(n) oblivious transfers given only n actual oblivious transfer calls and using one-way functions, where n is the security parameter. In addition, he showed that it is impossible to extend oblivious transfer information theoretically. The notion of extending oblivious transfer is important theoretically (to understand the complexity of computing this primitive) and practically (since oblivious transfers can be expensive and thus extending them using only one-way functions is very attractive). Despite its importance, very little is known about the feasibility of extending oblivious transfer, beyond the fact that it is impossible information theoretically. Specifically, it is not known whether or not one-way functions are actually necessary for extending oblivious transfer, whether or not it is possible to extend oblivious transfers with adaptive security, and whether or not it is possible to extend oblivious transfers when starting with just a few. In this paper, we address these questions and provide almost complete answers to all of them. We show that the existence of any oblivious transfer extension protocol with security for static semihonest adversaries implies one-way functions, that an oblivious transfer extension protocol with adaptive security implies oblivious transfer with static security, and that the existence of an oblivious transfer extension protocol from only O(log n) oblivious transfers implies oblivious transfer itself.
Which computational complexity assumptions are inherent to cryptography? We present a broad framework to pose and investigate this question. We first aim to understand the “cryptographic complexity ” of various tasks, independent of any computational assumptions. In our framework the cryptographic tasks are modeled as multi-party computation functionalities. We consider a universally composable secure protocol for one task given access to another task as the most natural complexity reduction between the two tasks. Some of these cryptographic complexity reductions are unconditional, others are unconditionally non-existent, but the vast majority appear to depend on computational assumptions; it is this relationship with computational assumptions that we study. In our detailed investigation of large classes of 2-party functionalities, we find that every reduction we are able to classify turns out to be unconditionally true or false, or else equivalent to the existence of one-way functions (OWF) or of semi-honest (equivalently, standalone-secure) oblivious transfer protocols (sh-OT). This leads us to conjecture that there are only a small finite number of distinct (i.e., black-box separable) computational assump-tions that are inherent among the infinite number of different cryptographic reductions in our framework. On moving to 3-party functionalities, there does exist a natural computational assumption, distinct from the OWF and sh-OT assumptions (namely, the existence of a key agreement protocol) that manifests itself as a cryptographic
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