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).
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).
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
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-black-box 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 semihonest parties. As a corollary, we obtain the first constructions of general multiparty protocols (with security against malicious adversaries and without an honest majority) which make only a black-box use of semihonest 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 its actions appear to be honest, then it is guaranteed that it learned nothing more than its prescribed output. We then show how to construct defensible oblivious transfer from semihonest oblivious transfer, and malicious oblivious transfer from defensible oblivious transfer, all in a black-box way.
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.
It is well known that the secure computation of non-trivial functionalities in the setting of no honest majority requires computational assumptions. We study the way such computational assumptions are used. Specifically, we ask whether the secure protocol can use the underlying primitive (e.g., one-way trapdoor permutation) in a black-box way, or must it be nonblack-box (by referring to the code that computes this primitive)? Despite the fact that many general constructions of cryptographic schemes (e.g., CPA-secure encryption) refer to the underlying primitive in a black-box way only, there are some constructions that are inherently nonblack-box. Indeed, all known constructions of protocols for general secure computation that are secure in the presence of a malicious adversary and without an honest majority use the underlying primitive in a nonblack-box way (requiring to prove in zero-knowledge statements that relate to the primitive).In this paper, we study whether such nonblack-box use is essential. We present protocols that use only black-box access to a family of (enhanced) trapdoor permutations or to a homomorphic public-key encryption scheme. The result is a protocol whose communication complexity is independent of the computational complexity of the underlying primitive (e.g., a trapdoor permutation) and whose computational complexity grows only linearly with that of the underlying primitive. This is the first protocol to exhibit these properties.
The seminal result of Impagliazzo and Rudich (STOC 1989) gave a black-box separation between one-way functions and public-key encryption: a public-key encryption scheme cannot be constructed using one-way functions in a black-box way. In addition, their result implied black-box separations between one-way functions and protocols for certain Secure Function Evaluation (SFE) functionalities (in particular, Oblivious Transfer). Surprisingly, however, since then there has been no further progress in separating one-way functions and SFE functionalities. In this work, we present the complete picture for finite deterministic 2-party SFE functionalities, vis a vis one-way functions. We show that in case of semi-honest adversaries, one-way functions are black-box separated from all such SFE functionalities, except the ones which have unconditionally secure protocols (and hence do not rely on any computational hardness). In the case of active adversaries, a black-box one-way function is indeed useful for SFE, but we show that it is useful only as much as access to an ideal commitment functionality is useful.
The seminal result of Impagliazzo and Rudich (STOC 1989) gave a black-box separation between one-way functions and public-key encryption: informally, a public-key encryption scheme cannot be constructed using one-way functions as the sole source of computational hardness. In addition, this implied a black-box separation between one-way functions and protocols for certain Secure Function Evaluation (SFE) functionalities (in particular, Oblivious Transfer). Surprisingly, however, {\em since then there has been no further progress in separating one-way functions and SFE functionalities} (though several other black-box separation results were shown). In this work, we present the complete picture for deterministic 2-party SFE functionalities. We show that one-way functions are black-box separated from {\em all such SFE functionalities}, except the ones which have unconditionally secure protocols (and hence do not rely on any computational hardness), when secure computation against semi-honest adversaries is considered. In the case of security against active adversaries, a black-box one-way function is indeed useful for SFE, but we show that it is useful only as much as access to an ideal commitment functionality is useful. Technically, our main result establishes the limitations of random oracles for secure computation.