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Annual Symposium on Foundations of Computer Science

Oblivious transfer with a memory-bounded receiver

Christian Cachin, Claude Crépeau, Julien Marcil

Published Nov 27, 2002• 111 citations
Publisher

Abstract

We propose a protocol for oblivious transfer that is unconditionally secure under the sole assumption that the memory size of the receiver is bounded. The model assumes that a random bit string slightly larger than the receiver's memory is broadcast (either by the sender or by a third party). In our construction, both parties need memory of size in /spl theta/(n/sup 2-2/spl alpha//) for some /spl alpha//spl beta/>0, whereas a malicious receiver can have up to /spl gamma/N bits of memory for any /spl gamma/<1. In the course of our analysis, we provide a direct study of an interactive hashing protocol closely related to that of M. Naor et al. (1998).

DOI
10.1109/sfcs.1998.743500

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We propose a protocol for oblivious transfer that is unconditionally secure under the sole assumption that the memory size of the receiver is bounded. The model assumes that a random bit string slightly larger than the receiver's memory is broadcast (either by the sender or by a third party). In our construction, both parties need memory of size in /spl theta/(n/sup 2-2/spl alpha//) for some /spl alpha//spl beta/>0, whereas a malicious receiver can have up to /spl gamma/N bits of memory for any /spl gamma/<1. In the course of our analysis, we provide a direct study of an interactive hashing protocol closely related to that of M. Naor et al. (1998).
Abstract. In this paper we propose an efficient OT N 1 scheme in the bounded storage model, which is provably secure without complexity assumptions. Under the assumption that a public random string of M bits is broadcasted, the protocol is secure against any computationally unbounded dishonest receiver who can store τM bits, τ < 1. The protocol requires the sender and the receiver to store N · O ( √ kM) bits, where k is a security parameter. When N = 2, our protocol is similar to that of Ding [10] but has more efficient round and communication complexities. Moreover, in case of N> 2, if the sender and receiver can store N ·O ( √ kM) bits, we are able to construct a protocol for OT N 1 which has almost the same complexity as in OT 2 1 scheme. Ding’s protocol was constructed by using the interactive hashing protocol which is introduced by Noar, Ostrovsky, Venkatesan and Yung [15] with very large roundcomplexity. We propose an efficiently extended interactive hashing and analyze its security. This protocol answers partially an open problem raised in [10]. 1
Interactive hashing has featured as an essential ingredient in protocols realizing a large variety of cryptographic tasks, notably oblivious transfer in the bounded storage model. In interactive hashing, a sender transfers a bit string to a receiver such that two strings are received, the original string and a second string that appear to be chosen at random. This paper presents a self-contained, information theoretic study of interactive hashing. We start by formalizing the notion of interactive hashing as a cryptographic primitive, disentangling it from the specifics of its various implementations. To this end, we present an application-independent set of information theoretic conditions that all interactive hashing protocols must ideally satisfy. We then provide a detailed analysis of a standard implementation of interactive hashing which is shown to satisfy all the conditions of our definition. Our analysis represents a significant improvement over previous attempts in more restricted contexts. Despite its generality, it offers a considerably simpler proof of security. Moreover, it establishes a tighter upper bound on the cheating probability of a dishonest sender, who wishes to manipulate the protocol so that both output strings have some rare desirable property. In particular, we prove that if the set of desirable strings for the dishonest sender represents a fraction f of all strings, then the probability that both outputs will be from this set is no larger than 15.6805 · f. This upper bound is valid for any f and is tight up to a small constant, since a sender acting honestly would get two outputs from this set with probability very close to f. We illustrate the power of interactive hashing as a cryptographic tool by surveying protocols achieving oblivious transfer in the bounded storage model, which typically rely heavily on interactive hashing.
Interactive Hashing has featured as an essential ingredient in protocols realizing a large variety of cryptographic tasks. We present a study of this important cryptographic tool in the information theoretic context. We start by presenting a security definition which is independent of any particular setting or application. We then show that a standard implementation of Interactive Hashing satisfies all the conditions of our definition. Our proof of security improves upon previous ones in several ways. Despite its generality, it is considerably simpler. Moreover, it establishes a tighter upper bound on the cheating probability of a dishonest sender. Specifically, we prove that if the fraction of good strings for a dishonest sender is f, then the probability that both outputs will be good is no larger than 15.6805· f. This upper bound is valid for any f and is tight up to a small constant since a sender acting honestly would get two good outputs with probability very close to f. We illustrate the potential of Interactive Hashing as a cryptographic primitive by demonstrating efficient reductions of String Oblivious Transfer with string length k to Bit Oblivious Transfer and several weaker variants. Our reductions incorporate tests based on Interactive Hashing that allow the sender to verify the receiver's adherence to the protocol without compromising the latter's privacy. This allows a much more efficient use of the available entropy without any appreciable impact on security. As a result, for Bit OT and most of its variants n = (1 + epsilon) k executions suffice, improving efficiency by a factor of two or more compared to the most efficient reductions that do not use Interactive Hashing. As it is theoretically impossible to achieve an expansion factor n/k smaller than 1, our reductions are in fact asymptotically optimal. They are also more general since they place no restrictions on the types of 2-universal hash families used for Privacy Amplification. Lastly, we present a direct reduction of String OT to Rabin OT which uses similar methods to achieve an expansion factor of 2 + epsilon which is again asymptotically optimal.
Two novel constructions of one out of n oblivious transfer protocols from bilinear maps between groups are proposed. The protocols satisfy the category of non-interactive, mean that any sender S can obliviously transfer some secret(s) to a receiver R, but R does not send any messages back to S. The choice of a receiver is unconditionally secure. A sender's secrecy is guaranteed if the receiver is semi-honest in a standard model since the decision bilinear Diffie-Hellman problem is hard and the sender's secrecy is achieved when the receiver is a malicious party in a random oracle model since the bilinear Diffie-Hellman problem assumption holds. When a sender is a cheating party, the receiver will detect him/her and halt the protocol. A precise proof of the security of the protocols is presented.
We initiate the study of two-party cryptographic primitives with unconditional security, assuming that the adversary's quantum memory is of bounded size. We show that oblivious transfer and bit commitment can be implemented in this model using protocols where honest parties need no quantum memory, whereas an adversarial player needs quantum memory of size at least n/2 in order to break the protocol, where n is the number of qubits transmitted. This is in sharp contrast to the classical bounded-memory model, where we can only tolerate adversaries with memory of size quadratic in honest players' memory size. Our protocols are efficient, non-interactive and can be implemented using today's technology. On the technical side, a new uncertainty relation is established
T wo non-interactive three parties k out of n oblivious transfer protocols from bilinear pairings are proposed i n this paper. In these protocols , a sender can obliviously transfer n messages to a receiver without communication with him/her. The public keys that a sender used to encrypt his/her messages are made by a third party. The receiver can extract k out of n messages at his/her choices by using the corresponding secret keys. The choices of the receiver are unconditionally secure. The sender's secrecy is guaranteed if the receiver is semi-honest in the standard model since the decision bilinear Diffie-Hellman problem (DBDH) is hard and the sender's secrecy is achieved when the receiver is a malicious party in the random oracle model since the bilinear Diffie-Hellman problem assumption (BDHA) holds. When a sender is a cheating party, the receiver will detect him/her and halt the protocol. A precise proof of the security of the protocols is presented.