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How to Derive Lower Bound on Oblivious Transfer Reduction.

Kaoru Kurosawa, Wataru Kishimoto, Takeshi Koshiba

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

Suppose that we are given an ideal oblivious transfer protocol (OT). We wish to construct a larger OT by using the above OT as a blackbox. Then how many instances of the given ideal OT should be invoked? For this problem, some lower bounds were derived using entropy. In this paper, we show more tight lower bounds by using combinatorial techniques. Roughly speaking, our lower bounds are two times larger than the previous bounds.

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Suppose that we are given an ideal oblivious transfer protocol (OT). We wish to construct a larger OT by using the above OT as a blackbox. Then how many instances of the given ideal OT should be invoked? For this problem, some lower bounds were derived using entropy. In this paper, we show more tight lower bounds by using combinatorial techniques. Roughly speaking, our lower bounds are two times larger than the previous bounds.
Consider the scenario where we are given an ideal functionality of oblivious transfer (OT), and we wish to construct a larger OT by invoking the above functionality as a black box. How many invocations of an ideal OT functionality are necessary? In tackling this problem, some lower bounds were derived using entropy previously. In this paper, we manage to achieve tighter lower bounds by employing a combinatorial approach. This new approach yields lower bounds which are two times larger than the existing bounds.
One of the most important primitives in two-party distrustful cryptography is oblivious transfer, a complete primitive for two-party computation. Recently introduced, the oblivious transfer capacity of a noisy channel measures an efficiency of information theoretical reductions from 1-out-of-k, l-string oblivious transfer to noisy channels. It is defined as the maximal achievable ratio l/n, where l is the length of the strings which are to be transferred and n is the number of times the noisy channel is invoked. This quantity is unknown in a general case. For discrete memoryless channels, it is known to be nonnegligible for honest-but-curious players, but the non-zero rates have not ever been proved achievable in the case of malicious players. Here, we show that in the particular case of the erasure channel, more precise answers can be obtained. We compute the OT capacity of the erasure channel for the case of honest-but-curious players and, for the fully malicious players, we give its lower bound.
We revisit the oblivious transfer (OT) capacities of noisy channels against the passive adversary, which have been identified only for a limited class of channels. In the literature, the general construction of oblivious transfer has been known only for generalized erasure channels (GECs); for other channels, we first convert a given channel to a GEC via alphabet extension and erasure emulation, and then apply the general construction for GEC. In this paper, we derive an improved lower bound on the OT capacity of the binary symmetric channel (BSC) and binary symmetric erasure channel (BSEC) by proposing a new protocol; by using interactive communication between the sender and the receiver, our protocol emulates erasure events recursively in multiple rounds. We also discuss a potential necessity of multiple rounds interactive communication to attain the OT capacity.
1 IntroductionOblivious Transfer (OT) protocols allow one party, the sender, to transmit part of its inputs to another party, the chooser, in a manner that protects both of them: the sender is assured that the chooser does not receive more information than it is entitled, while the chooser is assured that the sender does not learn which part of the inputs it received. OT is used as a key component in many applications of cryptography. Its computational requirements are quite demanding and they are likely to be the bottleneck in many applications that invoke it.1.1 Contributions.This paper presents several significant improvements to oblivious transfer (OT) protocols of strings, and in particular: (i) Improving the efficiency of applications which many invocations of oblivious transfer. (ii) Providing the first two-round OT protocol whose security analysis does not invoke the random oracle model.
Since oblivious transfer, a primitive of paramount importance in secure two- and multiparty computation, cannot be realized in an unconditionally secure way for both parties from scratch, reductions to weak information-theoretic primitives as well as between different variants of the functionality are of great interest. In this context, various monotones-quantities that cannot be increased by any protocol-are introduced and then used to derive lower bounds on the possibility and efficiency of such reductions.
We consider the oblivious transfer (OT) capacities of noisy channels against the passive adversary; this problem has not been solved even for the binary symmetric channel (BSC). In the literature, the general construction of OT has been known only for generalized erasure channels (GECs); for the BSC, we convert the channel to the binary symmetric erasure channel (BSEC), which is a special instance of the GEC, via alphabet extension and erasure emulation. In a previous paper by the authors, we derived an improved lower bound on the OT capacity of BSC by proposing a method to recursively emulate BSEC via interactive communication. In this paper, we introduce two new ideas of OT construction: (i) via “polarization” and interactive communication, we recursively emulate GECs that are not necessarily a BSEC; (ii) in addition to the GEC emulation part, we also utilize interactive communication in the key agreement part of OT protocol. By these methods, we derive lower bounds on the OT capacity of BSC that are superior to the previous one for a certain range of crossover probabilities of the BSC. Via our new lower bound, we show that, at the crossover probability being zero, the slope of tangent of the OT capacity is unbounded.