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IEEE Communications Letters

Oblivious Transfer Over Compound Binary Erasure Channels

Anuj Kumar Yadav, Preetam Kumar

Published Mar 3, 2022• 2 citations
Publisher

Abstract

Oblivious Transfer (OT) is a broadly studied cryptographic primitive which involves two mutually distrustful users who wish to interact with each other in order to transfer messages in an oblivious manner. Noise is an important resource to realize information-theoretically secure cryptographic protocols including oblivious transfer. However, oftentimes these noisy channels may be poorly characterized and the complete channel characteristics might be unknown to both users. Such channels are studied in the literature through the lens of compound channels. We study 1-out-of-2 oblivious transfer over a compound binary erasure channel (compound-BEC) and characterize the oblivious transfer capacity of compound-BEC. Our OT capacity characterization involves a general alphabet compound-DMC converse, subsequently specialized to compound-BEC along with an optimal and computationally-efficient achievability scheme for honest-but-curious users.

DOI
10.1109/lcomm.2022.3156704

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Oblivious Transfer (OT) is a broadly studied cryptographic primitive which involves two mutually distrustful users who wish to interact with each other in order to transfer messages in an oblivious manner. Noise is an important resource to realize information-theoretically secure cryptographic protocols including oblivious transfer. However, oftentimes these noisy channels may be poorly characterized and the complete channel characteristics might be unknown to both users. Such channels are studied in the literature through the lens of compound channels. We study 1-out-of-2 oblivious transfer over a compound binary erasure channel (compound-BEC) and characterize the oblivious transfer capacity of compound-BEC. Our OT capacity characterization involves a general alphabet compound-DMC converse, subsequently specialized to compound-BEC along with an optimal and computationally-efficient achievability scheme for honest-but-curious users.
Noisy channels are a powerful resource for cryptography as they can be used to obtain information-theoretically secure key agreement, commitment and oblivious transfer protocols, among others. Oblivious transfer (OT) is a fundamental primitive since it is complete for secure multi-party computation, and the OT capacity characterizes how efficiently a channel can be used for obtaining string oblivious transfer. Ahlswede and Csiszár (\emph{ISIT'07}) presented upper and lower bounds on the OT capacity of generalized erasure channels (GEC) against passive adversaries. In the case of GEC with erasure probability at least 1/2, the upper and lower bounds match and therefore the OT capacity was determined. It was later proved by Pinto et al. (\emph{IEEE Trans. Inf. Theory 57(8)}) that in this case there is also a protocol against malicious adversaries achieving the same lower bound, and hence the OT capacity is identical for passive and malicious adversaries. In the case of GEC with erasure probability smaller than 1/2, the known lower bound against passive adversaries that was established by Ahlswede and Csiszár does not match their upper bound and it was unknown whether this OT rate could be achieved against malicious adversaries as well. In this work we show that there is a protocol against malicious adversaries achieving the same OT rate that was obtained against passive adversaries. In order to obtain our results we introduce a novel use of interactive hashing that is suitable for dealing with the case of low erasure probability (p∗<1/2p^* <1/2p∗<1/2).
Noisy channels are a powerful resource for cryptography as they can be used to obtain information-theoretic secure key agreement, commitment, and oblivious transfer protocols, among others. Oblivious transfer (OT) is a fundamental primitive, since it is complete for secure multi-party computation, and the OT capacity characterizes how efficiently a channel can be used for obtaining string oblivious transfer. Ahlswede and Csiszár (ISIT'07) presented upper and lower bounds on the OT capacity of generalized erasure channels (GECs) against passive adversaries. In the case of GEC with erasure probability at least 1/2, the upper and lower bounds match and, therefore, the OT capacity was determined. It was later proved by Pinto et al. [IEEE Trans. Inf. Theory 57(8)] that the OT capacity is identical for passive and malicious adversaries. In the case of GEC with erasure probability smaller than 1/2, the known lower bound against passive adversaries that was established by Ahlswede and Csiszár does not match their upper bound and it was unknown whether this OT rate could be achieved against malicious adversaries as well. In this paper, we show that there is a protocol against malicious adversaries achieving the same OT rate that was obtained against passive adversaries. We obtain our results by a new combination of interactive hashing and typicality tests that are suitable for dealing with the case of low erasure probability (p* <;1/2 ).
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.
In the commitment problem, two mutually distrustful parties Alice and Bob interact in a two-phase protocol, viz., commit and reveal phase, to achieve commitment over a bit string that Alice possesses. The protocol successfully achieves commitment if, firstly, Alice can commit to sharing a string with Bob, with the guarantee that this string remains hidden from Bob until she chooses to reveal it to him. Secondly, when Alice does reveal a string, Bob is able to detect precisely whether the revealed string is different from the one Alice committed to sharing. Information-theoretically secure commitment is impossible if Alice and Bob communicate only noiselessly; however, communication using a noisy channel can be a resource to realize commitment. Even though a noisy channel may be available, it is possible that the corresponding channel law is imprecisely known or poorly characterized. We define and study a compound-binary symmetric channel (compound-BSC) which models such a scenario. A compound-BSC is a BSC whose transition probability is fixed but unknown to either party; the set of potential values which this transition probability can take, though, is known to both parties a priori. In this work, we completely characterize the maximum commitment throughput or commitment capacity of a compound-BSC. We provide an optimal, computationally-efficient scheme for our achievability, and we derive a converse for general alphabet compound DMCs, which is then specialized for compound-BSCs.
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.