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arXiv:2008.13551 · IEEE Transactions on Information Theory

Coding Constructions for Efficient Oblivious Transfer From Noisy Channels

Frédérique Oggier, Gilles Zémor

Published Jan 14, 2022• 3 citations
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Abstract

We consider oblivious transfer protocols performed over binary symmetric channels in a malicious setting where parties will actively cheat if they can. We provide constructions purely based on coding theory that achieve an explicit positive rate, the essential ingredient being the existence of linear codes whose Schur products are asymptotically good.

DOI
10.1109/tit.2022.3143310

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We consider oblivious transfer protocols performed over binary symmetric channels in a malicious setting where parties will actively cheat if they can. We provide constructions purely based on coding theory that achieve an explicit positive rate, the essential ingredient being the existence of linear codes whose Schur products are asymptotically good.
In this paper, we study the information-theoretic limits of oblivious transfer via noisy channels. We also investigate oblivious transfer over a noisy multiple-access channel with two non-colluding senders and a single receiver. The channel is modeled through correlations among the parties, who may be honest-but-curious or, in the case of the receiver, potentially malicious. We first revisit the information-theoretic limits of two-party oblivious transfer and then extend these results to the multiple-access setting. For honest-but-curious participants, we introduce a multiparty protocol that reduces a general multiple access channel to a suitable correlation model. In scenarios with a malicious receiver, we characterize an achievable oblivious transfer rate region.
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 a paper from EuroCrypt'99, Damgård, Kilian and Salvail show various positive and negative results on constructing Bit Commitment (BC) and Oblivious Transfer (OT) from Unfair Noisy Channels (UNC), i.e., binary symmetric channels where the error rate is only known to be in a certain interval [gamma ..delta] and can be chosen adversarily. They also introduce a related primitive called PassiveUNC. We prove in this paper that any OT protocol that can be constructed based on a PassiveUNC and is secure against a passive adversary can be transformed using a generic "compiler'' into an OT protocol based on a UNC which is secure against an active adversary. Apart from making positive results easier to prove in general, this also allows correcting a problem in the EuroCrypt'99 paper: There, a positive result was claimed on constructing from UNC an OT that is secure against active cheating. We point out that the proof sketch given for this was incomplete, and we show that a correct proof of a much stronger result follows from our general compilation result and a new technique for transforming between weaker versions of OT with different parameters.
Noisy channels are a valuable resource from a cryptographic point of view. They can be used for exchanging secret-keys as well as realizing other cryptographic primitives such as commitment and oblivious transfer. To be really useful, noisy channels have to be considered in the scenario where a cheating party has some degree of control over the channel characteristics. Damgård et al. (EUROCRYPT 1999) proposed a more realistic model where such level of control is permitted to an adversary, the so called unfair noisy channels, and proved that they can be used to obtain commitment and oblivious transfer protocols. Given that noisy channels are a precious resource for cryptographic purposes, one important question is determining the optimal rate in which they can be used. The commitment capacity has already been determined for the cases of discrete memoryless channels and Gaussian channels. In this work we address the problem of determining the commitment capacity of unfair noisy channels. We compute a single-letter characterization of the commitment capacity of unfair noisy channels. In the case where an adversary has no control over the channel (the fair case) our capacity reduces to the well-known capacity of a discrete memoryless binary symmetric channel.
We study quantum-state oblivious transfer over the quantum erasure channel under information-theoretic security guarantees. Specifically, Alice holds two quantum systems, Bob wishes to recover one of them, and the parties have access to independent channel uses and authenticated public classical communication, with no initial shared entanglement. We consider both honest-but-curious and malicious behavior. We establish finite-blocklength converse and achievability bounds, which match in the honest-but-curious setting. The malicious setting construction combines interactive hashing with tests of whether Bob received selected channel outputs, and distributes each input across several qubits so that erasures conceal an entire input from a dishonest receiver, even when the inputs are entangled with an external reference. We also derive fixed-error asymptotic expansions and show that both malicious and honest-but-curious settings have the same capacity.