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BRICS Report Series

On the (Im)possibility of Basing Oblivious Transfer and Bit Commitment on Weakened Security Assumptions

Ivan Damgård, Joe Kilian, Louis Salvail

Published Jun 7, 1998• 7 citations
Publisher

Abstract

We consider the problem of basing Oblivious Transfer (OT) and Bit Commitment (BC), with information theoretic security, on seemingly weaker primitives.We introduce a general model for describing such primitives, called Weak Generic Transfer (WGT). This model includes as important special cases Weak Oblivious Transfer (WOT), where both the sender and receiver may learn too much about the other party's input, and a new, more realistic model of noisy channels, called unfair noisy channels. An unfair noisy channel has a known range of possible noise levels; protocols must work for any level within this range against adversaries who know the actual noise level. We give a precise characterization for when one can base OT on WOT. When the deviation of the WOT from the ideal is above a certain threshold, we show that no information-theoretic reductions from OT (even against passive adversaries) and BC exist; when the deviation is below this threshold, we give a reduction from OT (and hence BC) that is information-theoretically secure against active adversaries. For unfair noisy channels we show a similar threshold phenomenon for bit commitment. If the upper bound on the noise is above a threshold (given as function of the lower bound) then no information-theoretic reduction from OT (even against passive adversaries) or BC exist; when it is below this threshold we give a reduction from BC. As a partial result, we give a reduction from OT to UNC for smaller noise intervals.

DOI
10.7146/brics.v5i37.19442

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Unfair Noisy Channels and Oblivious Transfer

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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.

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Abstract. Various results show that oblivious transfer can be implemented using the assumption of noisy channels. Unfortunately, this assumption is not as weak as one might think, because in a cryptographic setting, these noisy channels must satisfy very strong security requirements. Unfair noisy channels, introduced by Damg˚ard, Kilian and Salvail [Eurocrypt ’99], reduce these limitations: They give the adversary an unfair advantage over the honest player, and therefore weaken the security requirements on the noisy channel. However, this model still has many shortcomings: For example, the adversary’s advantage is only allowed to have a very special form, and no error is allowed in the implementation. In this paper we generalize the idea of unfair noisy channels. We introduce two new models of cryptographic noisy channels that we call the weak erasure channel and the weak binary symmetric channel, and show how they can be used to implement oblivious transfer. Our models are more general and use much weaker assumptions than unfair noisy channels, which makes implementation a more realistic prospect. For example, these are the first models that allows the parameters to come from experimental evidence. 1

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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.

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On the Oblivious Transfer Capacity of Generalized Erasure Channels against Malicious Adversaries

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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 ($p^* <1/2$).

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On the Oblivious Transfer Capacity of Generalized Erasure Channels Against Malicious Adversaries: The Case of Low Erasure Probability

Rafael Dowsley, Anderson C. A. Nascimento

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 ).

IEEE Transactions on Information Theory•Aug 3, 2017•9 citations
We consider the problem of basing Oblivious Transfer (OT) and Bit Commitment (BC), with information theoretic security, on seemingly weaker primitives.We introduce a general model for describing such primitives, called Weak Generic Transfer (WGT). This model includes as important special cases Weak Oblivious Transfer (WOT), where both the sender and receiver may learn too much about the other party's input, and a new, more realistic model of noisy channels, called unfair noisy channels. An unfair noisy channel has a known range of possible noise levels; protocols must work for any level within this range against adversaries who know the actual noise level. We give a precise characterization for when one can base OT on WOT. When the deviation of the WOT from the ideal is above a certain threshold, we show that no information-theoretic reductions from OT (even against passive adversaries) and BC exist; when the deviation is below this threshold, we give a reduction from OT (and hence BC) that is information-theoretically secure against active adversaries. For unfair noisy channels we show a similar threshold phenomenon for bit commitment. If the upper bound on the noise is above a threshold (given as function of the lower bound) then no information-theoretic reduction from OT (even against passive adversaries) or BC exist; when it is below this threshold we give a reduction from BC. As a partial result, we give a reduction from OT to UNC for smaller noise intervals.
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.
Abstract. Various results show that oblivious transfer can be implemented using the assumption of noisy channels. Unfortunately, this assumption is not as weak as one might think, because in a cryptographic setting, these noisy channels must satisfy very strong security requirements. Unfair noisy channels, introduced by Damg˚ard, Kilian and Salvail [Eurocrypt ’99], reduce these limitations: They give the adversary an unfair advantage over the honest player, and therefore weaken the security requirements on the noisy channel. However, this model still has many shortcomings: For example, the adversary’s advantage is only allowed to have a very special form, and no error is allowed in the implementation. In this paper we generalize the idea of unfair noisy channels. We introduce two new models of cryptographic noisy channels that we call the weak erasure channel and the weak binary symmetric channel, and show how they can be used to implement oblivious transfer. Our models are more general and use much weaker assumptions than unfair noisy channels, which makes implementation a more realistic prospect. For example, these are the first models that allows the parameters to come from experimental evidence. 1
Oblivious transfer is a primitive of paramount importance in cryptography or, more precisely, two- and multi-party computation due to its universality. Unfortunately, oblivious transfer cannot be achieved in an unconditionally secure way for both parties from scratch. Therefore, it is a natural question what information-theoretic primitives or computational assumptions oblivious transfer can be based on. The results in our thesis are threefold. First, we present a protocol that implements oblivious transfer from a weakened oblivious transfer called universal oblivious transfer, where one of the two players may get additional information. Our reduction is about twice as efficient as previous results. Weak oblivious transfer is an even weaker form of oblivious transfer, where both players may obtain additional information about the other player's input, and where the output can contain errors. We give a new, weaker definition of weak oblivious transfer, as well as new reductions with a more detailed analysis. Finally, we carry over our results to the computational setting and show how a weak oblivious transfer that is sometimes incorrect and only mildly secure against computationally bounded adversaries can be strengthened.
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.
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 ).