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Lecture notes in computer science

Bit Commitment in the Bounded Storage Model: Tight Bound and Simple Optimal Construction

Junji Shikata, Daisuke Yamanaka

Published Jan 1, 2011• 3 citations
Publisher
DOI
10.1007/978-3-642-25516-8_8

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We study the communication complexity of single-server Private Information Retrieval (PIR) protocolsthat are based on fundamental cryptographic primitives in a black-box manner. In this setting, we establish a tight lower bound on the number of bits communicated by the server in any polynomially-preservingconstruction that relies on trapdoor permutations. More specifically, our main result states that in such constructions \\Omega (n) bits must be communicated by the server, where n is the size of the server's database,and this improves the \\Omega (n / log n) lower bound due to Haitner, Hoch, Reingold and Segev (FOCS '07).Therefore, in the setting under consideration, the naive solution in which the user downloads the entire database turns out to be optimal up to constant multiplicative factors. We note that the lower boundwe establish holds for the most generic form of trapdoor permutations, including in particular enhanced trapdoor permutations.Technically speaking, this paper consists of two main contributions from which our lower bound is obtained. First, we derive a tight lower bound on the number of bits communicated by the sender duringthe commit stage of any black-box construction of a statistically-hiding bit-commitment scheme from a family of trapdoor permutations. This lower bound asymptotically matches the upper bound provided bythe scheme of Naor, Ostrovsky, Venkatesan and Yung (CRYPTO '92). Second, we improve the e fficiencyof the reduction of statistically-hiding commitment schemes to low-communication single-server PIR, due to Beimel, Ishai, Kushilevitz and Malkin (STOC '99). In particular, we present a reduction that essentiallypreserves the communication complexity of the underlying single-server PIR protocol.

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The Mayers-Lo-Chau theorem establishes that no quantum bit commitment protocol is unconditionally secure. Nonetheless, there can be non-trivial upper bounds on both Bob's probability of correctly estimating Alice's commitment and Alice's probability of successfully unveiling whatever bit she desires. In this paper, we seek to determine these bounds for generalizations of the BB84 bit commitment protocol. In such protocols, an honest Alice commits to a bit by randomly choosing a state from a specified set and submitting this to Bob, and later unveils the bit by announcing the chosen state, at which point Bob measures the projector onto the state. Bob's optimal cheating strategy can be easily deduced from well known results in the theory of quantum state estimation. We show how to understand Alice's most general cheating strategy, (which involves her submitting to Bob one half of an entangled state) in terms of the theorem of Hughston, Josza and Wootters. We also show how the problem of optimizing Alice's cheating strategy for a fixed submitted state can be mapped onto a problem of state estimation. Finally, using the Bloch ball representation of qubit states, we identify the optimal coherent attack for a class of protocols that can be implemented with just a single qubit. These results provide a tight upper bound on Alice's probability of successfully unveiling whatever bit she desires in the protocol proposed by Aharonov et al., and lead us to identify a qubit protocol with an even stronger bound.

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It is well known that no quantum bit commitment protocol is unconditionally secure. Nonetheless, there can be non-trivial upper bounds on both Bob's probability of correctly estimating Alice's commitment and Alice's probability of successfully unveiling whatever bit she desires. In this paper, we seek to determine these bounds for generalizations of the BB84 bit commitment protocol. In such protocols, an honest Alice commits to a bit by randomly choosing a state from a specified set and submitting this to Bob, and later unveils the bit to Bob by announcing the chosen state, at which point Bob measures the projector onto the state. Bob's optimal cheating strategy can be easily deduced from well known results in the theory of quantum state estimation. We show how to understand Alice's most general cheating strategy, (which involves her submitting to Bob one half of an entangled state) in terms of a theorem of Hughston, Jozsa and Wootters. We also show how the problem of optimizing Alice's cheating strategy for a fixed submitted state can be mapped onto a problem of state estimation. Finally, using the Bloch ball representation of qubit states, we identify the optimal coherent attack for a class of protocols that can be implemented with just a single qubit. These results provide a tight upper bound on Alice's probability of successfully unveiling whatever bit she desires in the protocol proposed by Aharonov et al., and lead us to identify a qubit protocol with even greater security.

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We study the communication complexity of single-server Private Information Retrieval (PIR) protocolsthat are based on fundamental cryptographic primitives in a black-box manner. In this setting, we establish a tight lower bound on the number of bits communicated by the server in any polynomially-preservingconstruction that relies on trapdoor permutations. More specifically, our main result states that in such constructions \\Omega (n) bits must be communicated by the server, where n is the size of the server's database,and this improves the \\Omega (n / log n) lower bound due to Haitner, Hoch, Reingold and Segev (FOCS '07).Therefore, in the setting under consideration, the naive solution in which the user downloads the entire database turns out to be optimal up to constant multiplicative factors. We note that the lower boundwe establish holds for the most generic form of trapdoor permutations, including in particular enhanced trapdoor permutations.Technically speaking, this paper consists of two main contributions from which our lower bound is obtained. First, we derive a tight lower bound on the number of bits communicated by the sender duringthe commit stage of any black-box construction of a statistically-hiding bit-commitment scheme from a family of trapdoor permutations. This lower bound asymptotically matches the upper bound provided bythe scheme of Naor, Ostrovsky, Venkatesan and Yung (CRYPTO '92). Second, we improve the e fficiencyof the reduction of statistically-hiding commitment schemes to low-communication single-server PIR, due to Beimel, Ishai, Kushilevitz and Malkin (STOC '99). In particular, we present a reduction that essentiallypreserves the communication complexity of the underlying single-server PIR protocol.
The Mayers-Lo-Chau theorem establishes that no quantum bit commitment protocol is unconditionally secure. Nonetheless, there can be non-trivial upper bounds on both Bob's probability of correctly estimating Alice's commitment and Alice's probability of successfully unveiling whatever bit she desires. In this paper, we seek to determine these bounds for generalizations of the BB84 bit commitment protocol. In such protocols, an honest Alice commits to a bit by randomly choosing a state from a specified set and submitting this to Bob, and later unveils the bit by announcing the chosen state, at which point Bob measures the projector onto the state. Bob's optimal cheating strategy can be easily deduced from well known results in the theory of quantum state estimation. We show how to understand Alice's most general cheating strategy, (which involves her submitting to Bob one half of an entangled state) in terms of the theorem of Hughston, Josza and Wootters. We also show how the problem of optimizing Alice's cheating strategy for a fixed submitted state can be mapped onto a problem of state estimation. Finally, using the Bloch ball representation of qubit states, we identify the optimal coherent attack for a class of protocols that can be implemented with just a single qubit. These results provide a tight upper bound on Alice's probability of successfully unveiling whatever bit she desires in the protocol proposed by Aharonov et al., and lead us to identify a qubit protocol with an even stronger bound.
It is well known that no quantum bit commitment protocol is unconditionally secure. Nonetheless, there can be non-trivial upper bounds on both Bob's probability of correctly estimating Alice's commitment and Alice's probability of successfully unveiling whatever bit she desires. In this paper, we seek to determine these bounds for generalizations of the BB84 bit commitment protocol. In such protocols, an honest Alice commits to a bit by randomly choosing a state from a specified set and submitting this to Bob, and later unveils the bit to Bob by announcing the chosen state, at which point Bob measures the projector onto the state. Bob's optimal cheating strategy can be easily deduced from well known results in the theory of quantum state estimation. We show how to understand Alice's most general cheating strategy, (which involves her submitting to Bob one half of an entangled state) in terms of a theorem of Hughston, Jozsa and Wootters. We also show how the problem of optimizing Alice's cheating strategy for a fixed submitted state can be mapped onto a problem of state estimation. Finally, using the Bloch ball representation of qubit states, we identify the optimal coherent attack for a class of protocols that can be implemented with just a single qubit. These results provide a tight upper bound on Alice's probability of successfully unveiling whatever bit she desires in the protocol proposed by Aharonov et al., and lead us to identify a qubit protocol with even greater security.