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arXiv:quant-ph/0209149

The quantum bit commitment: a finite open system approach for a complete classification of protocols

Giacomo Mauro D’Ariano

Published Sep 26, 2002• 1 citation
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Abstract

Mayers, Lo and Chau argued that all quantum bit commitment protocols are insecure, because there is no way to prevent an Einstein-Podolsky-Rosen (EPR) cheating attack. However, Yuen presented some protocols which challenged the previous impossibility argument. Up to now, it is still debated whether there exist or not unconditionally secure protocols. In this paper the above controversy is addressed. For such purpose, a complete classification of all possible bit commitment protocols is given, including all possible cheating attacks. Focusing on the simplest class of protocols (non-aborting and with complete and perfect verification), it is shown how naturally a game-theoretical situation arises. For these protocols, bounds for the cheating probabilities are derived, involving the two quantum operations encoding the bit values and their respective alternate Kraus decompositions. Such bounds are different from those given in the impossibility proof. The whole classification and analysis has been carried out using a "finite open system" approach. The discrepancy with the impossibility proof is explained on the basis of the implicit adoption of a "closed system approach"--equivalent to modeling the commitment as performed by two fixed machines interacting unitarily in a overall "closed system"--according to which it is possible to assume as "openly known" both the initial state and the probability distributions for all secret parameters, which can be then "purified". This approach is also motivated by existence of unitary extensions for any open system. However, it is shown that the closed system approach for the classification of commitment protocols unavoidably leads to infinite dimensions, which then invalidate the continuity argument at the basis of the impossibility proof.

DOI
10.48550/arxiv.quant-ph/0209149

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Mayers, Lo and Chau argued that all quantum bit commitment protocols are insecure, because there is no way to prevent an Einstein-Podolsky-Rosen (EPR) cheating attack. However, Yuen presented some protocols which challenged the previous impossibility argument. Up to now, it is still debated whether there exist or not unconditionally secure protocols. In this paper the above controversy is addressed. For such purpose, a complete classification of all possible bit commitment protocols is given, including all possible cheating attacks. Focusing on the simplest class of protocols (non-aborting and with complete and perfect verification), it is shown how naturally a game-theoretical situation arises. For these protocols, bounds for the cheating probabilities are derived, involving the two quantum operations encoding the bit values and their respective alternate Kraus decompositions. Such bounds are different from those given in the impossibility proof. The whole classification and analysis has been carried out using a "finite open system" approach. The discrepancy with the impossibility proof is explained on the basis of the implicit adoption of a "closed system approach"--equivalent to modeling the commitment as performed by two fixed machines interacting unitarily in a overall "closed system"--according to which it is possible to assume as "openly known" both the initial state and the probability distributions for all secret parameters, which can be then "purified". This approach is also motivated by existence of unitary extensions for any open system. However, it is shown that the closed system approach for the classification of commitment protocols unavoidably leads to infinite dimensions, which then invalidate the continuity argument at the basis of the impossibility proof.
This paper addresses the controversy between Mayers, Lo and Chau on one side, and Yuen on the opposite side, on whether there exist or not unconditionally secure protocols. For such purpose, a complete classification of all possible bit commitment protocols is given, including all possible cheating attacks. For the simplest class of protocols (non-aborting and with complete and perfect verification), it is shown how naturally a game-theoretical situation arises. For these protocols, bounds for the cheating probabilities are derived, which turn out to be different from those given in the impossibility proof. The whole classification and analysis has been carried out using a "finite open system" approach. The discrepancy with the impossibility proof is explained on the basis of the implicit adoption of a "closed system approach"--equivalent to modeling the commitment as performed by two fixed machines interacting unitarily in a overall "closed system". However, it is shown that the closed system approach for the classification of commitment protocols unavoidably leads to infinite dimensions, which then invalidate the continuity argument at the basis of the impossibility proof.
There had been well known claims of unconditionally secure quantum protocols for bit commitment. However, we, and independently Mayers, showed that all proposed quantum bit commitment schemes are, in principle, insecure because the sender, Alice, can almost always cheat successfully by using an Einstein-Podolsky-Rosen (EPR) type of attack and delaying her measurements. One might wonder if secure quantum bit commitment protocols exist at all. We answer this question by showing that the same type of attack by Alice will, in principle, break any bit commitment scheme. The cheating strategy generally requires a quantum computer. We emphasize the generality of this ``no-go theorem'': Unconditionally secure bit commitment schemes based on quantum mechanics---fully quantum, classical or quantum but with measurements---are all ruled out by this result. Since bit commitment is a useful primitive for building up more sophisticated protocols such as zero-knowledge proofs, our results cast very serious doubt on the security of quantum cryptography in the so-called ``post-cold-war'' applications. We also show that ideal quantum coin tossing is impossible because of the EPR attack. This no-go theorem for ideal quantum coin tossing may help to shed some lights on the possibility of non-ideal protocols.
It is generally believed that unconditionally secure quantum bit commitment is impossible, due to widespread acceptance of an impossibility proof that utilizes quantum entaglement cheating. In this paper, we delineate how the impossibiliy proof formulation misses various types of quantum bit commitment protocols based on two-way quantum communications. We point out some of the gaps in the impossibility proof reasoning, and present corresponding counterexamples. Four different types of bit commitment protocols are constructed with several new protocol techniques. A specific Type 4 protocol is described and proved unconditionally secure. Security analysis of a Type 1 protocol and a Type 2 protocol are also sketched. The security of Type 3 protocols is as yet open. A development of quantum statistical decision theory and quantum games is needed to provie a complete security analysis of many such protocols.
Bit commitment protocols whose security is based on the laws of quantum mechanics alone are generally held to be impossible. We give a strengthened and explicit proof of this result. We extend its scope to a much larger variety of protocols, which may have an arbitrary number of rounds, in which both classical and quantum information is exchanged, and which may include aborts and resets. Moreover, we do not consider the receiver to be bound to a fixed ``honest'' strategy, so that ``anonymous state protocols,'' which were recently suggested as a possible way to beat the known no-go results, are also covered. We show that any concealing protocol allows the sender to find a cheating strategy, which is universal in the sense that it works against any strategy of the receiver. Moreover, if the concealing property holds only approximately, the cheat goes undetected with a high probability, which we explicitly estimate. The proof uses an explicit formalization of general two-party protocols, which is applicable to more general situations, and an estimate about the continuity of the Stinespring dilation of a general quantum channel. The result also provides a natural characterization of protocols that fall outside the standard setting of unlimited available technology and thus may allow secure bit commitment. We present such a protocol whose security, perhaps surprisingly, relies on decoherence in the receiver's laboratory.
Quantum bit commitment has been known to be impossible by the independent proofs of Mayers, and Lo and Chau, under the assumption that the whole quantum states right before the unveiling phase are static to users. We here provide an unconditionally secure non-static quantum bit commitment protocol with a trusted third party, which is not directly involved in any communications between users and can be limited not to get any information of commitment without being detected by users. We also prove that our quantum bit commitment protocol is not secure without the help of the trusted third party. The proof is basically different from the Mayers-Lo-Chau's no-go theorem, because we do not assume the staticity of the finally shared quantum states between users.
This paper proposes a different approach to pinpoint the causes for which an unconditionally secure quantum bit commitment protocol cannot be realized, beyond the technical details on which the proof of Mayers' no-go theorem is constructed. We have adopted the tools of quantum entropy analysis to investigate the conditions under which the security properties of quantum bit commitment can be circumvented. Our study has revealed that cheating the binding property requires the quantum system acting as the safe to harbor the same amount of uncertainty with respect to both observers (Alice and Bob) as well as the use of entanglement. Our analysis also suggests that the ability to cheat one of the two fundamental properties of bit commitment by any of the two participants depends on how much information is leaked from one side of the system to the other and how much remains hidden from the other participant.