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arXiv:quant-ph/0508180

Secret parameters in quantum bit commitment

Chi-Yee Cheung

Published Aug 24, 2005• 5 citations
arXivHTMLPDF

Abstract

The no-go theorem of unconditionally secure quantum bit commitment depends crucially on the assumption that Alice knows in detail all the probability distributions generated by Bob. We show that if a protocol is concealing, then the cheating unitary transformation is independent of any parameters (including probability distributions) secretly chosen by Bob, so that Alice can calculate it without knowing Bob's secret choices. Otherwise the protocol cannot be concealing. Our result shows that the original impossibility proof was based on an incorrect assumption, despite the fact that its conclusion remains valid within the adopted framework. Furthermore, our result eliminates a potential loophole in the no-go theorem.

DOI
10.48550/arxiv.quant-ph/0508180

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The no-go theorem of unconditionally secure quantum bit commitment depends crucially on the assumption that Alice knows in detail all the probability distributions generated by Bob. We show that if a protocol is concealing, then the cheating unitary transformation is independent of any parameters (including probability distributions) secretly chosen by Bob, so that Alice can calculate it without knowing Bob's secret choices. Otherwise the protocol cannot be concealing. Our result shows that the original impossibility proof was based on an incorrect assumption, despite the fact that its conclusion remains valid within the adopted framework. Furthermore, our result eliminates a potential loophole in the no-go theorem.
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.
It is generally believed that unconditionally secure quantum bit commitment (QBC) is proven impossible by a "no-go theorem". We point out that the theorem only establishes the existence of a cheating unitary transformation in any QBC scheme secure against the receiver, but this fact alone is not sufficient to rule out unconditionally secure QBC as a matter of principle, because there exists no proof that the cheating unitary transformation must be known to the cheater in all possible cases. In this work, we show how to circumvent the "no-go theorem" and prove that unconditionally secure QBC is in fact possible.
We further study the security of the quantum bit commitment (QBC) protocol we previously proposed [Phys. Rev. A 74, 022332 (2006).], by analyzing the reduced density matrix ρ_{b}^{B} which describes the quantum state at Bob's side corresponding to Alice's committed bit b. It is shown that Alice will find ρ_{0}^{B}\perp ρ_{1}^{B} while the protocol remains concealing to Bob. On the contrary, the existing no-go theorem of unconditionally secure QBC is based on the condition ρ_{0}^{B}\simeq ρ_{1}^{B}. Thus the specific cheating strategy proposed in the no-go theorem does not necessarily applies to our protocol.
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.