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Recursive cheating strategies for the relativistic $\mathbb F_Q$ bit commitment protocol

Rémi Bricout, André Chailloux

Published Aug 1, 2017• 2 citations
Publisher

Abstract

In this paper, we study relativistic bit commitment, which uses timing and location constraints to achieve information theoretic security. We consider the FQ multi-round bit commitment scheme introduced by Lunghi et al. [LKB + 15]. This protocol was shown secure against classical adversaries as long as the number of rounds m is small compared to √ Q where Q is the size of the used field in the protocol [CCL15, FF16]. In this work, we study classical attacks on this scheme. We use classical strategies for the CHSHQ game described in [BS15] to derive cheating strategies for this protocol. In particular, our cheating strategy shows that if Q is an even power of any prime, then the protocol is not secure when the number of rounds m is of the order of √ Q. For those values of Q, this means that the upper bound of [CCL15, FF16] is essentially optimal.

DOI
10.3390/cryptography1020014

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Recursive cheating strategies for the relativistic FQ\mathbb F_QFQ​ bit commitment protocol
In this paper, we study relativistic bit commitment, which uses timing and location constraints to achieve information theoretic security. We consider the FQ multi-round bit commitment scheme introduced by Lunghi et al. [LKB + 15]. This protocol was shown secure against classical adversaries as long as the number of rounds m is small compared to √ Q where Q is the size of the used field in the protocol [CCL15, FF16]. In this work, we study classical attacks on this scheme. We use classical strategies for the CHSHQ game described in [BS15] to derive cheating strategies for this protocol. In particular, our cheating strategy shows that if Q is an even power of any prime, then the protocol is not secure when the number of rounds m is of the order of √ Q. For those values of Q, this means that the upper bound of [CCL15, FF16] is essentially optimal.
In this paper, we study relativistic bit commitment, which uses timing and location constraints to achieve information theoretic security. Using those constraints, we consider a relativistic bit commitment scheme introduced by Lunghi et al. This protocol was shown secure against classical adversaries as long as the number of rounds performed in the protocol is not too large. In this work, we study classical attacks on this scheme. We use the correspondence between this protocol and the CHSHQ game—which is a variant of the CHSH game—to derive cheating strategies for this protocol. Our attack matches the existing security bound for some range of parameters and shows that the scaling of the security in the number of rounds is essentially optimal.
Bit commitment is a fundamental cryptographic primitive in which Alice wishes to commit a secret bit to Bob. Perfectly secure bit commitment between two mistrustful parties is impossible through an asynchronous exchange of quantum information. Perfect security is, however, possible when Alice and Bob each split into several agents exchanging classical information at times and locations suitably chosen to satisfy specific relativistic constraints. In this Letter we first revisit a previously proposed scheme [C. Crépeau et al., Lect. Notes Comput. Sci. 7073, 407 (2011)] that realizes bit commitment using only classical communication. We prove that the protocol is secure against quantum adversaries for a duration limited by the light-speed communication time between the locations of the agents. We then propose a novel multiround scheme based on finite-field arithmetic that extends the commitment time beyond this limit, and we prove its security against classical attacks. Finally, we present an implementation of these protocols using dedicated hardware and we demonstrate a 2 ms-long bit commitment over a distance of 131 km. By positioning the agents on antipodal points on the surface of Earth, the commitment time could possibly be extended to 212 ms.
In this thesis we have studied how to exploit relativistic constraints such as the non-superluminal signalling principle to design secure cryptographic primitives like position-verification and bit commitment. According to non-superluminal signalling principle, no physical carrier of information can travel faster than the speed of light. This put a constraint on the communication time between two distant stations. One can consider this delay in information transfer as a temporal non-communication constraint. Cryptographic primitives like bit-commitment, oblivious transfer can be implemented with perfect secrecy under such non-communication assumption between the agents. The first part of this thesis has studied how non-signalling constraints can be used for secure position verification. Here, we have discussed about a strategy which can attack any position verification scheme. In the next part of this thesis we have discussed about the nonlocal games, relevant for studying relativistic bit commitment protocols. We have established an upper bound on the classical value of such family of games. The last part of this thesis discusses about two relativistic bit commitment protocols and their security against classical adversaries. We conclude this thesis by giving a brief summary of the content of each chapter and mentioning interesting open problems. These open problems can be very useful for better understanding of the role of spacetime constraints such as non-superluminal signalling in designing perfectly secure cryptographic primitives.
We consider the related notions of two-prover and of relativistic commitment schemes. In recent work, Lunghi et al. proposed a new relativistic commitment scheme with a multi-round sustain phase that enables to keep the binding property alive as long as the sustain phase is running. They prove security of their scheme against classical attacks; however, the proven bound on the error parameter is very weak: it blows up doubly exponentially in the number of rounds. In this work, we give a new analysis of the multi-round scheme of Lunghi et al., and we show a linear growth of the error parameter instead (also considering classical attacks only). Our analysis is based on a new and rather general composition theorem for two-prover commitment schemes. The proof of our composition theorem is based on a better understanding of the binding property of two-prover commitments that we provide in the form of new definitions and relations among them. These new insights are certainly of independent interest and are likely to be useful in other contexts as well. Finally, our work gives rise to several interesting open problems, for instance extending our results to the quantum setting, where the dishonest provers are allowed to perform measurements on an entangled quantum state in order to try to break the binding property.
We consider the recent relativistic bit commitment protocol introduced by Lunghi et al. [Phys. Rev. Lett. 115, 030502 (2015)] and present a new security analysis against classical attacks. In particular, while the initial complexity of the protocol scales double exponentially with the commitment time, our analysis shows that the correct dependence is only linear. This has dramatic implications in terms of implementation: in particular, the commitment time can easily be made arbitrarily long, by only requiring both parties to communicate classically and perform efficient classical computation.
Bit commitment is a fundamental cryptographic primitive with numerous applications. Quantum information allows for bit commitment schemes in the information theoretic setting where no dishonest party can perfectly cheat. The previously best-known quantum protocol by Ambainis achieved a cheating probability of at most 3/4. On the other hand, Kitaev showed that no quantum protocol can have cheating probability less than 1/√2 (his lower bound on coin flipping can be easily extended to bit commitment). Closing this gap has since been an important open question. In this paper, we provide the optimal bound for quantum bit commitment. First, we show a lower bound of approximately 0.739, improving Kitaev's lower bound. For this, we present some generic cheating strategies for Alice and Bob and conclude by proving a new relation between the trace distance and fidelity of two quantum states. Second, we present an optimal quantum bit commitment protocol which has cheating probability arbitrarily close to 0.739. More precisely, we show how to use any weak coin flipping protocol with cheating probability 1/2 + ε in order to achieve a quantum bit commitment protocol with cheating probability 0.739 + O(ε). We then use the optimal quantum weak coin flipping protocol described by Mochon. Last, in order to stress the fact that our protocol uses quantum effects beyond the weak coin flip, we show that any classical bit commitment protocol with access to perfect weak (or strong) coin flipping has cheating probability at least 3/4.