In the distrustful quantum cryptography model the parties have conflicting interests and do not trust one another. Nevertheless, they trust the quantum devices in their labs. The aim of the device-independent approach to cryptography is to do away with the latter assumption, and, consequently, significantly increase security. It is an open question whether the scope of this approach also extends to protocols in the distrustful cryptography model, thereby rendering them "fully" distrustful. In this Letter, we show that for bit commitment-one of the most basic primitives within the model-the answer is positive. We present a device-independent (imperfect) bit-commitment protocol, where Alice's and Bob's cheating probabilities are ≃0.854 and 3/4, which we then use to construct a device-independent coin flipping protocol with bias ≲0.336.
In the distrustful quantum cryptography model the parties have conflicting interests and do not trust one another. Nevertheless, they trust the quantum devices in their labs. The aim of the device-independent approach to cryptography is to do away with the latter assumption, and, consequently, significantly increase security. It is an open question whether the scope of this approach also extends to protocols in the distrustful cryptography model, thereby rendering them "fully" distrustful. In this Letter, we show that for bit commitment-one of the most basic primitives within the model-the answer is positive. We present a device-independent (imperfect) bit-commitment protocol, where Alice's and Bob's cheating probabilities are ≃0.854 and 3/4, which we then use to construct a device-independent coin flipping protocol with bias ≲0.336.
Unconditionally secure bit commitment and coin flipping are known to be impossible in the classical world. Bit commitment is known to be impossible also in the quantum world. We introduce a related new primitive - {\em quantum bit escrow}. In this primitive Alice commits to a bit b to Bob. The commitment is {\em binding} in the sense that if Alice is asked to reveal the bit, Alice can not bias her commitment without having a good probability of being detected cheating. The commitment is {\em sealing} in the sense that if Bob learns information about the encoded bit, then if later on he is asked to prove he was playing honestly, he is detected cheating with a good probability. Rigorously proving the correctness of quantum cryptographic protocols has proved to be a difficult task. We develop techniques to prove quantitative statements about the binding and sealing properties of the quantum bit escrow protocol.
A related primitive we construct is a quantum biased coin flipping protocol where no player can control the game, i.e., even an all-powerful cheating player must lose with some constant probability, which stands in sharp contrast to the classical world where such protocols are impossible.
Unconditionally secure bit commitment and coin flipping are known to be impossible in the classical world. Bit commitment is known to be impossible also in the quantum world. We introduce a related new primitive - {\em quantum bit escrow}. In this primitive Alice commits to a bit b to Bob. The commitment is {\em binding} in the sense that if Alice is asked to reveal the bit, Alice can not bias her commitment without having a good probability of being detected cheating. The commitment is {\em sealing} in the sense that if Bob learns information about the encoded bit, then if later on he is asked to prove he was playing honestly, he is detected cheating with a good probability. Rigorously proving the correctness of quantum cryptographic protocols has proved to be a difficult task. We develop techniques to prove quantitative statements about the binding and sealing properties of the quantum bit escrow protocol. A related primitive we construct is a quantum biased coin flipping protocol where no player can control the game, i.e., even an all-powerful cheating player must lose with some constant probability, which stands in sharp contrast to the classical world where such protocols are impossible.
Coin flipping and bit commitment are two fundamental cryptographic primitives with numerous applications. Quantum information allows for such protocols in the information theoretic setting where no dishonest party can perfectly cheat. The previously best-known quantum coin flipping and bit commitment protocol by Ambainis achieved a cheating probability of at most 3/4 [A. Ambainis, Proceedings of the 30th Annual ACM Symposium on Theory of Computing, Washington, DC, IEEE Computer Society, 2001]. On the other hand, Kitaev showed that no quantum coin flipping or bit commitment protocol can have cheating probability less than 1/2 [A. Kitaev, Presentation at the 6th Workshop on Quantum Information Processing (QIP), 2003]. Closing these gaps has been one of the important open questions in quantum cryptography. In this paper, we resolve both questions. First, we present a quantum strong coin flipping protocol with cheating probability arbitrarily close to 1/2. More precisely, we show how to use any weak coin flipping protocol with cheating probability 1/2+ε in order to achieve a strong coin flipping protocol with cheating probability 1/2+O(ε). The optimal quantum strong coin flipping protocol follows from our construction and the optimal quantum weak coin flipping protocol described by [C. Mochon, arXiv:0711.4114, 2007]. Second, we provide the optimal bound for quantum bit commitment. On the one hand, we show a lower bound of approximately γ≈0.739, improving Kitaev's lower bound. On the other hand, we present an optimal quantum bit commitment protocol which has cheating probability arbitrarily close to γ. 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 γ+O(ε). To obtain the final protocol, we then use the optimal quantum weak coin flipping protocol described by [C. Mochon, arXiv:0711.4114, 2007]. Unlike the previous protocol for coin flipping, our protocol uses quantum effects beyond the weak coin flip. To stress this fact, we additionally show that any classical bit commitment protocol with access to perfect weak (or strong) coin flipping has cheating probability at least 3/4.
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.
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[Amb01]. On the other hand, Kitaev showed that no quantum protocol can have cheating probability less than 1/sqrt{2} [Kit03] (his lower bound on coin flipping can be easily extended to bit commitment). Closing this gap has since been an important and open question. In this paper, we provide the optimal bound for quantum bit commitment. We first show a lower bound of approximately 0.739, improving Kitaev's lower bound. We then 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 + eps in order to achieve a quantum bit commitment protocol with cheating probability 0.739 + O(eps). We then use the optimal quantum weak coin flipping protocol described by Mochon[Moc07]. 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.
There had been well known claims of ``provably unbreakable'' quantum protocols for bit commitment and coin tossing. However, we, and independently Mayers, showed that all proposed quantum bit commitment (and therefore coin tossing) schemes are, in principle, insecure because the sender, Alice, can always cheat successfully by using an EPR-type of attack and delaying her measurements. One might wonder if secure quantum bit commitment and coin tossing protocols exist at all. Here we prove that an EPR-type of attack by Alice will, in principle, break {\em any} realistic quantum bit commitment and {\em ideal} coin tossing scheme. Therefore, provided that Alice has a quantum computer and is capable of storing quantum signals for an arbitrary length of time, all those schemes are insecure. Since bit commitment and coin tossing are useful primitives 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.