"God does not play dice. He flips coins instead." And though for some reason He has denied us quantum bit commitment. And though for some reason he has even denied us strong coin flipping. He has, in His infinite mercy, granted us quantum weak coin flipping so that we too may flip coins. Instructions for the flipping of coins are contained herein. But be warned! Only those who have mastered Kitaev's formalism relating coin flipping and operator monotone functions may succeed. For those foolhardy enough to even try, a complete tutorial is included.
"God does not play dice. He flips coins instead." And though for some reason He has denied us quantum bit commitment. And though for some reason he has even denied us strong coin flipping. He has, in His infinite mercy, granted us quantum weak coin flipping so that we too may flip coins. Instructions for the flipping of coins are contained herein. But be warned! Only those who have mastered Kitaev's formalism relating coin flipping and operator monotone functions may succeed. For those foolhardy enough to even try, a complete tutorial is included.
We investigate weak coin flipping, a fundamental cryptographic primitive where two distrustful parties need to remotely establish a shared random bit. A cheating player can try to bias the output bit towards a preferred value. For weak coin flipping the players have known opposite preferred values. A weak coin-flipping protocol has a bias ϵ if neither player can force the outcome towards their preferred value with probability more than 21+ϵ. While it is known that all classical protocols have ϵ=21, Mochon showed in 2007 [arXiv:0711.4114] that quantumly weak coin flipping can be achieved with arbitrarily small bias (near perfect) but the best known explicit protocol has bias 1/6 (also due to Mochon, 2005 [Phys. Rev. A 72, 022341]). We propose a framework to construct new explicit protocols achieving biases below 1/6. In particular, we construct explicit unitaries for protocols with bias approaching 1/10. To go below, we introduce what we call the Elliptic Monotone Align (EMA) algorithm which, together with the framework, allows us to numerically construct protocols with arbitrarily small biases.
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.
We investigate weak coin flipping, a fundamental cryptographic primitive where two distrustful parties need to remotely establish a shared random bit. A cheating player can try to bias the output bit towards a preferred value. For weak coin flipping the players have known opposite preferred values. A weak coin-flipping protocol has a bias є if neither player can force the outcome towards their preferred value with probability more than 1/2+є. While it is known that all classical protocols have є=1/2, Mochon showed in 2007 that quantumly weak coin flipping can be achieved with arbitrarily small bias (near perfect) but the former best known explicit protocol has bias 1/6 (also due to Mochon, 2005). We propose a framework to construct new explicit protocols achieving biases below 1/6. In particular, we construct explicit unitaries for protocols with bias down to 1/10. To go lower, we introduce what we call the Elliptic Monotone Align (EMA) algorithm which, together with the framework, allows us to construct protocols with arbitrarily small biases.
Mochon's proof [Quantum Weak Coin Flipping with Arbitrarily Small Bias, preprint, arXiv:0711.4114, 2007] of the existence of quantum weak coin flipping with arbitrarily small bias is a fundamental result in quantum cryptography, but at the same time one of the least understood. Though used several times as a black box in important follow-up results [M. Ganz, Quantum Leader Election, preprint, arXiv:0910.4952, 2009; A. Chailloux and I. Kerenidis, in Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2009, pp. 527--533; N. Aharon and J. Silman, New J. Phys., 12 (2010), 033027; A. Chailloux and I. Kerenidis, in Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2011; I. Kerenidis and S. Zhang, in Theory of Quantum Computation, Communication, and Cryptography, Lecture Notes in Computer Science 7582, Springer, Berlin, 2013, pp. 13--28], the result has not been peer reviewed, its novel techniques (and, in particular, Kitaev's point game formalism) have not been applied anywhere else, and an explicit protocol is missing. We believe that truly understanding the existence proof and the novel techniques it relies on would constitute a major step in quantum information theory, leading to deeper understanding of entanglement and of quantum protocols in general. In this work, we make a first step in this direction. We simplify parts of Mochon's construction considerably, making about 20 pages of analysis in the original proof superfluous, clarifying some other parts of the proof on the way, and presenting the proof in a way which is conceptually easier to grasp. We believe the resulting proof of existence is easier to understand, more readable, and certainly verifiable. Moreover, we analyze the resources neededto achieve a bias ε and show that the number of qubits is O(logε1), while the number of rounds is (ε1)O(ε1). A true understanding of the proof, including Kitaev's point-game techniques and their applicability, as well as completing the task of constructing an explicit (and also simpler and more efficient) protocol, are left to future work.
A coin is just a two sided dice. Recently, Mochon proved that quantum weak coin flipping with an arbitrarily small bias is possible. However, the use of quantum resources to allow N remote distrustful parties to roll an N-sided dice has yet to be addressed. In this paper we show that contrary to the classical case, N-sided dice rolling with arbitrarily small bias is possible for any N. In addition, we present a six-round three-sided dice rolling protocol, achieving a bias of 0.181, which incorporates weak imbalanced coin flipping.
The problem of quantum dice rolling (DR)—a generalization of the problem of quantum coin flipping (CF) to more than two outcomes and parties—is studied in both its weak and strong variants. We prove by construction that quantum mechanics allows for (i) weak N -sided DR admitting arbitrarily small bias for any N and (ii) two-party strong N -sided DR saturating Kitaev's bound for any N . To derive (ii) we also prove by construction that quantum mechanics allows for (iii) strong imbalanced CF saturating Kitaev's bound for any degree of imbalance. Furthermore, as a corollary of (ii) we introduce a family of optimal 2 m -party strong n m -sided DR protocols for any pair m and n .