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.
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.
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.
Weak coin flipping (WCF) is a fundamental cryptographic primitive for two-party secure computation, where two distrustful parties need to remotely establish a shared random bit whilst having opposite preferred outcomes. It is the strongest known primitive with arbitrarily close to perfect security quantumly while classically, its security is completely compromised (unless one makes further assumptions, such as computational hardness). A WCF protocol is said to have bias ϵ if neither party can force their preferred outcome with probability greater than 1/2+ϵ. Classical WCF protocols are shown to have bias 1/2, i.e., a cheating party can always force their preferred outcome. On the other hand, there exist quantum WCF protocols with arbitrarily small bias, as Mochon showed in his seminal work in 2007 [arXiv:0711.4114]. In particular, he proved the existence of a family of WCF protocols approaching bias ϵ(k)=1/(4k+2) for arbitrarily large k and proposed a protocol with bias 1/6. Last year, Arora, Roland and Weis presented a protocol with bias 1/10 and to go below this bias, they designed an algorithm that numerically constructs unitary matrices corresponding to WCF protocols with arbitrarily small bias [STOC'19, p.205-216]. In this work, we present new techniques which yield a fully analytical construction of WCF protocols with bias arbitrarily close to zero, thus achieving a solution that has been missing for more than a decade. Furthermore, our new techniques lead to a simplified proof of existence of WCF protocols by circumventing the non-constructive part of Mochon's proof. As an example, we illustrate the construction of a WCF protocol with bias 1/14.
Coin flipping is a fundamental cryptographic primitive that enables two distrustful and far apart parties to create a uniformly random bit. Quantum information allows for protocols in the information theoretic setting where no dishonest party can perfectly cheat. The previously best-known quantum protocol by Ambain is 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/sqrt{2}. Closing this gap has been one of the important open questions in quantum cryptography. In this paper, we resolve this question by presenting a quantum strong coin flipping protocol with cheating probability arbitrarily close to 1/sqrt{2}.More precisely, we show how to use any weak coin flipping protocol with cheating probability 1/2+epsilon in order to achieve a strong coin flipping protocol with cheating probability 1/sqrt{2}+O(epsilon). The optimal quantum strong coin flipping protocol follows from our construction and the optimal quantum weak coin flipping protocol described by Mochon.
Weak coin flipping is an important cryptographic primitive$\unicode{x2013}$it is the strongest known secure two-party computation primitive that classically becomes secure only under certain assumptions (e.g. computational hardness), while quantumly there exist protocols that achieve arbitrarily close to perfect security. This breakthrough result was established by Mochon in 2007 [arXiv:0711.4114]. However, his proof relied on the existence of certain unitary operators which was established by a non-constructive argument. Consequently, explicit protocols have remained elusive. In this work, we give exact constructions of related unitary operators. These, together with a new formalism, yield a family of protocols approaching perfect security thereby also simplifying Mochon's proof of existence. We illustrate the construction of explicit weak coin flipping protocols by considering concrete examples (from the aforementioned family of protocols) that are more secure than all previously known protocols. 76 pages. This is the TheoretiCS journal version
Coin-flipping is a fundamental task in two-party cryptography where two remote mistrustful parties wish to generate a shared uniformly random bit. While quantum protocols promising near-perfect security exist for weak coin-flipping -- when the parties want opposing outcomes -- it has been shown that they must be inefficient in terms of their round complexity, and it is an open question of how space efficient they can be. In this work, we consider a variant called cheat-penalised weak coin-flipping in which if a party gets caught cheating, they lose Λ points (compared to 0 in the standard definition). We find that already for a small cheating penalty, the landscape of coin-flipping changes dramatically. For example, with Λ=0.01, we exhibit a protocol where neither Alice nor Bob can bias the result in their favour beyond 1/2+10−8, which uses 24 qubits and 1016 rounds of communication (provably 107 times better than any weak coin-flipping protocol with matching security). For the same space requirements, we demonstrate how one can choose between lowering how much a malicious party can bias the result (down to 1/2+10−10) and reducing the rounds of communication (down to 25,180), depending on what is preferred. To find these protocols, we make two technical contributions. First, we extend the point game-protocol correspondence introduced by Kitaev and Mochon, to incorporate: (i) approximate point games, (ii) the cheat-penalised setting, and (iii) round and space complexity. Second, we give the first (to the best of our knowledge) numerical algorithm for constructing (approximate) point games that correspond to high security and low complexity. Our results open up the possibility of having secure and practical quantum protocols for multiparty computation.
Weak coin flipping (WCF) is a fundamental cryptographic primitive for two-party secure computation, where two distrustful parties need to remotely establish a shared random bit whilst having opposite preferred outcomes. It is the strongest known primitive with arbitrarily close to perfect security quantumly while classically, its security is completely compromised (unless one makes further assumptions, such as computational hardness). A WCF protocol is said to have bias ∊ if neither party can force their preferred outcome with probability greater than 1/2 + ∊. Classical WCF protocols are shown to have bias 1/2, i.e., a cheating party can always force their preferred outcome. On the other hand, there exist quantum WCF protocols with arbitrarily small bias, as Mochon showed in his seminal work in 2007 [arXiv:0711.4114]. In particular, he proved the existence of a family of WCF protocols approaching bias ∊(k) = 1/(4k +2) for arbitrarily large k and proposed a protocol with bias 1/6. Last year, Arora, Roland and Weis presented a protocol with bias 1/10 and to go below this bias, they designed an algorithm that numerically constructs unitary matrices corresponding to WCF protocols with arbitrarily small bias [STOC'19, p. 205–216]. In this work, we present new techniques which yield a fully analytical construction of WCF protocols with bias arbitrarily close to zero, thus achieving a solution that has been missing for more than a decade. Furthermore, our new techniques lead to a simplified proof of existence of WCF protocols by circumventing the non-constructive part of Mochon's proof. As an example, we illustrate the construction of a WCF protocol with bias 1/14.