Quantum Bit String Commitment

A bit string commitment protocol securely commits $N$ classical bits in such a way that the recipient can extract only $M<N$ bits of information about the string. Classical reasoning might suggest that bit string commitment implies bit commitment and hence, given the Mayers-Lo-Chau theorem, that non-relativistic quantum bit string commitment is impossible. Not so: there exist non-relativistic quantum bit string commitment protocols, with security parameters $\epsilon$ and $M$, that allow $A$ to commit $N = N(M, \epsilon)$ bits to $B$ so that $A$'s probability of successfully cheating when revealing any bit and $B$'s probability of extracting more than $N'=N-M$ bits of information about the $N$ bit string before revelation are both less than $\epsilon$. With a slightly weakened but still restrictive definition of security against $A$, $N$ can be taken to be $O(\exp (C N'))$ for a positive constant $C$. I briefly discuss possible applications.Comment: Published version. (Refs updated.

Cheat Sensitive Quantum Bit Commitment

We define cheat sensitive cryptographic protocols between mistrustful parties as protocols which guarantee that, if either cheats, the other has some nonzero probability of detecting the cheating. We give an example of an unconditionally secure cheat sensitive non-relativistic bit commitment protocol which uses quantum information to implement a task which is classically impossible; we also describe a simple relativistic protocol.Comment: Final version: a slightly shortened version of this will appear in PRL. Minor corrections from last versio

Exponential quantum enhancement for distributed addition with local nonlinearity

We consider classical and entanglement-assisted versions of a distributed computation scheme that computes nonlinear Boolean functions of a set of input bits supplied by separated parties. Communication between the parties is restricted to take place through a specific apparatus which enforces the constraints that all nonlinear, nonlocal classical logic is performed by a single receiver, and that all communication occurs through a limited number of one-bit channels. In the entanglement-assisted version, the number of channels required to compute a Boolean function of fixed nonlinearity can become exponentially smaller than in the classical version. We demonstrate this exponential enhancement for the problem of distributed integer addition.Comment: To appear in Quantum Information Processin

Unconditionally Secure Bit Commitment

We describe a new classical bit commitment protocol based on cryptographic constraints imposed by special relativity. The protocol is unconditionally secure against classical or quantum attacks. It evades the no-go results of Mayers, Lo and Chau by requiring from Alice a sequence of communications, including a post-revelation verification, each of which is guaranteed to be independent of its predecessor.Comment: Typos corrected. Reference details added. To appear in Phys. Rev. Let

Multipartite Nonlocal Quantum Correlations Resistant to Imperfections

We use techniques for lower bounds on communication to derive necessary conditions in terms of detector efficiency or amount of super-luminal communication for being able to reproduce with classical local hidden-variable theories the quantum correlations occurring in EPR-type experiments in the presence of noise. We apply our method to an example involving n parties sharing a GHZ-type state on which they carry out measurements and show that for local-hidden variable theories, the amount of super-luminal classical communication c and the detector efficiency eta are constrained by eta 2^(-c/n) = O(n^(-1/6)) even for constant general error probability epsilon = O(1)

Quantum Algorithm for the Collision Problem

In this note, we give a quantum algorithm that finds collisions in arbitrary r-to-one functions after only O((N/r)^(1/3)) expected evaluations of the function. Assuming the function is given by a black box, this is more efficient than the best possible classical algorithm, even allowing probabilism. We also give a similar algorithm for finding claws in pairs of functions. Furthermore, we exhibit a space-time tradeoff for our technique. Our approach uses Grover's quantum searching algorithm in a novel way.Comment: 8 pages, LaTeX2

Quantum Mechanics helps in searching for a needle in a haystack

Quantum mechanics can speed up a range of search applications over unsorted data. For example imagine a phone directory containing N names arranged in completely random order. To find someone's phone number with a probability of 50%, any classical algorithm (whether deterministic or probabilistic) will need to access the database a minimum of O(N) times. Quantum mechanical systems can be in a superposition of states and simultaneously examine multiple names. By properly adjusting the phases of various operations, successful computations reinforce each other while others interfere randomly. As a result, the desired phone number can be obtained in only O(sqrt(N)) accesses to the database.Comment: Postscript, 4 pages. This is a modified version of the STOC paper (quant-ph/9605043) and is modified to make it more comprehensible to physicists. It appeared in Phys. Rev. Letters on July 14, 1997. (This paper was originally put out on quant-ph on June 13, 1997, the present version has some minor typographical changes

Fair Loss-Tolerant Quantum Coin Flipping

Coin flipping is a cryptographic primitive in which two spatially separated players, who in principle do not trust each other, wish to establish a common random bit. If we limit ourselves to classical communication, this task requires either assumptions on the computational power of the players or it requires them to send messages to each other with sufficient simultaneity to force their complete independence. Without such assumptions, all classical protocols are so that one dishonest player has complete control over the outcome. If we use quantum communication, on the other hand, protocols have been introduced that limit the maximal bias that dishonest players can produce. However, those protocols would be very difficult to implement in practice because they are susceptible to realistic losses on the quantum channel between the players or in their quantum memory and measurement apparatus. In this paper, we introduce a novel quantum protocol and we prove that it is completely impervious to loss. The protocol is fair in the sense that either player has the same probability of success in cheating attempts at biasing the outcome of the coin flip. We also give explicit and optimal cheating strategies for both players.Comment: 12 pages, 1 figure; various minor typos corrected in version

Two-player quantum pseudo-telepathy based on recent all-versus-nothing violations of local realism

We introduce two two-player quantum pseudo-telepathy games based on two recently proposed all-versus-nothing (AVN) proofs of Bell's theorem [A. Cabello, Phys. Rev. Lett. 95, 210401 (2005); Phys. Rev. A 72, 050101(R) (2005)]. These games prove that Broadbent and Methot's claim that these AVN proofs do not rule out local-hidden-variable theories in which it is possible to exchange unlimited information inside the same light-cone (quant-ph/0511047) is incorrect.Comment: REVTeX4, 5 page

Quantum advantages in classically defined tasks

We analyze classically defined games for which a quantum team has an advantage over any classical team. The quantum team has a clear advantage in games in which the players of each team are separated in space and the quantum team can use unusually strong correlations of the Einstein-Podolsky-Rosen (EPR) type. We present an example of a classically defined game played at one location for which quantum players have a real advantage.Comment: 4 pages, revised version, to be published in PR