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arXiv:quant-ph/0504078 · Physical Review A

Possibility, impossibility, and cheat sensitivity of quantum-bit string commitment

Harry Buhrman, Matthias Christandl, Patrick M. Hayden, Hoi‐Kwong Lo, Stephanie Wehner

Published Aug 11, 2008• 63 citations
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Abstract

Unconditionally secure nonrelativistic bit commitment is known to be impossible in both the classical and the quantum worlds. But when committing to a string of $n$ bits at once, how far can we stretch the quantum limits? In this paper, we introduce a framework for quantum schemes where Alice commits a string of $n$ bits to Bob in such a way that she can only cheat on $a$ bits and Bob can learn at most $b$ bits of information before the reveal phase. Our results are twofold: we show by an explicit construction that in the traditional approach, where the reveal and guess probabilities form the security criteria, no good schemes can exist: $a+b$ is at least $n$. If, however, we use a more liberal criterion of security, the accessible information, we construct schemes where $a=4\text{ }{\text{log}}_{2}\text{ }n+O(1)$ and $b=4$, which is impossible classically. We furthermore present a cheat-sensitive quantum bit string commitment protocol for which we give an explicit tradeoff between Bob's ability to gain information about the committed string, and the probability of him being detected cheating.

DOI
10.1103/physreva.78.022316

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Unconditionally secure nonrelativistic bit commitment is known to be impossible in both the classical and the quantum worlds. But when committing to a string of nnn bits at once, how far can we stretch the quantum limits? In this paper, we introduce a framework for quantum schemes where Alice commits a string of nnn bits to Bob in such a way that she can only cheat on aaa bits and Bob can learn at most bbb bits of information before the reveal phase. Our results are twofold: we show by an explicit construction that in the traditional approach, where the reveal and guess probabilities form the security criteria, no good schemes can exist: a+ba+ba+b is at least nnn. If, however, we use a more liberal criterion of security, the accessible information, we construct schemes where a=4 log2 n+O(1)a=4\text{ }{\text{log}}_{2}\text{ }n+O(1)a=4 log2​ n+O(1) and b=4b=4b=4, which is impossible classically. We furthermore present a cheat-sensitive quantum bit string commitment protocol for which we give an explicit tradeoff between Bob's ability to gain information about the committed string, and the probability of him being detected cheating.
Unconditionally secure nonrelativistic bit commitment is known to be impossible in both the classical and the quantum world. However, when committing to a string of n bits at once, how far can we stretch the quantum limits? In this Letter, we introduce a framework of quantum schemes where Alice commits a string of n bits to Bob, in such a way that she can only cheat on a bits and Bob can learn at most b bits of information before the reveal phase. Our results are twofold: we show by an explicit construction that in the traditional approach, where the reveal and guess probabilities form the security criteria, no good schemes can exist: a + b is at least n. If, however, we use a more liberal criterion of security, the accessible information, we construct schemes where a = 4log2(n) + O(1) and b = 4, which is impossible classically. Our findings significantly extend known no-go results for quantum bit commitment.
Unconditionally secure non-relativistic bit commitment is known to be impossible in both the classical and quantum worlds. However, when committing to a string of n bits at once, how far can we stretch the quantum limits? We consider quantum schemes where Alice commits a string of n bits to Bob, in such a way that she can only cheat on a bits and Bob can learn at most b bits of ''information'' before the reveal phase. We show a negative and a positive result, depending on how we measure Bob's information. If we use the Holevo quantity, no good schemes exist: a+b is at least n. If, however, we use accessible information, there exists a scheme where a=4 log n+O(1) and b=4. This is classically impossible. Our protocol is not efficient, however, we also exhibit an efficient scheme when Bob's measurement circuit is restricted to polynomial size. Our scheme also implies a protocol for n simultaneous coin flips which achieves higher entropy of the resulting string than any previously known protocol.
It is well known that unconditionally secure bit commitment is impossible even in the quantum world. In this paper a weak variant of quantum bit commitment, introduced independently by Aharonov et al. [STOC, 2000] and Hardy and Kent [Phys. Rev. Lett. 92 (2004)] is investigated. In this variant, the parties require some nonzero probability of detecting a cheating, i.e. if Bob, who commits a bit b to Alice, changes his mind during the revealing phase then Alice detects the cheating with a positive probability (we call this property binding); and if Alice gains information about the committed bit before the revealing phase then Bob discovers this with positive probability (sealing). In our paper we give quantum bit commitment scheme that is simultaneously binding and sealing and we show that if a cheating gives epsilon advantage to a malicious Alice then Bob can detect the cheating with a probability Omega(epsilon^2). If Bob cheats then Alice's probability of detecting the cheating is greater than some fixed constant lambda>0. This improves the probabilities of cheating detections shown by Hardy and Kent and the scheme by Aharonov et al. who presented a protocol that is either binding or sealing, but not simultaneously both. To construct a cheat sensitive quantum bit commitment scheme we use a protocol for a weak quantum one-out-of-two oblivious transfer.
A bit string commitment protocol securely commits N classical bits so that the recipient can extract only M<N bits of information about the string. Classical reasoning suggests that bit string commitment implies bit commitment and hence, given the Mayers-Lo-Chau theorem, that nonrelativistic quantum bit string commitment is impossible. Not so: there exist nonrelativistic 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 restrictive definition of security against A, N can be taken to be O(exp(CN')) for a positive constant C. I briefly discuss possible applications.
This paper proposes a cheat-sensitive quantum bit commitment scheme based on single photons, in which Alice commits a bit to Bob. Here, Bob's probability of success at cheating as obtains the committed bit before the opening phase becomes close to (just like performing a guess) as the number of single photons used is increased. And if Alice alters her committed bit after the commitment phase, her cheating will be detected with a probability that becomes close to 1 as the number of single photons used is increased. The scheme is easy to realize with present day technology.