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International Journal of Modern Physics B

TWO-WAY QUANTUM-BIT-STRING COMMITMENT

Chuan-Ming Li, Tzonelih Hwang

Published Apr 20, 2010• 0 citations
Publisher

Abstract

Due to the high cost of the qubit generating device (QGD) (the single photon source), QGD may not be available at the same time in both communicating parties (Alice and Bob) who have to execute the quantum bit string commitment (QBSC) protocol for some application environments. All the existing QBSC protocols allow one-way commitment i.e., only the party (Bob) who possesses the QGD is allowed to commit a bit string to Alice who does not have the QGD, but not the other way around. However, what happens if Alice, who does not have the QGD, is required to commit a bit string to Bob? The current QBSC protocols fail to provide a solution in this case. Therefore, the purpose of this paper is to allow Alice, who is not equipped with the QGD, to commit a bit string to Bob who has the QGD. With our proposed protocol, together with previously proposed QBSC protocols, Alice and Bob can perform two-way commitment, even though Bob is the only party who is equipped with the QGD.

DOI
10.1142/s0217979210048764

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Due to the high cost of the qubit generating device (QGD) (the single photon source), QGD may not be available at the same time in both communicating parties (Alice and Bob) who have to execute the quantum bit string commitment (QBSC) protocol for some application environments. All the existing QBSC protocols allow one-way commitment i.e., only the party (Bob) who possesses the QGD is allowed to commit a bit string to Alice who does not have the QGD, but not the other way around. However, what happens if Alice, who does not have the QGD, is required to commit a bit string to Bob? The current QBSC protocols fail to provide a solution in this case. Therefore, the purpose of this paper is to allow Alice, who is not equipped with the QGD, to commit a bit string to Bob who has the QGD. With our proposed protocol, together with previously proposed QBSC protocols, Alice and Bob can perform two-way commitment, even though Bob is the only party who is equipped with the QGD.
Although Mayers, Lo and Chau successfully showed that unconditionally secure quantum bit commitment (QBC) is impossible, we present a secure QBC protocol which is quite different from the model used in their proofs. If Alice wants to commit a bit b, she first encodes b into a series of classical bit strings of the same length. The requirement is that the modulo 2 sum of all bits in every bit string equals to b. Then Alice uses quantum one-way function to produce quantum states which are corresponding to the encoded classical bit strings and will be sent to Bob as evidence. The quantum one-way function used here is just that was successfully applied in Gottesman and Chuang's quantum digital signatures. When it comes to the unveilingtime, Alice only sends the bit strings to Bob. Finally, Bob generates quantum states form the bit strings by the same quantum one-way function and uses controlled-swap circuit to check whether the quantum states are identical with the evidence.
It is well known, due to the no-go theorems, that perfectly secure bit commitment is impossible. For that reason, there exist many proposals for bit commitment protocols that are secure under certain assumptions (such as the existence of a trusted party, computational hardness, or noisy memories). In this paper, we investigate the minimal assumptions required to obtain a secure bit commitment protocol. We show that it is possible to construct a bit commitment scheme with two parties Alice and Bob as long as there is a third party that is willing to broadcast either entangled qubits for the parties or to produce classical strings that are xor-correlated. We also show that the bit commitment scheme proposed is universally composable.
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.
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.