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Analysis of one quantum bit string commitment.

Zhengjun Cao

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

Abstract A. Kent proposed a quantum bit string commitment protocol in 2003. Not using the standard two conjugate states |0〉, |1〉, |+ 〉 and |−〉, the protocol uses ψ0 = |0 〉 and ψ1 = sin θ|0 〉 + cos θ|1〉, where θ ̸ = 0. In this paper, we show that the protocol can not guarantee security to the receiver. (1 − sin2 θ 2)n bits are definitely exposed to the receiver, where n is the length of the committed string.

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Abstract A. Kent proposed a quantum bit string commitment protocol in 2003. Not using the standard two conjugate states |0〉, |1〉, |+ 〉 and |−〉, the protocol uses ψ0 = |0 〉 and ψ1 = sin θ|0 〉 + cos θ|1〉, where θ ̸ = 0. In this paper, we show that the protocol can not guarantee security to the receiver. (1 − sin2 θ 2)n bits are definitely exposed to the receiver, where n is the length of the committed string.
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.
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.
In a secure bit commitment protocol involving only classical physics, A commits either a 000 or a 111 to B. If quantum information is used in the protocol, A may be able to commit a state of the form $\ensuremath{\alpha}|0〉+\ensuremath{\beta}|1〉.$ If so, she can also commit mixed states in which the committed bit is entangled with other quantum states under her control. We introduce here a quantum cryptographic primitive, bit commitment with a certificate of classicality (BCCC), which differs from standard bit commitment in that it guarantees that the committed state has a fixed classical value. We show that no unconditionally secure BCCC protocol based on special relativity and quantum theory exists. We also propose complete definitions of security for quantum and relativistic bit commitment.
Bit commitment is an essential primitive in quantum cryptography. It is the basic building block of many cryptographic protocols in which it provides security over distrustful parties. This paper proposes a secured quantum two-bit commitment protocol for any two classical bits. The protocol contains two parties - a committer and a receiver - who share both quantum and classical communication channels. The main phases of this protocol are commitment and revealing phases where the concealing and binding conditions are proved in order to accept the commitment message. The proposed framework utilizes different security layers to hide specific quantum states in a superposition alongside trivial one qubit operations. The secured unitary transformations are applied by the two parties to meet the concealing and binding conditions. To verify the success of the proposed protocol, the measured output of the two parties are compared. The protocol operates under the assumption that the two parties might cheat and the possible existence of an eavesdropper.