This paper proposes a quantum protocol for a cheat-sensitive commitment of a classical bit. Alice, the receiver of the bit, can examine dishonest Bob, who changes or postpones his choice. Bob, the sender of the bit, can examine dishonest Alice, who violates concealment. For each round-trip case, Alice sends one of two spin states |
S\ifmmode\pm\else\textpm\fi{}⟩ by choosing basis
S at random from two conjugate bases
X and
Y. Bob chooses basis
C \ensuremath{\in} {
X,
Y} to perform a measurement and returns a resultant state |
C\ifmmode\pm\else\textpm\fi{}⟩. Alice then performs a measurement with the other basis
R (\ensuremath{\ne}
S) and obtains an outcome |
R\ifmmode\pm\else\textpm\fi{}⟩. In the opening phase, she can discover dishonest Bob, who unveils a wrong basis with a faked spin state, or Bob can discover dishonest Alice, who infers basis
C but destroys |
C\ifmmode\pm\else\textpm\fi{}⟩ by setting
R to be identical to
S in the commitment phase. If a classical bit is coded in a block of
m \ifmmode\times\else\texttimes\fi{}
n qubit particles, impartial examinations and probabilistic security criteria can be achieved.
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.
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.
Bit commitment is a fundamental cryptographic primitive with numerous applications. Quantum information allows for bit commitment schemes in the information theoretic setting where no dishonest party can perfectly cheat. The previously best-known quantum protocol by Ambainis 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/√2 (his lower bound on coin flipping can be easily extended to bit commitment). Closing this gap has since been an important open question. In this paper, we provide the optimal bound for quantum bit commitment. First, we show a lower bound of approximately 0.739, improving Kitaev's lower bound. For this, we present some generic cheating strategies for Alice and Bob and conclude by proving a new relation between the trace distance and fidelity of two quantum states. Second, we present an optimal quantum bit commitment protocol which has cheating probability arbitrarily close to 0.739. More precisely, we show how to use any weak coin flipping protocol with cheating probability 1/2 + ε in order to achieve a quantum bit commitment protocol with cheating probability 0.739 + O(ε). We then use the optimal quantum weak coin flipping protocol described by Mochon. Last, in order to stress the fact that our protocol uses quantum effects beyond the weak coin flip, we show that any classical bit commitment protocol with access to perfect weak (or strong) coin flipping has cheating probability at least 3/4.
Cheat sensitive quantum bit commitment (CSQBC) loosens the security requirement of quantum bit commitment (QBC), so that the existing impossibility proofs of unconditionally secure QBC can be evaded. But here we analyze the common features in all existing CSQBC protocols, and show that in any CSQBC having these features, the receiver can always learn a non-trivial amount of information on the sender's committed bit before it is unveiled, while his cheating can pass the security check with a probability not less than 50%. The sender's cheating is also studied. The optimal CSQBC protocols that can minimize the sum of the cheating probabilities of both parties are found to be trivial, as they are practically useless. We also discuss the possibility of building a fair protocol in which both parties can cheat with equal probabilities.