Insecurity of a relativistic quantum commitment scheme

We propose a cheating strategy to a relativistic quantum commitment scheme [Sci Rep 2014;4:6774] which was claimed to be unconditionally secure. It is shown that the sender Alice can cheat successfully with probability 100%, thus disproving the security claim.

In these relativistic schemes, the participants are split into "agents", so that it is no longer a 2-party cryptographic task. This makes it much easier to prevent the participants from cheatings. But if the scheme is not properly designed, cheatings could still be possible even in this multi-party scenario. Here we will show that the scheme in Ref. [13] is such an example.

The scheme
The original QBC is defined as a cryptographic task between two parties Alice and Bob. It can be divided into two phases. In the commit phase, Alice decides the value of the bit b that she wants to commit, and sends Bob some evidence, e.g., quantum states. Later, in the revealing phase, Alice announces the value of b, and Bob checks it with the evidence. An unconditionally secure QBC scheme needs to be both binding (i.e., Alice cannot change the value of b after the commit phase) and concealing (Bob cannot know b before the revealing phase).
The relativistic commitment scheme proposed in Ref. [13] is carried on among three parties, the sender Alice, the receiver Bob and his agent C (who is separated far away from Bob so that they cannot communicate efficiently during the commit phase since no signal can travel faster than the speed of light). The coding method of the scheme is as follows. Denote the Bell states as (Eq. (1) of Ref. [13]) (1) * Electronic address: hegp@mail.sysu.edu.cn To make a commitment, Alice secretly prepares N Bell pairs |β ac = N n=1 |u a u c n and sends the second qubit of each pair to agent C, while keeping the first qubit of each pair to herself. The values of u a and u c are chosen according to the value she wants to commit. To commit to the bit b = 0, Alice initiates with |β ac = N n=1 |00 n ; to commit to b = 1, Alice initiates with |β ac = N n=1 |01 n . Besides b = 0, 1, this scheme also allows Alice to commit to a qubit. To commit to the qubit (|0 + |1 )/ √ 2, Alice prepares |β ac = N n=1 |10 n ; while for committing (|0 − |1 )/ √ 2, Alice starts with |β ac = N n=1 |11 n . Bob and agent C then perform some operations on their half of |β ac and some other qubits at their side, which are basically parts of the quantum teleportation process [14]. But the details of these operations are not important to us, because we shall show that Alice can cheat anyway regardless what Bob and agent C do at their side.
In the revealing phase, Alice reveals her commitment by announcing the values of u a u c . Also, to justify that she is honest, she sends to Bob the other half of |β ac that she kept.

The cheating strategy
Though the above scheme was claimed to be unconditionally secure in Ref. [13], here we show that a dishonest Alice can always alter her commitment in the revealing phase by applying local operations on the qubits she kept, while passing the security checks of Bob and agent C with probability 100%.
Her cheating strategy is simple. From Eq. (1) we can see that the Pauli matrix σ z acting on the first qubit alone can flip the value of the bit u i , while (−1) ui σ x acting on the first qubit alone can flip the value of the bit u j , i.e., Here I is the identity operator on the second qubit. That is, Alice can change |u a u c n among the four Bell states freely with a local unitary transformation of her own, without the help of Bob and agent C. Therefore, Alice can always starts the commitment phase with |β ac = N n=1 |00 n . Later, in the revealing phase, if she wants to convince Bob and agent C that she has committed to the bit 0, she simply follows the original scheme honestly. But if she wants to show that she has committed to the bit 1 (or the qubits (|0 + |1 )/ √ 2 or (|0 − |1 )/ √ 2), she applies σ x (σ z or σ z σ x ) on each of the N qubits she kept, before sending them to B. Note that the local operations on the two qubits of a Bell state, respectively, always commute with each other. Therefore, no matter what operations Bob and agent C had performed on their share of the qubits, Alice's above operation will make |β ac appear as if it was originally prepared as N n=1 |01 n ( N n=1 |10 n or N n=1 |11 n ), without causing any conflict with the measurement results of Bob and agent C. Thus her cheating can never be detected.
To summarize, the commitment scheme in Ref. [13] cannot meet the binding condition against Alice's cheating, so that it is not unconditionally secure.

Funding:
This work was supported in part by Guangdong Basic and Applied Basic Research Foundation, China (Grant No. 2019A1515011048).