Quantum Bit Commitment

This example protocol achieves the task of Bit Commitment securely by using a relativistic scheme. In bit commitment, the commiter "commits" to a particular bit value. The receiver knows nothing about the committed bit value until the commiter chooses to do so. The receiver has a guarantee that once committed, the commiter cannot change the committed bit value. Bit commitment cannot be done with non-relativistic schemes as proved by Mayers in his paper.

Assumptions

• Quantum theory is correct.
• The background space-time is approximately Minkowski.
• The commmiter can signal at precisely light speed.
• All information processing is instantaneous.

Outline

The receiver securely preprepares a set of qubits randomly from the BB84 states and sends them to the commiter. To commit to the bit 0, the commiter measures the received qubits in the standard basis and in Hadamard basis to commit to 1. The commiter then sends the outcomes to their agents over secure classical channels. To unveil the committed bit, the commiter's agents reveal the outcomes to the receiver's agents. The receiver's agents then check if the outcomes they have received are the same and consistent with the states sent to the commiter. If the check passes, the receiver accepts the commitment.

Notation

• $N$: Number of random qubits used in the commitment.
• $|\psi _{i}\rangle$: Random BB84 qubit with index $i$.
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} : Space-time origin point for the Minkowski space.
• $Q_{0}$: Commiter's first agent.
• $Q_{1}$: Commiter's second agent.
• $Q_{0}^{'}$: Receiver's first agent.
• $Q_{1}^{'}$: Receiver's second agent.

Hardware Requirements

• Secure classical channels between the parties and their agents.
• Basic state preparation abilities for the receiver.
• Instantaneous measurement capabilities for the commiter.

Properties

• There is no need of quantum memory for the parties.
• The protocol is unconditionally secure.

Pseudocode

The commiter and the receiver agree on the space-time origin point P and two light-like separated points where the two agents of each party will be stationed.

Commitment Phase

Receiver

1. Prepare a set of $N$ qubits Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi_i\rangle_{i=1}^N} chosen independently and randomly from the BB84 states - $\{|0\rangle ,|1\rangle ,|+\rangle ,|-\rangle \}$.
2. Send the qubits to the commiter at point P.

Commiter

1. To commit to 0, measure in the $\{|0\rangle ,|1\rangle \}$ basis.
2. To commit to 1, measure in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{|+\rangle, |-\rangle\}} basis.
3. Send the measurement outcomes to your agents $Q_{0}$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_1} via the secure classical channels.

Unveiling Phase

Commiter

1. The commiter's agents reveal the measurement outcomes to the receiver's agents.

Receiver

1. Check if the revealed outcomes of both the agents are same, if not, then abort.
2. Check if the revealed outcomes are consistent with the sent states, if not, then abort.
3. If the checks pass, accept the commitment.

Further Information

*contributed by Natansh Mathur