Quantum Bit Commitment

2019-11-12T14:23:29Z

132.227.102.87: /* Protocol Description */

This [https://arxiv.org/abs/1108.2879 example protocol] achieves the task of [[bit commitment]] securely by using a relativistic scheme.
In bit commitment, the committer "commits" to a particular bit value.
The receiver knows nothing about the committed bit value until the committer chooses to do so (''hiding property'').
The receiver has a guarantee that once committed, the committer cannot change the committed bit value (''binding property'').
Information-theoretic secure bit commitment cannot be done with non-relativistic schemes see this review paper [https://arxiv.org/abs/quant-ph/9712023].

'''Tags:''' [[:Category:Two Party Protocols|Two Party Protocols]], [[:Category:Quantum Enhanced Classical Functionality|Quantum Enhanced Classical Functionality]], [[:Category:Specific Task|Specific Task]],
[[:Category:Information-theoretic security|Information-theoretic security]],
[[Category:Two Party Protocols]] [[Category:Quantum Enhanced Classical Functionality]][[Category:Specific Task]]
[[Category:Information-theoretic security]]

==Assumptions==

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

==Outline==

Both the receiver and the committer have 2 agents each which are the parties they send their qubits to and receive the committed value from. The agents are light-like separated from the committer.

The receiver securely pre-prepares a set of qubits randomly chosen from the BB84 states and sends them to the committer.
To commit to the bit 0, the committer measures the received qubits in the standard basis and in Hadamard basis to commit to 1.
The committer then sends the outcomes to their agents over secure classical channels.
To unveil the committed bit, the committer'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 committer.
If the check passes, the receiver accepts the commitment.

==Notation==

* <math>N</math>: Number of random qubits used in the commitment.
* <math>|\psi_i\rangle</math>: Random BB84 qubit with index <math>i</math>.
* <math>P</math>: Space-time origin point for the Minkowski space which is the position of the committer.
* <math>Q_0</math>: Commiter's first agent.
* <math>Q_1</math>: Commiter's second agent.
* <math>Q^{'}_0</math>: Receiver's first agent.
* <math>Q^{'}_1</math>: Receiver's second agent.

==Requirements==

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

==Knowledge Graph==

{{graph}}

==Properties==

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

==Protocol Description==
[https://github.com/quantumprotocolzoo/protocols/tree/master/QuantumBitCommitment <u>Click here for Python code</u>]</br>
The committer 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''
# Prepare a set of <math>N</math> qubits <math>|\psi_i\rangle_{i=1}^N</math> chosen independently and randomly from the BB84 states - <math>\{|0\rangle, |1\rangle, |+\rangle, |-\rangle\}</math>.
# Send the qubits to the commiter at point P.

''Commiter''
# To commit to 0, measure in the <math>\{|0\rangle, |1\rangle\}</math> basis.
# To commit to 1, measure in the <math>\{|+\rangle, |-\rangle\}</math> basis.
# Send the measurement outcomes to the agents <math>Q_0</math> and <math>Q_1</math> via the secure classical channels.

===Unveiling Phase===

''Committer''
# The committer's agents reveal the measurement outcomes to the receiver's agents <math>Q'_0</math> and <math>Q'_1</math>.

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

==Further Information==
<div style='text-align: right;'>''*contributed by Natansh Mathur''</div>

GHZ-based Quantum Anonymous Transmission

2019-10-16T09:11:58Z

132.227.102.87:

This [https://arxiv.org/abs/quant-ph/0409201 example protocol] implements the task of [[Anonymous Transmission]] in a multiple node quantum network. The protocol uses <math>n</math>-partite [https://en.wikipedia.org/wiki/Greenberger%E2%80%93Horne%E2%80%93Zeilinger_state GHZ state] to enable two nodes, sender and receiver, to establish a link which they use to transmit a quantum message. Importantly, the quantum message is transmitted in a way that the identity of the sender is unknown to every other node, and the identity of the receiver is known only to the sender.

'''Tags:''' [[:Category: Quantum Enhanced Classical Functionality|Quantum Enhanced Classical Functionality]][[Category: Quantum Enhanced Classical Functionality]], [[:Category: Multi Party Protocols|Multi Party Protocols]] [[Category: Multi Party Protocols]], [[:Category:Specific Task|Specific Task]][[Category:Specific Task]], GHZ state, anonymous transmission
==Assumptions==
* '''Network:''' The network consists of <math>n</math> nodes that are fully identified and completely connected with pairwise [[authenticated]] classical channels. Additionally, there is a secure classical [https://en.wikipedia.org/wiki/Broadcasting_(networking) broadcast] channel.
* '''Source:''' [[Trusted]] [[multipartite]] state source.
* '''Adversarial model:''' [[active adversary]] who does not control the source.

==Outline==
The presented GHZ-based quantum anonymous transmission protocol is based on the work of [[GHZ State based Quantum Anonymous Transmission#References|[6]]]. The goal of the protocol is to transmit a quantum state <math>|\psi \rangle</math> from the sender <math>S</math> to the receiver <math>R</math>, while keeping the identities of <math>S</math> and <math>R</math> anonymous. We assume that there is exactly one receiver which is determined before the start of the protocol. The protocol consists of the following steps:
* ''Collision detection:'' Nodes run a collision detection protocol to determine a single sender <math>S</math>.
* ''State distribution:'' A trusted source distributes the <math>n</math>-partite GHZ state.
* ''Anonymous entanglement:'' <math>n-2</math> nodes (all except for <math>S</math> and <math>R</math>) measure in the <math>X</math> basis and broadcast their measurement outcome. <math>S</math> and <math>R</math> broadcast random dummy bits. The parity of measurement outcomes allows to establish an entangled link between <math>S</math> and <math>R</math> which is called [[anonymous entanglement]] (AE).
* ''Teleportation:'' Sender <math>S</math> teleports the message state <math>|\psi\rangle</math> to the receiver <math>R</math> using the established anonymous entanglement. Classical message <math>m</math> associated with teleportation is also sent anonymously.

==Notation==
* <math>n</math>: number of network nodes taking part in the anonymous transmission.
* <math>|\psi\rangle</math>: quantum message which the sender wants to send anonymously
* <math>S</math>: the sender of the quantum message
* <math>R</math>: the receiver of the quantum message

==Requirements==
*Network stage: [[:Category: Quantum Memory Network Stage|quantum memory network]][[Category:Quantum Memory Network Stage]].
* Relevant parameters to establish one anonymous link: <math>k=1</math> round of quantum communication per node, circuit depth <math>m=1</math>, <math>q=1</math> physical qubits per node.
* Quantum memories, single-qubit Pauli gates and single-qubit measurements at the end nodes.

<br/>

[[File:GHZQAT_Nodes.PNG|center|GHZ-based Quantum Anonymous Transmission (Nodes)]]

<br/>

[[File:GHZQAT_TD.PNG|center|GHZ-based Quantum Anonymous Transmission (Trusted Distributor)]]

{{graph}}

==Properties==
See [[Quantum Anonymous Transmission]] for the precise security definition. [[GHZ State based Quantum Anonymous Transmission#Pseudocode|Pseudocode]] given below implements secure anonymous transmission, i.e. it hides the identities of the sender and the receiver from other nodes in the network. That is, the maximum probability that adversaries guess the identity of <math>S</math> or <math>R</math> given all the classical and quantum information they have available at the end of the protocol is no larger than the uncertainty the adversaries have about the identities of <math>S</math> and <math>R</math> before the protocol begins. More formally, the anonymous transmission protocol with the GHZ state is sender- and receiver-secure: </br>
<math>P_{\text{guess}}[S|C,S\notin \mathcal{A}] \leq \max_{i\in[n]} P[S=i|S\notin \mathcal{A}] = \frac{1}{n-t},</math></br>
<math>P_{\text{guess}}[R|C,S\notin \mathcal{A}] \leq \max_{i\in[n]} P[R=i|S\notin \mathcal{A}] = \frac{1}{n-t},</math></br>
where <math>\mathcal{A}</math> is the subset of <math>t</math> adversaries among <math>n</math> nodes and <math>C</math> is the register that contains all classical and quantum side information accessible to the adversaries. Note that this implies that the protocol is also trace-less, since even if the adversary hijacks any <math>t\leq n-2</math> players and gains access to all of their classical and quantum information after the end of the protocol, she cannot learn the identities of <math>S</math> and <math>R</math>. For a formal argument see [[GHZ State based Quantum Anonymous Transmission#References|[6]]].

==Protocol Description==
Receiver <math>R</math> is determined before the start of the protocol. <math>S</math> holds a message qubit <math>|\psi\rangle</math>.
# Nodes run a collision detection protocol and determine a single sender <math>S</math>.
# A trusted source distributes <math>n</math>-partite GHZ state to every player, <math>|GHZ\rangle = \frac{1}{\sqrt{2}} (|0^n\rangle + |1^n\rangle)</math>.
# Anonymous entanglement:
## Sender <math>S</math> and receiver <math>R</math> do not do anything to their part of the state.
## Every player <math>j \in [n] \setminus \{S,R\}</math>:
### Applies a Hadamard transform to her qubit,
### Measures this qubit in the computational basis with outcome <math>m_j</math>,
### Broadcasts <math>m_j</math>.
## <math>S</math> picks a random bit <math>b \in_R \{ 0,1 \}</math> and broadcasts <math>b</math>.
## <math>S</math> applies a phase flip <math>Z</math> to her qubit if <math>b=1</math>.
## <math>R</math> picks a random bit <math>b' \in_R \{ 0,1 \}</math> and broadcasts <math>b'</math>.
## <math>R</math> applies a phase flip <math>Z</math> to her qubit, if <math>b \oplus \bigoplus_{j \in [n] \setminus \{S,R\}} m_j = 1</math>. </br> <math>S</math> and <math>R</math> share anonymous entanglement <math>|\Gamma\rangle_{SR} = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle)</math>.
# <math>S</math> uses the quantum teleportation circuit with input <math>|\psi\rangle</math> and anonymous entanglement <math>|\Gamma\rangle_{SR}</math>, and obtains measurement outcomes <math>m_0, m_1</math>.
# The players run a protocol to anonymously send bits <math>m_0,m_1</math> from <math>S</math> to <math>R</math> (see [[GHZ-based Quantum Anonymous Transmission #Further Information|Further Information]] for details).
# <math>R</math> applies the transformation described by <math>m_0,m_1</math> on his part of <math>|\Gamma\rangle_{SR}</math> and obtains <math>|\psi\rangle_{SR}</math>.

==Further Information==
* To determine the sender <math>S</math> (Step 1) one can run either a classical collision detection protocol of [[GHZ State based Quantum Anonymous Transmission#References|[4] ]] or a quantum collision detection protocol of [[GHZ State based Quantum Anonymous Transmission#References|[6] ]]. The quantum version of the protocol requires additional <math>(\left\lceil \log n \right\rceil + 1)</math> GHZ states.
* To determine the receiver <math>R</math> during the protocol one can incorporate an additional step using a classical receiver notification protocol of [[GHZ State based Quantum Anonymous Transmission#References|[4] ]].
* To send classical teleportation bits <math>m_0,m_1</math> (Step 5) the players can run a classical logical OR protocol of [[GHZ State based Quantum Anonymous Transmission#References|[4] ]] or anonymous transmission protocol for classical bits with quantum resources of [[GHZ State based Quantum Anonymous Transmission#References|[6] ]]. The quantum protocol requires one additional GHZ state for transmitting one classical bit.
* The anonymous transmission of quantum states was introduced in [[GHZ State based Quantum Anonymous Transmission#References|[6] ]].
* The problem was subsequently developed to consider the preparation and certification of the GHZ state [[GHZ State based Quantum Anonymous Transmission#References|[3], [5], [7] ]].
* In [[GHZ State based Quantum Anonymous Transmission#References|[5] ]], it was first shown that the proposed protocol is information-theoretically secure against an active adversary.
* In [[GHZ State based Quantum Anonymous Transmission#References|[1] ]] a protocol using another multipartite state, the W state, was introduced. The reference discusses the noise robustness of both GHZ-based and W-based protocols and compares the performance of both protocols.
* Other protocols were proposed, which do not make use of multipartite entanglement, but utilise solely Bell pairs to create anonymous entanglement [[GHZ State based Quantum Anonymous Transmission#References|[2] ]].

==References==
#[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.98.052320 Lipinska et al (2018)]
#[https://europepmc.org/abstract/med/27247078 Yang et al (2016)]
#[https://ieeexplore.ieee.org/document/4077005 Bouda et al (2007)]
#[https://arxiv.org/abs/0706.2010 Broadbent et al (2007)]
#[https://arxiv.org/abs/0706.2356 Brassard et al (2007)]
#[https://arxiv.org/abs/quant-ph/0409201 Christandl et al (2005)]
#[https://arxiv.org/abs/1811.04729 Unnikrishnan et al (2018)]
<div style='text-align: right;'>''contributed by Victoria Lipinska''</div>

Quantum Protocol Zoo - User contributions [en]

Quantum Bit Commitment

GHZ-based Quantum Anonymous Transmission