Anonymous Conference Key Agreement using GHZ states: Difference between revisions

From Quantum Protocol Zoo
Jump to navigation Jump to search
(Created page with "<!-- This is a comment. You can erase them or write below --> <!-- Intro: brief description of the protocol --> This [https://arxiv.org/abs/2007.07995 example protocol] achie...")
 
No edit summary
Line 2: Line 2:


<!-- Intro: brief description of the protocol -->
<!-- Intro: brief description of the protocol -->
This [https://arxiv.org/abs/2007.07995 example protocol] achieves the functionality of quantum conference key agreement anonymously. This protocol allows multiple parties in a quantum network to establish a shared secret key anonymously.
This [https://arxiv.org/abs/2007.07995 example protocol] achieves the functionality of quantum conference key agreement. This protocol allows multiple parties in a quantum network to establish a shared secret key anonymously.


<!--Tags: related pages or category -->
<!--Tags: related pages or category -->
Line 8: Line 8:
==Assumptions==
==Assumptions==
<!-- It describes the setting in which the protocol will be successful. -->
<!-- It describes the setting in which the protocol will be successful. -->
 
We require the following for this protocol:
# A source of n-party GHZ states
# Private randomness sources
# A randomness source that is not associated with any party
# A classical broadcasting channel
# Pairwise private communication channels


==Outline==
==Outline==
Line 44: Line 49:


===Protocol 2: Notification===
===Protocol 2: Notification===
''Input: '' Sender's choice of <math>m</math> receivers
''Goal: '' The <math>m</math> receivers get notified
''Requirements: '' Private pairwise classical communication channels and randomness sources
For agent <math>i = 1,...,n</math>:
# All agents <math>j \in \{1,...,n\}</math> do the following:
#* '''When agent <math>j</math> is the sender''': If <math>i</math> is not a receiver, the sender chooses <math>n</math> random bits <math>\{r_{j,k}^i\}_{k = 1}^n</math> such that <math>\bigoplus_{k=1}^n r_{j,k}^i = 0</math>. Otherwise, if <math>i</math> is a receiver, the sender chooses <math>n</math> random bits such that <math>\bigoplus_{k=1}^n r_{j,k}^i = 1</math>. The sender sends bit <math>r_{j,k}^i</math> to agent <math>k</math>
#* '''When agent <math>j</math> is not the sender''':  The agent chooses <math>n</math> random bits <math>\{r_{j,k}^i\}_{k = 1}^n</math> such that <math>\bigoplus_{k=1}^n r_{j,k}^i = 0</math> and sends bit <math>r_{j,k}^i</math> to agent <math>k</math>
# All agents <math>k \in \{1,...,n\}</math> receive <math>\{r_{j,k}^i\}_{j = 1}^n</math>, and compute <math>z_k^i = \bigoplus_{j=1}^n r_{j,k}^i</math> and send it to agent <math>i</math>
# Agent <math>i</math> takes the received <math>\{z_k^i\}_{k=1}^n</math> to compute <math>z^i = \bigoplus_{k=1}^nz_k^i</math>. If <math>z^i = 1</math>, they are thereby notified to be a designated receiver.
===Protocol 3: Anonymous Multiparty Entanglement===
===Protocol 3: Anonymous Multiparty Entanglement===
''Input: '' <math>n</math>-partite GHZ state <math>\frac{1}{\sqrt{2}}(|0\rangle^{\otimes n} + |1\rangle^{\otimes n})</math>
''Output: '' <math>(m+1)</math>-partite GHZ state <math>\frac{1}{\sqrt{2}}(|0\rangle^{\otimes (m+1)} + |1\rangle^{\otimes (m+1)})</math> shared between the sender and receivers
''Requirements: '' A broadcast channel; private randomness sources
# Sender and receivers draw a random bit each. Everyone else measures their qubits in the X-basis, yielding a measurement outcome bit <math>x_i</math>
# All parties broadcast their bits in a random order, or if possible, simultaneously.
# The sender applies a Z gate to their qubit if the parity of the non-participating parties' bits is odd.


===Protocol 4: Verification===
===Protocol 4: Verification===


''Input: '' A verifier V; a shared state between <math>k</math> parties
''Goal: '' Verification or rejection of the shared state as the GHZ<math>_k</math> state by V
''Requirements: '' Private randomness sources; a classical broadcasting channel
# Everyone but V draws a random bit <math>b_i</math> and measures in the X or Y basis if their bit equals 0 or 1 respectively, obtaining a measurement outcome <math>m_i</math>. V chooses both bits at random
# Everyone (including V) broadcasts <math>(b_i,m_i)</math>
# V resets her bit such that <math>\sum_ib_i = 0 (</math>mod <math>2)</math>. She measures in the X or Y basis if her bit equals 0 or 1 respectively, thereby also resetting her <math>m_i = m_v</math>
# V accepts the state if and only if <math>\sum_im_i = \frac{1}{2}\sum_ib_i (</math>mod <math>2)</math>
==Properties==
==Properties==
<!-- important information on the protocol: parameters (threshold values), security claim, success probability... -->
<!-- important information on the protocol: parameters (threshold values), security claim, success probability... -->

Revision as of 20:59, 7 January 2022


This example protocol achieves the functionality of quantum conference key agreement. This protocol allows multiple parties in a quantum network to establish a shared secret key anonymously.


Assumptions

We require the following for this protocol:

  1. A source of n-party GHZ states
  2. Private randomness sources
  3. A randomness source that is not associated with any party
  4. A classical broadcasting channel
  5. Pairwise private communication channels

Outline

  • First, the sender notifies each receiver in the network anonymously
  • The entanglement source generates and distributes sufficient GHZ states to all nodes in the network
  • The GHZ states are distilled to establish multipartite entanglement shared only by the participating parties (the sender and receivers)
  • Each GHZ state is randomly chosen to be used for either Verification or Key Generation. For Key Generation rounds, a single bit of the key is established using one GHZ state by measuring in the Z-basis
  • If the sender is content with the Verification results, they can anonymously validate the protocol and conclude that the key has been established successfully.

Notation

Protocol Description

Protocol 1: Anonymous Verifiable Conference Key Agreement

Input: Parameters 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 L} 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 D}

Requirements: A source of n-party GHZ states; private randomness sources; a randomness source that is not associated with any party; a classical broadcasting channel; pairwise private communication channels

Goal: Anonymoous generation of key between sender 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 m} receivers

  1. The sender notifies 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 m} receivers by running the Notification protocol
  2. The source generates and shares 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 L} GHZ states
  3. The parties run the Anonymous Multipartite Entanglement protocol on the GHZ states
  4. For each -partite GHZ state, the parties do the following:
    • They ask a source of randomness to broadcast a bit such that PrFailed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [b=1]={\frac {1}{D}}}
    • Verification round: If b = 0, the sender runs Verification as verifier on the state corresponding to that round, while only considering the announcements of the Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle m} receivers. The remaining parties announce random values.
    • KeyGen round: If b = 1, the sender and receivers measure in the Z-basis.
  5. If the sender is content with the checks of the Verification protocol, they can anonymously validate the protocol

Protocol 2: Notification

Input: Sender's choice of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle m} receivers

Goal: The receivers get notified

Requirements: Private pairwise classical communication channels and randomness sources

For agent 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 i = 1,...,n} :

  1. All agents 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 j \in \{1,...,n\}} do the following:
    • When agent 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 j} is the sender: If 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 i} is not a receiver, the sender chooses 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 n} random bits 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 \{r_{j,k}^i\}_{k = 1}^n} such that 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 \bigoplus_{k=1}^n r_{j,k}^i = 0} . Otherwise, if 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 i} is a receiver, the sender chooses 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 n} random bits such that 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 \bigoplus_{k=1}^n r_{j,k}^i = 1} . The sender sends bit 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 r_{j,k}^i} to agent 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 k}
    • When agent 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 j} is not the sender: The agent chooses 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 n} random bits 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 \{r_{j,k}^i\}_{k = 1}^n} such that 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 \bigoplus_{k=1}^n r_{j,k}^i = 0} and sends bit 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 r_{j,k}^i} to agent 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 k}
  2. All agents 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 k \in \{1,...,n\}} receive 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 \{r_{j,k}^i\}_{j = 1}^n} , and compute 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 z_k^i = \bigoplus_{j=1}^n r_{j,k}^i} and send it to agent 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 i}
  3. Agent 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 i} takes the received 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 \{z_k^i\}_{k=1}^n} to compute 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 z^i = \bigoplus_{k=1}^nz_k^i} . If 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 z^i = 1} , they are thereby notified to be a designated receiver.

Protocol 3: Anonymous Multiparty Entanglement

Input: 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 n} -partite GHZ state 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 \frac{1}{\sqrt{2}}(|0\rangle^{\otimes n} + |1\rangle^{\otimes n})}

Output: 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 (m+1)} -partite GHZ state 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 \frac{1}{\sqrt{2}}(|0\rangle^{\otimes (m+1)} + |1\rangle^{\otimes (m+1)})} shared between the sender and receivers

Requirements: A broadcast channel; private randomness sources

  1. Sender and receivers draw a random bit each. Everyone else measures their qubits in the X-basis, yielding a measurement outcome bit 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 x_i}
  2. All parties broadcast their bits in a random order, or if possible, simultaneously.
  3. The sender applies a Z gate to their qubit if the parity of the non-participating parties' bits is odd.

Protocol 4: Verification

Input: A verifier V; a shared state between 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 k} parties

Goal: Verification or rejection of the shared state as the GHZFailed 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 _k} state by V

Requirements: Private randomness sources; a classical broadcasting channel

  1. Everyone but V draws a random bit 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 b_i} and measures in the X or Y basis if their bit equals 0 or 1 respectively, obtaining a measurement outcome 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 m_i} . V chooses both bits at random
  2. Everyone (including V) broadcasts 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 (b_i,m_i)}
  3. V resets her bit such that 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 \sum_ib_i = 0 (} mod 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 2)} . She measures in the X or Y basis if her bit equals 0 or 1 respectively, thereby also resetting her 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 m_i = m_v}
  4. V accepts the state if and only if 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 \sum_im_i = \frac{1}{2}\sum_ib_i (} mod 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 2)}

Properties

Further Information

References

Retrieved from "https://wiki.veriqloud.fr/index.php?title=Anonymous_Conference_Key_Agreement_using_GHZ_states&oldid=4428"