Anonymous Conference Key Agreement using GHZ states

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.

Tags: Multi Party Protocols, Quantum Enhanced Classical Functionality, Specific Task

Requirements

We require the following resources 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

• 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} : Total number of nodes in the network

• $m$: Number of receiving nodes

• $L$: Number of GHZ states used

• $D$: Security parameter; expected number of GHZ states used to establish one bit of key

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

Protocol Description

Protocol 1: Anonymous Verifiable Conference Key Agreement

Input: Parameters $L$ and $D$

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 $m$ receivers

1. The sender notifies the $m$ receivers by running the Notification protocol
2. The source generates and shares $L$ GHZ states
3. The parties run the Anonymous Multipartite Entanglement protocol on the GHZ states
4. For each 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, the parties do the following:
   • They ask a source of randomness to broadcast a bit $b$ such that Pr$[b=1]={\frac {1}{D}}$
   • 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 $m$ 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 $m$ receivers

Goal: The $m$ receivers get notified

Requirements: Private pairwise classical communication channels and randomness sources

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

1. All agents $j\in \{1,...,n\}$ do the following:
   • When agent $j$ is the sender: If $i$ is not a receiver, the sender chooses $n$ random bits $\{r_{j,k}^{i}\}_{k=1}^{n}$ such that $\bigoplus _{k=1}^{n}r_{j,k}^{i}=0$. Otherwise, if $i$ is a receiver, the sender chooses $n$ random bits such that $\bigoplus _{k=1}^{n}r_{j,k}^{i}=1$. The sender sends bit $r_{j,k}^{i}$ to agent $k$
   • When agent $j$ is not the sender: The agent chooses $n$ random bits $\{r_{j,k}^{i}\}_{k=1}^{n}$ such that $\bigoplus _{k=1}^{n}r_{j,k}^{i}=0$ and sends bit $r_{j,k}^{i}$ to agent $k$
2. All agents $k\in \{1,...,n\}$ receive $\{r_{j,k}^{i}\}_{j=1}^{n}$, and compute $z_{k}^{i}=\bigoplus _{j=1}^{n}r_{j,k}^{i}$ and send it to agent $i$
3. Agent $i$ 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 $z^{i}=\bigoplus _{k=1}^{n}z_{k}^{i}$. 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: $n$-partite GHZ state ${\frac {1}{\sqrt {2}}}(|0\rangle ^{\otimes n}+|1\rangle ^{\otimes n})$

Output: $(m+1)$-partite GHZ state ${\frac {1}{\sqrt {2}}}(|0\rangle ^{\otimes (m+1)}+|1\rangle ^{\otimes (m+1)})$ 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 $x_{i}$
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 GHZ$_{k}$ state by V

Requirements: Private randomness sources; a classical broadcasting channel

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

Properties

• Protocol 1 has an asymptotic key rate of ${\frac {L}{D}}$
• This protocol satisfies the following notions of anonymity:
  • Sender Anonymity: A protocol allows a sender to remain anonymous sending a message to $m$ receivers, if an adversary who corrupts $t\leq n-2$ players, cannot guess the identity of the sender with probability higher than ${\frac {1}{n-t}}$
  • Receiver Anonymity: A protocol allows a receiver to remain anonymous receiving a message, if an adversary who corrupts $t\leq n-2$ players, cannot guess the identity of the receiver with probability higher than ${\frac {1}{n-t}}$
• Error correction and privacy amplification must be carried out anonymously and are not considered in the analysis of this protocol.

References

• The protocols and their security analysis, along with an experimental implementation for $n=4$ can be found in Hahn et al.(2020)

*contributed by Chirag Wadhwa