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.

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.

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 $m$ 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle b} 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 $m$ 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{r_{j,k}^{i}\}_{k=1}^{n}} such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \bigoplus _{k=1}^{n}r_{j,k}^{i}=0} . Otherwise, if $i$ is a receiver, the sender chooses $n$ random bits such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \bigoplus _{k=1}^{n}r_{j,k}^{i}=1} . The sender sends bit Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r_{j,k}^{i}} to agent $k$
   • When agent $j$ is not the sender: The agent chooses $n$ random bits Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{r_{j,k}^{i}\}_{k=1}^{n}} such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \bigoplus _{k=1}^{n}r_{j,k}^{i}=0} and sends bit Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r_{j,k}^{i}} to agent $k$
2. All agents $k\in \{1,...,n\}$ receive Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{r_{j,k}^{i}\}_{j=1}^{n}} , and compute Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{z_{k}^{i}\}_{k=1}^{n}} to compute Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle z^{i}=\bigoplus _{k=1}^{n}z_{k}^{i}} . If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle z^{i}=1} , they are thereby notified to be a designated receiver.

Protocol 3: Anonymous Multiparty Entanglement

Input: $n$-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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (m+1)} -partite GHZ state Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 $x_{i}$
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 $k$ parties

Goal: Verification or rejection of the shared state as the GHZFailed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle _{k}} state by V

Requirements: Private randomness sources; a classical broadcasting channel

1. Everyone but V draws a random bit Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (b_{i},m_{i})}
3. V resets her bit such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sum _{i}b_{i}=0(} mod Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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)}