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 $L$ 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 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 $m$ 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 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 (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} 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i=1,...,n} :

1. All agents $j\in \{1,...,n\}$ do the following:
   • When agent Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle k}
   • When agent Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 $\{z_{k}^{i}\}_{k=1}^{n}$ to compute $z^{i}=\bigoplus _{k=1}^{n}z_{k}^{i}$. If $z^{i}=1$, 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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 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)}