Editing Anonymous Conference Key Agreement using GHZ states (section)

==Protocol Description==
<!-- Mathematical step-wise protocol algorithm helpful to write a subroutine. -->
===Protocol 1: Anonymous Verifiable Conference Key Agreement===
''Input'': Parameters <math>L</math> and <math>D</math>

''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 <math>m</math> receivers

# The sender notifies the <math>m</math> receivers by running the ''Notification'' protocol
# The source generates and shares <math>L</math> GHZ states
# The parties run the ''Anonymous Multipartite Entanglement'' protocol on the GHZ states
# For each <math>(m+1)</math>-partite GHZ state, the parties do the following:
#* They ask a source of randomness to broadcast a bit <math>b</math> such that Pr<math>[b=1] = \frac{1}{D}</math>
#*  '''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 <math>m</math> receivers. The remaining parties announce random values.
#* '''KeyGen round: '''If b = 1, the sender and receivers measure in the Z-basis.
# 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 <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===
''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===

''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>