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The protocol for quantum anonymous transmission consists of the following steps: | The protocol for quantum anonymous transmission consists of the following steps: | ||
# | # ''{Receiver notification}: The Sender <math>\mathcal{S}</math> notifies the Receiver <math>R</math> by running <span style="font-variant:small-caps">Notification</span>. | ||
# | # ''{State distribution}: A source, who may be untrusted, distributes a state claiming to be the GHZ state. | ||
# | # ''{Verification or anonymous transmission}: <math>\mathcal{S}</math> anonymously chooses whether to verify the state or use it for anonymous transmission, using <span style="font-variant:small-caps">RandomBit</span>. | ||
If verification is chosen, a player is chosen to run <span style="font-variant:small-caps">Verification</span>, using <math>\log_2 n</math> repetitions of <span style="font-variant:small-caps">RandomBit</span>. | If verification is chosen, a player is chosen to run <span style="font-variant:small-caps">Verification</span>, using <math>\log_2 n</math> repetitions of <span style="font-variant:small-caps">RandomBit</span>. | ||
If the test passes, the protocol goes back to the | If the test passes, the protocol goes back to the ''{State distribution} stage and runs again. If the test fails, the players abort. | ||
If anonymous transmission is chosen, the players run <span style="font-variant:small-caps">Anonymous Entanglement</span>, establishing an anonymous entanglement link between <math>\mathcal{S}</math> and <math>\mathcal{R}</math>. | If anonymous transmission is chosen, the players run <span style="font-variant:small-caps">Anonymous Entanglement</span>, establishing an anonymous entanglement link between <math>\mathcal{S}</math> and <math>\mathcal{R}</math>. | ||
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* <math>n</math>: number of network nodes taking part in the anonymous transmission. | * <math>n</math>: number of network nodes taking part in the anonymous transmission. | ||
* <math>t</math>: number of adversarial network nodes taking part in the anonymous transmission. | * <math>t</math>: number of adversarial network nodes taking part in the anonymous transmission. | ||
* <math>\ | * <math>|\psi\rangle</math>: quantum message which the Sender wants to send anonymously. | ||
* <math>\ | * <math>|GHZ\rangle = \frac{1}{\sqrt{2</span> (|0^n\rangle + |1^n\rangle)</math>: GHZ state. | ||
* <math>\ | * <math>|\Psi\rangle</math>: state provided by the untrusted source for anonymous transmission (in the ideal case, this is the GHZ state). | ||
* <math>\mathcal{S}</math>: the Sender of the quantum message. | * <math>\mathcal{S}</math>: the Sender of the quantum message. | ||
* <math>\mathcal{R}</math>: the Receiver of the quantum message. | * <math>\mathcal{R}</math>: the Receiver of the quantum message. | ||
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====<span style="font-variant:small-caps"><math>\epsilon<math>-anonymous transmission of a quantum message</span>==== | ====<span style="font-variant:small-caps"><math>\epsilon<math>-anonymous transmission of a quantum message</span>==== | ||
\noindent | \noindent ''{Input}: Security parameter <math>q<math>. \\ | ||
''{Goal}: <math>\mathcal{S}<math> sends message qubit <math>\ket{\psi}<math> to <math>\mathcal{R}<math> with <math>\epsilon<math>-anonymity. | |||
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====<span style="font-variant:small-caps">Parity</span>==== | ====<span style="font-variant:small-caps">Parity</span>==== | ||
\noindent | \noindent ''{Input}: <math>\{ x_i \}_{i=1}^n<math>. \\ | ||
''{Goal}: Each player gets <math>y_i = \bigoplus_{i=1}^n x_i<math>. | |||
# Every player <math>i<math> chooses random bits <math>\{r_i^j \}_{j=1}^n<math> such that <math>\bigoplus_{j=1}^n r_i^j = x_i<math>. | # Every player <math>i<math> chooses random bits <math>\{r_i^j \}_{j=1}^n<math> such that <math>\bigoplus_{j=1}^n r_i^j = x_i<math>. | ||
# Every player <math>i<math> sends their <math>j<math>th bit <math>r_i^j<math> to player <math>j<math> (<math>j<math> can equal <math>i<math>). | # Every player <math>i<math> sends their <math>j<math>th bit <math>r_i^j<math> to player <math>j<math> (<math>j<math> can equal <math>i<math>). | ||
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====<span style="font-variant:small-caps">LogicalOR</span>==== | ====<span style="font-variant:small-caps">LogicalOR</span>==== | ||
\noindent | \noindent ''{Input}: <math>\{ x_i \}_{i=1}^n<math>, security parameter <math>q<math>. \\ | ||
''{Goal}: Each player gets <math>y_i = \bigvee_{i=1}^n x_i<math>. | |||
# The players agree on <math>n<math> orderings, with each ordering having a different last participant. | # The players agree on <math>n<math> orderings, with each ordering having a different last participant. | ||
# For each ordering: | # For each ordering: | ||
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====<span style="font-variant:small-caps">Notification</span>==== | ====<span style="font-variant:small-caps">Notification</span>==== | ||
\noindent | \noindent ''{Input}: Security parameter <math>q<math>, <math>\mathcal{S}<math>'s choice of <math>\mathcal{R}<math> is player <math>r<math>. \\ | ||
''{Goal}: <math>\mathcal{S}<math> notifies <math>\mathcal{R}<math>. \\ | |||
For each player <math>i<math>: | For each player <math>i<math>: | ||
# For each player <math>i<math>: | # For each player <math>i<math>: | ||
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====<span style="font-variant:small-caps">RandomBit</span>==== | ====<span style="font-variant:small-caps">RandomBit</span>==== | ||
\noindent | \noindent ''{Input:} All: parameter <math>q<math>. <math>\mathcal{S}<math>: distribution <math>D<math>. \\ | ||
''{Goal:} <math>\mathcal{S}<math> chooses a bit according to <math>D<math>. | |||
# The players pick bits <math>\{ x_i \}_{i=1}^n<math> as follows: <math>\mathcal{S}<math> picks bit <math>x_i<math> to be 0 or 1 according to <math>D<math>; all other players pick <math>x_i = 0<math>. | # The players pick bits <math>\{ x_i \}_{i=1}^n<math> as follows: <math>\mathcal{S}<math> picks bit <math>x_i<math> to be 0 or 1 according to <math>D<math>; all other players pick <math>x_i = 0<math>. | ||
# Run <span style="font-variant:small-caps">LogicalOR</span> with input <math>\{ x_i \}_{i=1}^n<math> and security parameter <math>q<math> and output its outcome. | # Run <span style="font-variant:small-caps">LogicalOR</span> with input <math>\{ x_i \}_{i=1}^n<math> and security parameter <math>q<math> and output its outcome. | ||
====<span style="font-variant:small-caps">Verification</span>==== | ====<span style="font-variant:small-caps">Verification</span>==== | ||
\noindent | \noindent ''{Input}: <math>n<math> players share state <math>\ket{\Psi}<math>. \\ | ||
''{Goal}: GHZ verification of <math>\ket{\Psi}<math> for <math>n-t<math> honest players. | |||
# The Verifier generates random angles <math>\theta_j \in [0,\pi)<math> for all players including themselves (<math>j\in[n]<math>), such that <math>\sum_j \theta_j<math> is a multiple of <math>\pi<math>. The angles are then sent out to all the players in the network. | # The Verifier generates random angles <math>\theta_j \in [0,\pi)<math> for all players including themselves (<math>j\in[n]<math>), such that <math>\sum_j \theta_j<math> is a multiple of <math>\pi<math>. The angles are then sent out to all the players in the network. | ||
# Player <math>j<math> measures in the basis <math>\{\ket{+_{\theta_j</span>,\ket{-_{\theta_j</span>\}=\{\frac{1}{\sqrt{2</span>(\ket{0}+e^{i\theta_j}\ket{1}),\frac{1}{\sqrt{2</span>(\ket{0}-e^{i\theta_j}\ket{1})\}<math>, and sends the outcome <math>Y_j=\{0,1\}<math> to the Verifier. | # Player <math>j<math> measures in the basis <math>\{\ket{+_{\theta_j</span>,\ket{-_{\theta_j</span>\}=\{\frac{1}{\sqrt{2</span>(\ket{0}+e^{i\theta_j}\ket{1}),\frac{1}{\sqrt{2</span>(\ket{0}-e^{i\theta_j}\ket{1})\}<math>, and sends the outcome <math>Y_j=\{0,1\}<math> to the Verifier. | ||
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====<span style="font-variant:small-caps">Anonymous Transmission</span>==== | ====<span style="font-variant:small-caps">Anonymous Transmission</span>==== | ||
\noindent | \noindent ''{Input}: <math>n<math> players share a GHZ state. \\ | ||
''{Goal}: Anonymous transmission of quantum message <math>\ket{\psi}<math> from <math>\mathcal{S}<math> to <math>\mathcal{R}<math>. | |||
# <math>\mathcal{S}<math> and <math>\mathcal{R}<math> do not do anything to their part of the state. | # <math>\mathcal{S}<math> and <math>\mathcal{R}<math> do not do anything to their part of the state. | ||
# Every player <math>j \in [n] \backslash \{ \mathcal{S}, \mathcal{R} \}<math>: | # Every player <math>j \in [n] \backslash \{ \mathcal{S}, \mathcal{R} \}<math>: | ||
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\begin{thebibliography}{9} | \begin{thebibliography}{9} | ||
\bibitem{Unnikrishnan} | \bibitem{Unnikrishnan} | ||
A. Unnikrishnan, I. J. MacFarlane, R. Yi, E. Diamanti, D. Markham, and I. Kerenidis. | A. Unnikrishnan, I. J. MacFarlane, R. Yi, E. Diamanti, D. Markham, and I. Kerenidis. ''{Anonymity for practical quantum networks.} To be published in Physical Review Letters. arXiv:1811.04729 (2018). | ||
\bibitem{Wehner} | \bibitem{Wehner} | ||
M. Christandl and S. Wehner. | M. Christandl and S. Wehner. ''{Quantum anonymous transmissions.} Proceedings of ASIACRYPT (2005). | ||
\bibitem{Broadbent} | \bibitem{Broadbent} | ||
A. Broadbent and A. Tapp. | A. Broadbent and A. Tapp. ''{Information-theoretic security without an honest majority.} Proceedings of ASIACRYPT (2007). | ||
\bibitem{Pappa} | \bibitem{Pappa} | ||
A. Pappa, A. Chailloux, S. Wehner, E. Diamanti, and I. Kerenidis. | A. Pappa, A. Chailloux, S. Wehner, E. Diamanti, and I. Kerenidis. ''{Multipartite entanglement verification resistant against dishonest parties.} Physical Review Letters, 108 (2012). | ||
\bibitem{McCutcheon} | \bibitem{McCutcheon} | ||
W. McCutcheon, A. Pappa, B. A. Bell, A. McMillan, A. Chailloux, T. Lawson, M. Mafu, D. Markham, E. Diamanti, I. Kerenidis, J. G. Rarity, and M. S. Tame. | W. McCutcheon, A. Pappa, B. A. Bell, A. McMillan, A. Chailloux, T. Lawson, M. Mafu, D. Markham, E. Diamanti, I. Kerenidis, J. G. Rarity, and M. S. Tame. ''{Experimental verification of multipartite entanglement in quantum networks.} Nature Communications (2016). | ||
\bibitem{Brassard} | \bibitem{Brassard} | ||
G. Brassard, A. Broadbent, J. Fitzsimons, S. Gambs, and A. Tapp. | G. Brassard, A. Broadbent, J. Fitzsimons, S. Gambs, and A. Tapp. ''{Anonymous quantum communication.} Proceedings of ASIACRYPT (2007). | ||
\bibitem{Lipinska} | \bibitem{Lipinska} | ||
V. Lipinska, G. Murta, and S. Wehner. | V. Lipinska, G. Murta, and S. Wehner. ''{Anonymous transmission in a noisy quantum network using the W state.} Physical Review A, 98 (2018). | ||
\end{thebibliography} | \end{thebibliography} | ||