Polynomial Code based Quantum Authentication: Difference between revisions

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==Protocol Description==
==Protocol Description==
Add Input and Output for each subroutine
'''Input:''' <math>\rho</math> owned by <math>\mathcal{S}</math>; <math>k</math>, <math>x</math>, <math>y</math> shared among <math>\mathcal{S}</math> and <math>\mathcal{A}</math></br></br>
*'''Input:''' <math>\rho</math> owned by <math>\mathcal{S}</math>; <math>k</math>, <math>x</math>, <math>y</math> shared among <math>\mathcal{S}</math> and <math>\mathcal{A}</math>
'''Output:''' Receiver accepts or aborts the quantum state <math>\rho^\prime</math>
*'''Output:''' Receiver: accepts or aborts <math>\rho^\prime</math>
*'''''Encryption and encoding:'''''
**'''''Encryption and encoding:'''''
#<math>\mathcal{S}</math> q-encrypts the <math>m</math>-qubit original message <math>\rho</math> as <math>\tau</math> using the classical key <math>x</math> and a [[Quantum One-Time Pad | quantum one-time pad]]. This encryption is given by <math>\tau = \sigma_x^{\vec{t}_1}\sigma_z^{\vec{t}_2}\rho\sigma_z^{\vec{1}_1}\sigma_x^{\vec{t}_1}</math>, where <math>\vec{t}_1</math> and <math>\vec{t}_2</math> are <math>m</math>-bit vectors and given by the random binary key <math>x</math>.
#<math>\mathcal{S}</math> q-encrypts the <math>m</math>-qubit original message <math>\rho</math> as <math>\tau</math> using the classical key <math>x</math> and a [[Quantum One-Time Pad | quantum one-time pad]]. This encryption is given by <math>\tau = \sigma_x^{\vec{t}_1}\sigma_z^{\vec{t}_2}\rho\sigma_z^{\vec{1}_1}\sigma_x^{\vec{t}_1}</math>, where <math>\vec{t}_1</math> and <math>\vec{t}_2</math> are <math>m</math>-bit vectors and given by the random binary key <math>x</math>.
#<math>\mathcal{S}</math> then encodes <math>\tau</math> according to <math>Q_k</math> with syndrome <math>y</math>, which results in the <math>n</math>-qubit state <math>\sigma</math>. This means <math>\mathcal{S}</math> encodes <math>\rho</math> in <math>n</math> qubits using <math>Q_k</math>, and then "applies" errors according to the random syndrome.
#<math>\mathcal{S}</math> then encodes <math>\tau</math> according to <math>Q_k</math> with syndrome <math>y</math>, which results in the <math>n</math>-qubit state <math>\sigma</math>. This means <math>\mathcal{S}</math> encodes <math>\rho</math> in <math>n</math> qubits using <math>Q_k</math>, and then "applies" errors according to the random syndrome.
#<math>\mathcal{S}</math> sends <math>\sigma</math> to <math>\mathcal{A}</math>.
#<math>\mathcal{S}</math> sends <math>\sigma</math> to <math>\mathcal{A}</math>.
**'''''Decoding and decryption:'''''
*'''''Decoding and decryption:'''''
#<math>\mathcal{A}</math> receives the <math>n</math> qubits, whose state is denoted by <math>\sigma^\prime</math>.
#<math>\mathcal{A}</math> receives the <math>n</math> qubits, whose state is denoted by <math>\sigma^\prime</math>.
#<math>\mathcal{A}</math> measures the syndrome <math>y^\prime</math> of the code <math>Q_k</math> on his <math>n</math> qubits in state <math>\sigma^\prime</math>.
#<math>\mathcal{A}</math> measures the syndrome <math>y^\prime</math> of the code <math>Q_k</math> on his <math>n</math> qubits in state <math>\sigma^\prime</math>.
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==Further Information==
==Further Information==
#[https://ieeexplore.ieee.org/abstract/document/4031361?casa_token=j0BWLVeqOZkAAAAA:T19kamFiwuoLaEbL_bESvUendLVhWzsXWZpegOxPADA_PjSobjg4Wyo8ZmV92qvfVF3Pc7_v| Ben-Or et al. (2006).]
#[https://arxiv.org/pdf/0810.5375.pdf%7C| Aharonov et al. (2008).]


==References==
==References==
#[https://arxiv.org/pdf/quant-ph/0205128.pdf| Barnum et al. (2002).]
#[https://arxiv.org/pdf/quant-ph/0205128.pdf| Barnum et al. (2002).]
<div style='text-align: right;'>''contributed by Shraddha Singh and Isabel Nha Minh Le''</div>
 
<div style='text-align: right;'>''Contributed by Isabel Nha Minh Le and Shraddha Singh''</div>
<div style='text-align: right;'>''This page was created within the [https://www.qosf.org/qc_mentorship/| QOSF Mentorship Program Cohort 4]''</div>
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