Editing Polynomial Code based Quantum Authentication

The paper [https://arxiv.org/pdf/quant-ph/0205128.pdf Authentication of Quantum Messages by Barnum et al.] provides a non-interactive scheme with classical keys for the sender to encrypt as well as [[Authentication of Quantum Messages|authenticate quantum messages]]. It was the first protocol designed to achieve the task of authentication for quantum states, i.e. it gives the guarantee that the message sent by a party (suppliant) over a communication line is received by a party on the other end (authenticator) without having been tampered with or modified by the dishonest party (eavesdropper).

'''Tags:''' [[:Category:Two Party Protocols|Two Party Protocol]][[Category:Two Party Protocols]], [[:Category:Quantum Functionality|Quantum Functionality]][[Category:Quantum Functionality]], [[:Category:Specific Task|Specific Task]][[Category:Specific Task]], [[:Category:Building Blocks|Building Block]][[Category:Building Blocks]]

==Outline==
The polynomial code consists of three steps: preprocessing, encryption and encoding, and decoding and decryption. Within the preprocessing, sender and receiver agree on a [[Stabilizer Purity Testing Code | stabilizer purity testing code]] and three private, random binary keys. Within the encryption and encoding step, the sender uses one of these keys to encrypt the original message. Consequently, a second key is used to choose a specific quantum error correction code out of the [[Stabilizer Purity Testing Code | stabilizer purity testing code]]. The chosen quantum error correction code is then used, together with the last key, to encode the encrypted quantum message. Within the last step, the decoding and decryption step, the respective keys are used by the receiver to decide whether to abort or not, and if not, to decode and decrypt the received quantum message.

==Assumptions==
*The sender and the receiver share a private, classical random key drawn from a probability distribution

==Notations==
*<math>\mathcal{S}</math>: suppliant (sender)
*<math>\mathcal{A}</math>: authenticator (prover)
*<math>\rho</math>: quantum message to be sent
*<math>m</math>: number of qubits in the message <math>\rho</math>
*<math>\{Q_k\}</math>: [[Stabilizer Purity Testing Code | stabilizer purity testing code]], each stabilizer code is identified by index <math>k</math>
*<math>n</math>: number of qubits used to encode the message with <math>\{Q_k\}</math>
*<math>x</math>: random binary <math>2m</math>-bit key
*<math>y</math>: random syndrome for a specific <math>Q_k</math>

==Protocol Description==
'''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>
'''Output:''' Receiver accepts or aborts the quantum state <math>\rho^\prime</math>
*'''''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> 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>.
*'''''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> 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> compares the syndromes <math>y</math> and <math>y^\prime</math> and aborts the process if they are different.
#<math>\mathcal{A}</math> decodes his <math>n</math>-qubit word according to <math>Q_k</math> obtaining <math>\tau^\prime</math>.
#<math>\mathcal{A}</math> q-decrypts <math>\tau^\prime</math> using the random binary strings <math>x</math> obtaining <math>\rho^\prime</math>.

==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==
#[https://arxiv.org/pdf/quant-ph/0205128.pdf| Barnum et al. (2002).]

<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>