Polynomial Code based Quantum Authentication: Difference between revisions

Polynomial Code based Quantum Authentication (edit)

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-The [https://arxiv.org/pdf/quant-ph/0205128.pdf example protocol] provides a non-interactive scheme 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) as it is and, has not been tampered with or modified by the dishonest party (eavesdropper).
+The paper [https://arxiv.org/pdf/quant-ph/0205128.pdf Authentication of Quantum Messages by Barnum et al.] provides a non-interactive scheme 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) as it is and, has not 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]]
 ==Assumptions==
-*The sender and the receiver share a private (known to only the two of them), classical random key drawn from a probability distribution.
+*The sender and the receiver share a private, classical random key drawn from a probability distribution
-==Outline==
 ==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>s</math>: security parameter
-*<math>m</math>: number of qubits in the message.
 ==Properties==
-*For an <math>m</math> qubit message, the protocol requires <math>m+s</math> qubits encoded state, and a private key of <math>2m+O(s)</math>.
+*For a <math>m</math>-qubit message, the protocol requires <math>m+s</math> qubits to encode the quantum message.
+*The protocol requires a private key of size <math>2m+O(s)</math>.
 ==Protocol Description==
+*'''''Preprocessing:''''' <math>\mathcal{S}</math> and <math>\mathcal{A}</math> agree on some [[Stabilizer Purity Testing Code | stabilizer purity testing code]] <math>\{Q_k\}</math> and some private and random binary strings <math>k, x, y</math>.
+**<math>k</math> is used to choose a random stabilizer code <math>Q_k</math>
+**<math>x</math> is a <math>2m</math>-bit random key used for q-encryption
+**<math>y</math> is a random syndrome
+*'''''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==
 ==References==
-<div style='text-align: right;'>''contributed by Shraddha Singh''</div>
+#[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>