Polynomial Code based Quantum Authentication: Difference between revisions

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Polynomial Code based Quantum Authentication (edit)

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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).
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]]
'''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==
==Assumptions==
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*<math>n</math>: number of qubits used to encode the message with <math>\{Q_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>x</math>: random binary <math>2m</math>-bit key
*<math>s</math>: security parameter
*<math>y</math>: random syndrome for a specific <math>Q_k</math>
 
 
==Properties==
*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==
==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>.
Add Input and Output for each subroutine
**<math>k</math> is used to choose a random stabilizer code <math>Q_k</math>
*'''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>
**<math>x</math> is a <math>2m</math>-bit random key used for q-encryption
*'''Output:''' Receiver: accepts or aborts <math>\rho^\prime</math>
**<math>y</math> is a random syndrome
**'''''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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