Polynomial Code based Quantum Authentication: Difference between revisions

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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.
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==
==Assumptions==
*The sender and the receiver share a classical key drawn from a probability distribution.
*The sender and the receiver share a private, classical random key drawn from a probability distribution
==Outline==
 
==Notations==
==Notations==
==Properties==
*<math>\mathcal{S}</math>: suppliant (sender)
==Pseudo Code==
*<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==
==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==
<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 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>

Latest revision as of 19:49, 16 January 2022

The paper Authentication of Quantum Messages by Barnum et al. provides a non-interactive scheme with classical keys for the sender to encrypt as well as 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: Two Party Protocol, Quantum Functionality, Specific Task, Building Block

Outline[edit]

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 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. 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[edit]

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

Notations[edit]

  • : suppliant (sender)
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} : authenticator (prover)
  • : quantum message to be sent
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} : number of qubits in the message Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho}
  • : stabilizer purity testing code, each stabilizer code is identified by index Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k}
  • : number of qubits used to encode the message with
  • : random binary -bit key
  • : random syndrome for a specific

Protocol Description[edit]

Input: owned by ; , , shared among and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}}

Output: Receiver accepts or aborts the quantum state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^\prime}

  • Encryption and encoding:
  1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{S}} q-encrypts the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} -qubit original message Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau} using the classical key Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and a quantum one-time pad. This encryption is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau = \sigma_x^{\vec{t}_1}\sigma_z^{\vec{t}_2}\rho\sigma_z^{\vec{1}_1}\sigma_x^{\vec{t}_1}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{t}_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{t}_2} are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} -bit vectors and given by the random binary key Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} .
  2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{S}} then encodes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau} according to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_k} with syndrome Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} , which results in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -qubit state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} . This means Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{S}} encodes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} qubits using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_k} , and then "applies" errors according to the random syndrome.
  3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{S}} sends Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} .
  • Decoding and decryption:
  1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} receives the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} qubits, whose state is denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^\prime} .
  2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} measures the syndrome Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^\prime} of the code Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_k} on his Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} qubits in state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^\prime} .
  3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} compares the syndromes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^\prime} and aborts the process if they are different.
  4. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} decodes his Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -qubit word according to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_k} obtaining Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau^\prime} .
  5. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}} q-decrypts Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau^\prime} using the random binary strings Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} obtaining Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^\prime} .

Further Information[edit]

  1. Ben-Or et al. (2006).
  2. Aharonov et al. (2008).

References[edit]

  1. Barnum et al. (2002).
Contributed by Isabel Nha Minh Le and Shraddha Singh
This page was created within the QOSF Mentorship Program Cohort 4
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