Polynomial Code based Quantum Authentication

Revision as of 18:24, 8 December 2021 by 137.226.108.44 (talk)

The paper Authentication of Quantum Messages by Barnum et al. provides a non-interactive scheme 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) as it is and, has not been tampered with or modified by the dishonest party (eavesdropper).

Tags: Two Party Protocol, Quantum Functionality, Specific Task, Building Block

Assumptions

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

Notations

  • 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}} : suppliant (sender)
  •  : authenticator (prover)
  •  : quantum message to be sent
  •  : 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}
  • 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\}} : stabilizer purity testing code, each stabilizer code is identified by index  
  •  : number of qubits used to encode the message with 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\}}
  •  : random binary  -bit key
  •  : security parameter


Properties

  • For a  -qubit message, the protocol requires   qubits to encode the quantum message.
  • The protocol requires a private key of size  .

Protocol Description

  • Preprocessing:   and   agree on some stabilizer purity testing code   and some private and random binary strings  .
    •   is used to choose a random stabilizer code  
    •   is a  -bit random key used for q-encryption
    •   is a random syndrome
  • Encryption and encoding:
  1.   q-encrypts the  -qubit original message   as   using the classical key   and a quantum one-time pad. This encryption is given by  , where   and   are  -bit vectors and given by the random binary key  .
  2.   then encodes   according to   with syndrome  , which results in the  -qubit state  . This means   encodes   in   qubits using  , and then "applies" errors according to the random syndrome.
  3.   sends   to  .
  • Decoding and decryption:
  1.   receives the   qubits, whose state is denoted by  .
  2.   measures the syndrome   of the code   on his   qubits in state  .
  3.   compares the syndromes   and   and aborts the process if they are different.
  4.   decodes his  -qubit word according to   obtaining  .
  5.   q-decrypts   using the random binary strings   obtaining  .

Further Information

References

  1. Barnum et al. (2002).
contributed by Shraddha Singh and Isabel Nha Minh Le