Polynomial Code based Quantum Authentication

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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)
  • : authenticator (prover)
  • : quantum message to be sent
  • : number of qubits in the message
  • : stabilizer purity testing code, each stabilizer code is identified by index
  • : 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

Output: Receiver accepts or aborts the quantum state

  • 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[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