Polynomial Code based Quantum Authentication

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

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

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

Notations

• ${\mathcal {S}}$: suppliant (sender)
• ${\mathcal {A}}$: authenticator (prover)
• $\rho$: quantum message to be sent
• $m$: number of qubits in the message $\rho$
• $\{Q_{k}\}$: stabilizer purity testing code, each stabilizer code is identified by index $k$
• $n$: 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\}}
• $x$: random binary $2m$-bit key
• $y$: random syndrome for a specific $Q_{k}$

Protocol Description

Input: $\rho$ owned by ${\mathcal {S}}$; 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} , $x$, $y$ shared among ${\mathcal {S}}$ and ${\mathcal {A}}$

Output: Receiver accepts or aborts the quantum state $\rho ^{\prime }$

• Encryption and encoding:

1. ${\mathcal {S}}$ q-encrypts the $m$-qubit original message $\rho$ 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 ${\vec {t}}_{1}$ and ${\vec {t}}_{2}$ are $m$-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. ${\mathcal {S}}$ then encodes $\tau$ according to $Q_{k}$ with syndrome $y$, 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 $\sigma$. 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 $\rho$ in $n$ qubits using $Q_{k}$, and then "applies" errors according to the random syndrome.
3. ${\mathcal {S}}$ 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 ${\mathcal {A}}$.

• Decoding and decryption:

1. ${\mathcal {A}}$ receives the $n$ qubits, whose state is denoted by $\sigma ^{\prime }$.
2. ${\mathcal {A}}$ measures the syndrome $y^{\prime }$ of the code $Q_{k}$ on his $n$ qubits in state $\sigma ^{\prime }$.
3. ${\mathcal {A}}$ compares the syndromes $y$ and $y^{\prime }$ and aborts the process if they are different.
4. ${\mathcal {A}}$ decodes his $n$-qubit word according to $Q_{k}$ obtaining $\tau ^{\prime }$.
5. ${\mathcal {A}}$ q-decrypts $\tau ^{\prime }$ using the random binary strings $x$ obtaining $\rho ^{\prime }$.

Further Information

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

References

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