# Authentication of Quantum Messages

## Functionality

If a person sends some information over an insecure channel (a dishonest/malicious party has access to the channel), what is the guarantee that the receiver on the other end will receive the same information as sent and not something which is modified or replaced by the dishonest party? Authentication of quantum channels/quantum states/quantum messages provides this guarantee to the users of a quantum communication line/ channel. The sender is called the suppliant (prover) and the receiver is called the authenticator.
Note that, it is different from the functionality of digital signatures, a multi-party (more than two) protocol, which comes with additional properties (non-repudiation, unforgeability and transferability). Authenticating quantum states is possible but signing quantum states is impossible, as concluded in (1). Also, unlike classical message authentication, quantum messages authentication requires encryption. However, classical messages can be publicly readable (not encrypted) and yet authenticated.

## Use Case

• No classical analogue

## Protocols

• Non-interactive Protocols
• Interactive Protocols

## Properties

• Any scheme which authenticates quantum messages must also encrypt them. (1)
• Definition 1: A quantum authentication scheme (QAS) is a pair of polynomial time quantum algorithms ${\displaystyle {\mathcal {S}}}$ (suppliant) and ${\displaystyle {\mathcal {A}}}$ (authenticator) together with a set of classical keys ${\displaystyle K}$ such that:
1. ${\displaystyle {\mathcal {S}}}$ takes as input an ${\displaystyle m}$-qubit message system ${\displaystyle M}$ and a key ${\displaystyle k\epsilon K}$ and outputs a transmitted system ${\displaystyle T}$ of ${\displaystyle m+t}$ qubits.
2. ${\displaystyle {\mathcal {A}}}$ takes as input the (possibly altered) transmitted system ${\displaystyle T}$' and a classical key ${\displaystyle k\epsilon K}$ and outputs two systems: a ${\displaystyle m}$-qubit message state ${\displaystyle M}$, and a single qubit ${\displaystyle V}$ which indicates acceptance or rejection. The classical basis states of ${\displaystyle V}$ are called ${\displaystyle |ACC\rangle ,|REJ\rangle }$ by convention. For any fixed key ${\displaystyle k}$, we denote the corresponding super-operators by ${\displaystyle S_{k}}$ and ${\displaystyle A_{k}}$.
• For non-interactive protocols, a QAS is secure with error ${\displaystyle \epsilon }$ for a state ${\displaystyle |\psi \rangle }$ if it satisfies:
1. Completeness: For all keys ${\displaystyle k\epsilon K:A_{k}(S_{k}(|\psi \rangle \langle \psi |)=|\psi \rangle \langle \psi |\otimes |ACC\rangle \langle ACC|}$
2. Soundness: : For all super-operators ${\displaystyle {\mathcal {O}}}$, let ${\displaystyle \rho _{auth}}$ be the state output be ${\displaystyle {\mathcal {A}}}$ when the adversary’s intervention

is characterized by ${\displaystyle {\mathcal {O}}}$, that is:

## Further Information

1. Barnum et al (2002) First protocol on authentication of quantum messages. It is also used later for verification of quantum computation in Interactive Proofs for Quantum Computation. Protocol file for this article is given as the Polynomial Code based Quantum Authentication