# Quantum Digital Signature

## Contents

## Functionality

Digital Signatures (QDS) allow the exchange of classical messages from sender to multiple recipients, with a guarantee that the signature has come from a genuine sender. Additionally, it comes with the properties of transferability, non-repudiation and unforgeability. In contrast, classical digital signatures rely on authentication (taken as an assumption for some QDS protocols) i.e. the message has come from the claimed party; integrity i.e. the message has not been altered (if authentication is confirmed, this property is unforgeability) and non-repudiation (same as QDS). These properties distinguish quantum digital signatures from quantum authentication. Quantum messages can be authenticated but not signed (1), (2). Note that QDS schemes sign classical messages and not quantum messages.

**Tags:** Multi Party (three), Quantum Enhanced Classical Functionality, Specific Task

## Use Case

- Classical task
- Classical analogue: RSA, Post-Quantum Secure analogue: XMSS
- QDS implementation specifications (best achieved) per half bit message (0 or 1):
- best estimated time: 3.5 secs
- key length: 2Mbits
- maximum transmission distance: 200 kms
- scalability: linear in time, not linear in key length

## Protocols

- Gottesman and Chuang Quantum Digital Signature: Quantum Memory Network Stage
- Prepare and Measure Quantum Digital Signature: Prepare and Measure Network Stage
- Measurement Device Independent Quantum Digital Signature (MDI-QDS): Prepare and Measure Network Stage
- Arbitrated Quantum Digital Signature: Quantum Memory Network Stage

Below are few signature protocols with slightly modified functionality but same properties and requirements.

- Blind Delegation of Quantum Digital Signature
- Designated Verifiable Quantum Signature
- Limited Delegation of Quantum Signature
- Quantum Proxy Signature

## Properties

All QDS protocols are divided into two phases, distribution and messaging. Distribution phase enables sender to generate private keys (kept secret with sender) and public keys (information distributed to recipients) while messaging phase enables exchange of messages using the above keys. For simplicity, most protocols use the case of three parties, one sender (Seller) and two recipients (Buyer and Verifier) exchanging one-bit classical messages signed by Quantum Digital Signatures (QDS).

- A QDS scheme is correct if a message signed by a genuine sender is accepted by a recipient with unit probability.
- A QDS scheme is secure if no one but the sender can sign a message such that it is accepted by a recipient with non-negligible probability.
**Transferability**means that at any point a recipient (buyer) can prove it to another recipient (verifier) that the concerned message has been signed by the claimed sender (Seller).**Unforgeability**ensures that a dishonest recipient (buyer) can neither alter a DS nor sign a message with a fake DS (DS that has not come from a genuine sender) and forward it to other recipients (verifier) successfully.**Non-Repudiation**implies that at any point a dishonest sender (seller) cannot deny having signed the message sent to a genuine recipient (Buyer).

## Further Information

Unlike classical digital signature schemes which generalize a two party model, QDS protocols always study a three party model as transferability is not inherent and has to be proved in the quantum case. Given this situation, usually, the third party acts as the judge (a verififer) who would gain nothing out of cheating, and hence, cheating strategy is only studied for seller (repudiation) and buyer (forgery). Quantum digital signatures provide unconditional security, not relying on any computational assumption which is its basic advantage over the classical schemes. However, over time classical unconditionally secure digital signature schemes have been realized. These classical protocols take extra some assumptions like trusted omnipotent (one who distributes everyone signatures) or authenticated message broadcast. QDS does not require any such assumption. Yet, the low key rate could render QDS impractical over classical digital signature schemes. At the same time, there exist post quantum secure Digital signature schemes based on hash-key cryptography which cannot be broken by quantum computers. Still, if someone requires a lifetime security without the above mentioned assumptions, QDS is the answer. Areas to improve QDS could be addressing the key rate and scalability of key length with length of message. Following are a few articles useful for those interested in a more detailed overview of QDS.

## Knowledge Graph

- Review Papers

- AA (2015) Discusses various classical and quantum digital signature schemes
- Wallden P. (2018) (In preparation): Discusses the development of Quantum Digital Signatures from the first protocol by Gottesman and Chuang, elaborating advancements in further protocols to turn it into a practical QDS scheme.

## References

- Barum et al (2002) First intuition towards impossibility of signing quantum states
- Li et al (2013) Discusses the possibility of arbitrated Quantum Signature schemes for quantum signatures
- Alagic et al (2018) Impossibility result of signing quantum states established and the article rules out all weak schemes for quantum messages

*contributed by Shraddha Singh*