Efficient quantum digital signatures over long distances with likely bit strings

Quantum digital signatures (QDSs) can provide information-theoretic security of messages against forgery and repudiation. Compared with previous QDS protocols that focus on signing one-bit messages, hash function–based QDS protocols can save quantum resources and are able to sign messages of arbitrary length. Using the idea of likely bit strings, we propose an efficient QDS protocol with hash functions over long distances. Our method of likely bit strings can be applied to any quantum key distribution–based QDS protocol to significantly improve the signature rate and dramatically increase the secure signature distance of QDS protocols. In order to save computing resources, we propose an improved method where Alice participates in the verification process of Bob and Charlie. This eliminates the computational complexity relating to the huge number of all likely strings. We demonstrate the advantages of our method and our improved method with the example of sending-or-not-sending QDS. Under typical parameters, both our method and our improved method can improve the signature rate by more than $100$ times and increase the signature distance by about $150\mathrm{km}$ compared with hash function–based QDS protocols without likely bit strings.