Efficient quantum digital signatures over long distances with likely bit strings

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