  Quantum digital signatures (QDS), generating correlated bit strings among
three remote parties for signatures through quantum law, can guarantee
non-repudiation, authenticity, and integrity of messages. Recently, one-time
universal hashing QDS framework, exploiting the quantum asymmetric encryption
and universal hash functions, has been proposed to significantly improve the
signature rate and ensure unconditional security by directly signing the hash
value of long messages. However, similar to quantum key distribution, this
framework utilizes keys with perfect secrecy by performing privacy
amplification that introduces cumbersome matrix operations, thereby consuming
large computational resources, causing delays and increasing failure
probability. Here, we prove that, different from private communication,
imperfect quantum keys with limited information leakage can be used for digital
signatures and authentication without compromising the security while having
eight orders of magnitude improvement on signature rate for signing a megabit
message compared with conventional single-bit schemes. This study significantly
reduces the delay for data postprocessing and is compatible with any quantum
key generation protocols. In our simulation, taking two-photon twin-field key
generation protocol as an example, QDS can be practically implemented over a
fiber distance of 650 km between the signer and receiver. For the first time,
this study offers a cryptographic application of quantum keys with imperfect
secrecy and paves a way for the practical and agile implementation of digital
signatures in a future quantum network.
