Security of quantum digital signatures for classical messages

Quantum digital signatures can be used to authenticate classical messages in an information-theoretically secure way. Previously, a novel quantum digital signature for classical messages has been proposed and gave an experimental demonstration of distributing quantum digital signatures from one sender to two receivers. Some improvement versions were subsequently presented, which made it more feasible with present technology. These proposals for quantum digital signatures are basic building blocks which only deal with the problem of sending single bit messages while no-forging and non-repudiation are guaranteed. For a multi-bit message, it is only mentioned that the basic building blocks must be iterated, but the iteration of the basic building block still does not suffice to define the entire protocol. In this paper, we show that it is necessary to define the entire protocol because some attacks will arise if these building blocks are used in a naive way of iteration. Therefore, we give a way of defining an entire protocol to deal with the problem of sending multi-bit messages based on the basic building blocks and analyse its security.