# Quantum Dual Signature with Coherent States Based on Chained Phase-Controlled Operations
## 1 Introduction  
The digital signature, which is an important way to realize identity and message authentication, is one of the significant cryptographic building blocks and has been extensively employed in secure electronic commerce. A digital signature is considered to be additional information to the message [1] and it can prevent cheating legitimate users, including forging the sender's signature by the receiver, and repudiating the signature by the sender [2]. There are lots of classical digital signature schemes [3; 4; 5] including post-quantum signature [6], whose security depends on complex algorithms or unresolved mathematical problems, such as the discrete logarithm problem [7] and the prime factorization problem [8]. However, with the advent of quantum computing and quantum algorithms [9; 10], these classical signature schemes are likely to be compromised [11; 12]. For example, the security of the signature with prime factorization may be quickly broken [13; 14] by [PERSON]'s algorithm [15] in the near future. Hence research imminently needs to blaze a trail in the security improvement of signature schemes with another way. The product of combination of quantumcryptography and classical signature schemes, the so-called quantum signature, can effectively contend with the attacks brought by quantum computation. With physical properties, such as Heisenberg uncertainty [16] and the quantum no-cloning theorem [17], a quantum signature can have unconditional security in theory, so that more and more scholars invest in research on this topic [18; 19; 20]. For adjusting to diverse application scenarios, arbitrated quantum signature (AQS) [11; 12; 21], quantum group signature (QGS) [22; 23], quantum proxy signature (QPS) [24; 25], quantum blind signature (QBS) [26; 27] or their combinations [28; 29; 30], have been proposed, which are apparently corresponding to various classical digital signature schemes. In 2001, [PERSON] et al. [31] first proposed a quantum digital signature based on fundamental principles of quantum physics, i.e., a quantum analogue of one-way functions, which required \(\mathbf{O}(m)\) qubits to encrypt \(m\)-bit message. On the basis of Greenberger-Horne-Zeilinger (GHZ) triplet states, [PERSON] et al. [32] and [PERSON] [33] put forward a quantum signature scheme, whose realization depends upon a trusty arbitrator, namely AQS schemes [21; 34] which can be adapted to both known and unknown quantum states and still provide certain degree of security by employing quantum key distribution (QKD) protocols and quantum one-time pads (QOTPs) [35; 36]. [PERSON]. [2; 34] pointed out that forgery attacks may occur in AQS schemes which use QOTP. On the one hand, the QOTP encrypts data qubit by qubit and the key bits are independent of each other. On the other hand, Pauli operations commute or anticommute with each other, i.e., the Pauli operators can exchange each other. Therefore, the recipient Bob may forge the sender's signature under known message by using the encryption features of QOTP. [PERSON] et al. [37] designed a quantum group blind signature scheme to provide anonymity of voters in an e-voting system and [PERSON]. [22] proposed a QGS scheme without the help of arbitrator, which can be used in e-government and e-business. [PERSON] et al. [38] introduced a quantum digital signature scheme applying coherent states which is experimentally feasible to access. For pushing the limit of single stream in quantum networks, [PERSON] et al. [39] first proposed a quantum homomorphic signature protocol by using entanglement swapping, which is used to authenticate data packets of multiple stream for quantum networks. Then other quantum homomorphic [40; 41] signatures have been proposed successively. It is common knowledge that account information and trading activities of a consumer are required to not be leaked to others.