## 1 Introduction
Soon later, [PERSON] and [PERSON] [24] proposed an arbitrated quantum signature scheme based on three-qubit Green-Horne-Zeilinger (GHZ) states, which provides an elegant framework for designing quantum signature schemes with the participation of a trusted arbitrator. Notably, quantum proxy signature is a special class in arbitrated quantum signature with distinct original signatory and proxy signatory. In 2010, [PERSON] et al. [25] presented a proxy signature scheme by employing Einstein-Podolsky-Rosen (EPR) states as the quantum channel for teleportation. In 2011, [PERSON] et al. [26] proposed a quantum proxy signature scheme based on public verifiability, in which EPR states are combined with the unitary transformation to generate proxy signature. In 2014, [PERSON] et al. [27] raised a quantum weak blind signature scheme with a genuinely entangled six-qubit state. Subsequently, [PERSON] and [PERSON] [28] analyzed the cryptanalysis of [PERSON] et al. 's work [27] and pointed out that the verifier can forge the signature by modifying the received message without being caught. Next year, [PERSON] et al. [29] put forward a proxy weak blind signature using the controlled teleportation scheme with five-qubit entangled states as quantum channels. Based on the above-described research, it is known that entangled states take up a significant position in designing quantum signature schemes. Yet, the challenge is that the generation of the ideal entanglement resource is difficult in experiments. To this end, many scholars began to seek for other methods to evade this challenge. For example, in 2015, [PERSON] et al. [30] brought forward a quantum proxy signature scheme in line with single-particle states instead of entangled states. In the next year, [PERSON] et al. [31] suggested a strong blind quantum signature scheme with multi-proxy by executing appropriate unitary operators. In 2018, [PERSON] et al. [32] brought forward a batch quantum multi-proxy signature, in which quantum controlled-not (CNOT) gates are employed to encode the information to be signed. Recently, [PERSON] et al. [33] developed a quantum proxy blind signature based on superdense coding, where various unitary operators are used to blind two-bit classical information. Compared to quantum proxy signature schemes without entanglement, it can be seen that the entanglement-based quantum proxy signature schemes have more ability to resist risks or attacks due to the disturbance detection owing to the existence of entanglement. In addition, we mentioned that the most challenge is the difficult generation of entanglement resource with the state-of-the-art technology, fortunately, which can be efficiently addressed using the models of quantum walks. On the other hand, in the existing quantum proxy signature schemes, the involved encryption algorithm is quantum one-time pad (QOTP), which may lead to different aspects of disavowal and forgery attacks [34; 35]. To solve this issue, we improve the QOTP by introducing the CNOT operation. Therefore, motivated by quantum walk-based teleportation, we present a proxy signature scheme using quantum walks and QOTP-CNOT operation. The presented quantum proxy signature scheme makes the following contributions. * Before generating the proxy signature, the random binary sequence is circularly used to encrypt the original message. Then the QOTP-CNOT operation is used to generate the proxy signature state with the length of the secret keys being the same as that of the message to be encrypted, which reduces the length of the required keys by three times in terms of efficiency and improves the security of the presented scheme. The introduction of CNOT operation into the QOTP makes the encrypted qubit related to not only the qubit and the key of the current position but also other qubits and keys of other positions, which can resist against the proxy signatory's disavowal attack and the receiver's forgery attack on the proxy signature by modifying the qubits of particular positions in it.