Today, E-commerce is favored by the majority of individuals for its convenience and speediness with technological advancements. Hence, it is particularly important to choose an appropriate model of payment. In 1983, Chaum proposed the first concept of E-cash.^[1] Subsequently, E-cash system has attracted much attention and researchers proposed a number of E-cash payment schemes.^[2–6] However, the security of the most existing E-cash payment schemes depends on some computational problems, which may be not difficult to solve with quantum algorithms.^[7–8] The application of quantum signature in E-payment has attracted much attention because that the security of quantum cryptography relies on the principles of quantum mechanics.

In 2010, Wen and Nie proposed an E-payment system based on quantum blind and group signature, employing two TTPs (Third Trusted Party) to enhance the system’s robustness.^[9] In succession, Wen et al. proposed an inter-bank E-payment protocol based on quantum proxy blind signature.^[10] However, Cai et al.^[11] showed that the dishonest merchant can succeed to change the purchase information of the customer in this protocol. In 2017, Shao et al.^[12] proposed an E-payment protocol based on quantum multi-proxy blind signature. However, these protocols can not be applied to the business transactions, which use mobile E-payment. In real life, there are many business transactions need mobile E-payment, such as pos terminals. And an E-payment system supports secure mobile transactions is desired from the view of practical application.

In this paper, we proposed a new E-payment protocol to solve the problem and not only support secure mobile business transactions but also secure business transactions between two different banks. In our system, quantum multi-proxy blind signature is adopted to implement mobile payment and inter-bank payment. To the best of our knowledge, this is the first time to proposed a mobile quantum E-payment scheme, which not only supports mobile payment and inter-bank payment but also enhances the transaction credibility and the success of business rate. The property of multi-proxy blind signature could protect the anonymity of E-payment system, while the quantum signature could guarantee unconditionally security. Our scheme only needs Bell state measurement, which can be easily implemented under the current experimental conditions.

In this section, we introduce the basic theories of the quantum multi-proxy blind signature.

2.2 Controlled Quantum Teleportation

Our quantum multi-proxy blind signature is based on controlled teleportation, which takes a genuinely entangled six qubits as its quantum channel. It is given by

As shown in Fig. 1, this controlled quantum teleportation involves the following three partners: a sender Alice, a controller Charlie, and a receiver Bob. Alice holds particle 1, Charlie and Bob own particles (2,3) and particles (4,5,6), respectively.

Fig. 1

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	Fig. 1 The model of quantum teleportation. (BM represents Bell state measurement).

Suppose that the quantum state of particle M carrying message in Alice is

where the coefficients α and β are unknown and satisfy |α|² + |β|² = 1.

The system quantum state composed of particle M and particles (1,2,3,4,5,6) are given by

The details of the controlled teleportation are as follows.

(i) Alice performs Bell state measurement on particles M and 1. The measurement will collapse the state of |ψ〉_M123456 into one of the following states

Then Alice sends her measurement result to Charlie and Bob.

(ii) If Charlie agrees Alice and Bob to complete their teleportation, he performs Bell state measurement on particles (2,3).

Suppose that Alice’s measurement outcome is , and Charlie’s measurement result is , , , and , the whole state of particles (4,5,6) will be projected into the following states, respectively.

Then Charlie sends his measurement result to Bob.

(iii) Bob performs Bell-state measurement on particles (5,6). Suppose that Alice’s measurement outcome is and Charlie’s measurement result is , and Bob’s measurement result is , , , and , particle 4 will be projected into the following states, respectively.

According to Alice’s, Charlie’s measurement outcomes, Bob performs an appropriate unitary operator on particle 4 to rebuild the original state |ψ〉_M. For example, when particle 4 is in the state of Eq. (6), the unitary operators are −σ_x, −iσ_y, I, iσ_z.

To clarify our mobile quantum E-payment system, five characters are defined as follows:

(i) Alice: the customer.

(ii) Bob1: the bank where Alice opens her account.

(iii) U_j: one number of the mobile terminals (j = 1,2,. . .,t).

(iv) Charlie: the merchant.

(v) Bob2: the bank where Charlie opens his account.

As shown in Fig. 2, at the beginning of this transaction, the customer Alice informs U_j about her purchase. Then U_j applies to the bank Bob1 for deducting the corresponding amount of money from Alice’s account. Bob1 transfers the money to the bank Bob2 and delegates Bob2 sign to merchant Charlie, such that Charlie should receive the proper money in his account. After verifying, Charlie gives the corresponding goods to Alice.

Fig. 2

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	Fig. 2 Framework of the mobile E-payment system.

The detailed procedure of our scheme can be described as follows.

3.1 Initial Phase

Fig. 3

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	Fig. 3 Schematic of transmission on the i-th particles.

3.2 Blind the Message M2 Phase

Table 1

Table 1

Table 1 The measuring and encoding rules for quantum states. . M2(i) Measuring base Measuring result Encoding rule 0 {|0〉, |1〉} |0〉 00 |1〉 01 1 {|+〉, |−〉} |+〉 10 |−〉 11

Table 1 The measuring and encoding rules for quantum states. .

In this section, we will prove that this scheme satisfies the properties of blindness, unforgeability, undeniability, and unconditionally security.

In this paper, we propose a practical E-payment protocol based on quantum multi-proxy blind signature. Compared with the previous works,^[9–12] our scheme has some advantages. Firstly, our scheme could not only implement mobile E-payment but also realize the inter-bank payment. Secondly, our protocol has arranged eavesdropping checks to ensure a stronger security. Thirdly, our scheme only relies on Bell state measurement which is easy to implement with current technologies and experimental conditions. However, the received state’s fidelity may reduce because of the experimental environment,^[19–20] with the development of the quantum information technique, our protocol can be applied successfully.