A New Privacy Aware Payment Scheme for Wireless Charging of Electric Vehicles

Abstract

Electric vehicles (EVs) can be considered as a revolution in the combustion industry with significant improvement in fuel utilization and decrease in pollution compared to combustion engines. However, by decreasing the size of the battery to reduce the cost, the frequency of charging EVs in a day increases. Therefore, to reduce the downtime required for charging EVs, wireless charging on the move can be an effective solution. In such a situation, paying for wireless charging on the move is an important issue. However, it can endanger the location privacy of users, since the EVs need to charge frequently in a day. In this paper, we first explain different methods of payment and problems with such payment methods in the case of wireless charging on the move. Then, we propose an efficient payment method based on ‘tokens’ for wireless charging on the move, which minimizes the communications between service providers and users during the charging process. The proposed scheme prevents users and service providers from cheating, and it is robust to support different values for the price. Finally, we compare it with other payment methods that have been proposed for plug-in electric vehicles.

Appendix

1.1 Signature Verification

As mentioned in [16], an RSU or a service provider with access to the $\left\{ {{\mathbb{G}} , {\mathbb{G}}_{T} , q, P,P_{Pub} } \right\}$ parameters can verify the signature $\alpha_{i} = E_{i} ,F_{i}$ on the message $M_{i}$ as follows:

$$\hat{e}\left( {F_{i} ,P} \right) = \hat{e}\left( {E_{i} + h\left( {M_{i} ,E_{i} } \right)H\left( {PID_{i} } \right),P_{Pub} } \right).$$

This is proved below:

$$\begin{aligned} \hat{e}\left( {F_{i} ,P} \right) & = \hat{e}\left( {r_{i} P_{Pub} + h\left( {M_{i} ,E_{i} } \right)SK_{i} , P} \right) = \hat{e}\left( {r_{i} P_{Pub} , P} \right) \cdot \hat{e}\left( {h\left( {M_{i} ,E_{i} } \right)SK_{i} , P} \right) \\ & = \hat{e}\left( {r_{i} P, P_{Pub} } \right) \cdot \hat{e}\left( {h\left( {M_{i} ,E_{i} } \right) sH\left( {PID_{i} } \right), P} \right) = \hat{e}\left( {E_{i} , P_{Pub} } \right) \cdot \hat{e}\left( {h\left( {M_{i} ,E_{i} } \right) H\left( {PID_{i} } \right),P_{Pub} } \right) \\ & = \hat{e}\left( {E_{i} + h\left( {M_{i} ,E_{i} } \right)H\left( {PID_{i} } \right),P_{Pub} } \right). \\ \end{aligned}$$

The computation cost to verify the above signature is one multiplication and two pairing operations.

1.2 Verifying the Group Signature

To reduce the computation cost and improve the efficiency, Jian et al. [16] introduce the BAT signature in which $n = 2^{h}$ vehicles $\left\{ {V_{1} ,V_{2} ,V_{3} , \ldots ,V_{n} } \right\}$ with corresponding signatures $\left\{ {\alpha_{1} ,\alpha_{2} ,\alpha_{3} , \ldots ,\alpha_{n} } \right\}$ can construct a Binary Authentication Tree. In this tree, each leaf node contains the signatures of a vehicle, and each inner node is associated with an aggregate signature that contains signatures of the whole leaf nodes in this sub-tree. Moreover, the root of the tree includes an accumulation of all signatures at the leaf-nodes. For verifying all the signatures $\left\{ {\alpha_{{k_{1} }} ,} \right.\alpha_{{k_{1} + 1}} , \ldots ,\left. {\alpha_{{k_{2} }} } \right\}$, the following equation should hold:

$$\hat{e}\left( {\mathop \sum \limits_{{i = k_{1} }}^{{k_{2} }} F_{i} ,P} \right) = \hat{e}\left\{ {\mathop \sum \limits_{{i = k_{1} }}^{{k_{2} }} \left[ {E_{i} + h\left( {M_{i} ,E_{i} } \right)H\left( {PID_{i} } \right)],} \right.P_{Pub} } \right\},$$

which can be proven as follows:

$$\begin{aligned} \hat{e}\left( {\mathop \sum \limits_{{i = k_{1} }}^{{k_{2} }} F_{i} ,P} \right) & = \hat{e}\left\{ {\mathop \sum \limits_{{i = k_{1} }}^{{k_{2} }} r_{i} P_{Pub} + h\left( {M_{i} ,E_{i} } \right)SK_{i} , P} \right\} = \hat{e}\left( {\mathop \sum \limits_{{i = k_{1} }}^{{k_{2} }} r_{i} P_{Pub} , P} \right) \cdot \hat{e}\left( {\mathop \sum \limits_{{i = k_{1} }}^{{k_{2} }} h\left( {M_{i} ,E_{i} } \right)SK_{i} , P} \right) \\ & = \hat{e}\left( {\mathop \sum \limits_{{i = k_{1} }}^{{k_{2} }} r_{i} P_{Pub} , P} \right) \cdot \hat{e}\left( {\mathop \sum \limits_{{i = k_{1} }}^{{k_{2} }} h\left( {M_{i} ,E_{i} } \right) sH\left( {PID_{i} } \right), P} \right) \\ & = \hat{e}\left( {\mathop \sum \limits_{{i = k_{1} }}^{{k_{2} }} E_{i} , P_{Pub} } \right) \cdot \hat{e}\left( {\mathop \sum \limits_{{i = k_{1} }}^{{k_{2} }} h\left( {M_{i} ,E_{i} } \right) H\left( {PID_{i} } \right),P_{Pub} } \right) \\ & = \hat{e}\left\{ {\mathop \sum \limits_{{i = k_{1} }}^{{k_{2} }} \left[ {E_{i} + h\left( {M_{i} ,E_{i} } \right)H\left( {PID_{i} } \right)],} \right.P_{Pub} } \right\}. \\ \end{aligned}$$

The computation cost for verifying the $k$ aggregate signatures contains $k$ multiplications, $k$ one-way hash, and 2 pairing operations. Cleary, using the BAT signature can effectively lower the computation cost.