In 2001, Wong et al. proposed a linear Mutual Authentication and Key Exchange Protocol (linear MAKEP). It can easily establish a secure communication between a low-power wireless device (client) and a powerful base station (server). Unfortunately, Shim found that the scheme was vulnerable to the unknown key-share attack, and he proposed an improved scheme to modify this weakness. However, in this paper, we provide another attack that is the man-in-the-middle attack, which can still attack Shim's improved scheme. Therefore, we shall propose a new efficient MAKEP in spirit of Girault's method, which can overcome the attacks of both aforementioned. At the same time, besides the information of pre-computation and some system parameters, the scheme just needs to keep one user's secret key in client's memory. It is much better than the Shim's improved linear MAKEP, which keeps too many pairs of private keys (Certificates).