Two innovative sliding mode control laws based on the convergence principle of reaching law are presented in this work. These control laws are used to achieve both finite-time and fixed-time synchronization for a specific class of memristive chaotic system, which are known for their intricate and complex dynamical behaviors. By utilizing these control strategies, we can effectively manage the synchronization process and ensure rapid convergence. Firstly, for the finite-time synchronization issue, a novel power reaching law is derived. Compared with the conventional reaching law, the reaching law presented in this work has a prominent advantage that the chattering of the sliding mode control is reduced to a lesser extent and the speed of reaching the sliding surface is quicker. An upper bound of the stabilization time, which is dependent on the initial conditions of the system, is obtained and the system is proved stable. For the fixed time synchronization problem, a new double power reaching law is put forward to minimize the chattering and accelerate the convergence. Then, by utilizing the fixed time stability theory, the upper bound of the convergence time that remains invariant with the initial value of the system is derived. Finally, in order to verify the effectiveness and feasibility of the theoretical derivation in this paper, two sets of control experiments are set up and the influences of the two control laws on the system synchronization state are compared. The experimental phenomenon strongly proves the accuracy of the proposed theorem.