This paper presents two innovative sliding mode control laws that are meticulously designed based on the reaching law convergence principle. These control laws aim to achieve both finite-time and fixed-time synchronization for a specific class of memristive chaotic systems, which are known for their intricate and complex dynamical behaviors. By leveraging these control strategies, we can effectively manage the synchronization process, ensuring rapid convergence. Firstly, for the finite-time synchronization issue, a novel power reaching law is devised. Compared with the conventional reaching law, the prominent advantage of this reaching law is 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 on the stabilization time, which is dependent on the initial conditions of the system, is obtained and the stability of the system is proved. 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 in the numerical simulation part to compare the influence of the two control laws on the system synchronization state. The experimental phenomenon strongly proves the accuracy of the proposed theorem.