The new emerging non-volatile memory (NVM) devices known as memristors could be the promising candidate for future digital architecture, owing to their nanoscale size and its ability to integrate with the exciting CMOS technology. In this paper, a combination of memristor devices and CMOS transistors are working together to form a hybrid CMOS-memristor circuit for XAX- Module, a core element for the finite field multiplier. The proposed design was implemented using Pt /TaOx/Ta memristor device and simulated in Cadence Virtuoso. The simulation results demonstrate the design functionality. The proposed module appears to be efficient in terms of layout area, delay and power consumption since the design utilizes the hybrid CMOS/memristor gates.