The fourth fundamental circuit element, memristor, attracts increasing attention because of its memory characteristic. The special memory behavior of the memristor has been exploited to design control systems, memory arrays, logic gate, and security primitives in previous work. However, the power characteristics of the memristor have not been widely studied yet. In this work, we used a memristor model that is suitable for circuit simulation to investigate the power characteristics of the memristor itself and memristor-based logic gates. Our simulation results indicate that memristor has different power characteristic compared with CMOS devices. The peak power of memristor-based gates does not monotonically increases with input voltage amplitude; instead, the combination of input period length and voltage amplitude determines the occurrence of power peak. The reason is that the power consumption of memristors depends on the effective memristor width, which is controlled by the input. We further examine the feasibility of utilizing memristors to implement a new block cipher, SIMON. Our studies show that the unique power characteristic of memristor based SIMON may add extra challenges for extraction of the secret key from the cipher as it introduces 94% power deviation while power sampling.