Power consumption limits the application of public key cryptosystem in portable devices. This paper proposes a low power design of 1,024-bit RSA. In algorithm, the Chinese Remainder Theorem (CRT) and an improved Montgomery algorithm are selected to decrease the computation of RSA. In architecture and circuit, the operand isolation technique is applied to avoid unnecessary flip-flops of the combinational logic, and the clock gating technique is used to reduce the power dissipation of the registers. The proposed design is functionally verified on Altera FPGA EP2C8Q208C8N device. With SMIC 0.18m CMOS process, the Synopsys synthesizing result shows that the area and the critical path are 7.1k gates and 5.3ns respectively, while the power is 2.56mW and the throughput can reach 49 kbps. Thus the proposed design requires lower power than previous designs.