The Internet of Things requires greater attention to the security and privacy of the network. Compared to other public-key cryptosystems, elliptic curve cryptography can provide better security and lower latency with shorter keys, rendering it more suitable for IoT security. This paper presents a high-efficiency and low-delay elliptic curve cryptographic architecture based on the NIST-p256prime field for IoT security applications. A modular square unit utilizes a fast partial Montgomery reduction algorithm, demanding just a mere four clock cycles to complete a modular square operation. The modular square unit can be computed simultaneously with the modular multiplication unit, consequently improving the speed of point multiplication operations. Synthesized on the Xilinx Virtex-7 FPGA platform, the proposed architecture completes one PM operation in 0.08 ms using 23.1 k LUTs at 105.3 MHz. These results show significantly better performance compared to that in previous works.