E\({^2}\)CSM: efficient FPGA implementation of elliptic curve scalar multiplication over generic prime field GF(p)

Elliptic curve scalar multiplication (ECSM) is the primitive operation that is also the main computational hurdle in almost all protocols based on elliptic curve cryptography (ECC). This work proposes a novel ECSM hardware architecture by adopting several optimization strategies at circuit and system levels. On the circuit level, it is based on an efficient finite field multiplier that takes fewer clock cycles, produces low latency, and consumes fewer hardware resources. On the system level, Jacobian coordinates with the Montgomery laddering algorithm and a fast scheduling mechanism to execute group operations are adopted. The proposed ECSM design is synthesized and implemented targeting different FPGAs using Xilinx ISE Design Suite. It takes 1.01 ms on the Virtex-7 FPGA to compute a single ECSM operation, occupies 7.1K slices, and achieves 187 MHz frequency. This provides a 30% improvement in computational time with a significantly lower area-time product with better efficiency. Therefore, the proposed ECSM design is better optimized in terms of speed, area-time product, and throughput per slice and hence is suitable for many ECC applications.