High-Throughput Finite Field Multipliers Using Redundant Basis for FPGA and ASIC Implementations

Redundant basis (RB) multipliers over Galois Field ( GF(2m)) have gained huge popularity in elliptic curve cryptography (ECC) mainly because of their negligible hardware cost for squaring and modular reduction. In this paper, we have proposed a novel recursive decomposition algorithm for RB multiplication to obtain high-throughput digit-serial implementation. Through efficient projection of signal-flow graph (SFG) of the proposed algorithm, a highly regular processor-space flow-graph (PSFG) is derived. By identifying suitable cut-sets, we have modified the PSFG suitably and performed efficient feed-forward cut-set retiming to derive three novel multipliers which not only involve significantly less time-complexity than the existing ones but also require less area and less power consumption compared with the others. Both theoretical analysis and synthesis results confirm the efficiency of proposed multipliers over the existing ones. The synthesis results for field programmable gate array (FPGA) and application specific integrated circuit (ASIC) realization of the proposed designs and competing existing designs are compared. It is shown that the proposed high-throughput structures are the best among the corresponding designs, for FPGA and ASIC implementation. It is shown that the proposed designs can achieve up to 94% and 60% savings of area-delay-power product (ADPP) on FPGA and ASIC implementation over the best of the existing designs, respectively.