FPGA Realization of Low Register Systolic All-One-Polynomial Multipliers Over $GF(2^{m})$ and Their Applications in Trinomial Multipliers

Systolic all-one-polynomial (AOP) multipliers usually suffer from the problem of high register complexity, especially in field-programmable gate array (FPGA) platforms where the register resources are not that abundant. In this paper, we have shown that the AOP-based systolic multipliers can easily achieve low register-complexity implementations and the proposed architectures can be employed as computation cores to derive efficient implementations of systolic Montgomery multipliers based on trinomials. First, we propose a novel data broadcasting scheme in which the register complexity involved within existing AOP-based systolic multipliers is significantly reduced. We have found out that the modified AOP-based structure can be packed as a standard computation core. Next, we propose a novel Montgomery multiplication algorithm that can fully employ the proposed AOP-based computation core. The proposed Montgomery algorithm employs a novel precomputed-modular operation, and the systolic structures based on this algorithm fully inherit the advantages brought from the AOP-based core (low register complexity, low critical-path delay, and low latency) except some marginal hardware overhead brought by a precomputation unit. The proposed architectures are then implemented by Xilinx ISE 14.1 and it is shown that compared with the existing designs, the proposed designs achieve at least 61.8% and 47.6% less area-delay product and power-delay product than the best of competing designs, respectively.