Complexity Analysis and Efficient Implementations of Bit Parallel Finite Field Multipliers Based on Karatsuba-Ofman Algorithm on FPGAs

This paper presents complexity analysis [both in application-specific integrated circuits (ASICs) and on field-programmable gate arrays (FPGAs)] and efficient FPGA implementations of bit parallel mixed Karatsuba-Ofman multipliers (KOM) over GF(2^m) . By introducing the common expression sharing and the complexity analysis on odd-term polynomials, we achieve a lower gate bound than previous ASIC discussions. The analysis is extended by using 4-input/6-input lookup tables (LUT) on FPGAs. For an arbitrary bit-depth, the optimum iteration step is shown. The optimum iteration steps differ for ASICs, 4-input LUT-based FPGAs and 6-input LUT-based FPGAs. We evaluate the LUT complexity and area-time product tradeoffs on FPGAs with different computer-aided design (CAD) tools. Furthermore, the experimental results on FPGAs for bit parallel modular multipliers are shown and compared with previous implementations. To the best of our knowledge, our bit parallel multipliers consume the least resources among known FPGA implementations to date.