Extension of Montgomery multiplication algorithms in GF(p) are studied and analyzed. The time and space requirements of various state-of-the-art algorithms are presented. We propose Modified Montgomery Modular Multiplication Algorithms that reduces the number of computational operations such as number of additions, memory reads and writes involved in the existing algorithms, thereby, saving considerable time and area for execution. Many design examples has been solved to prove the theoretical correctness of the proposed algorithms. Complexity analysis shows that Modified Coarsely Integrated Scanning (MCIOS) consume less space and time compared to other modified Montgomery Algorithms. To verify the logical correctness, the proposed MCIOS algorithm was implemented in Xilinx Spartan3E FPGA. The total memory for execution of 64-bit operand is 135484 KB for MCIOS and 140496 KB for existing Coarsely Integrated Scanning (CIOS) method. The proposed algorithm can be changed to be suitable for any arbitrary Galois field size with little modifications. Also the proposed algorithm can be developed as architecture suitable for System on Chip (SoC) implementations of Elliptic curve cryptosystem. Subsequently, the system can be developed as a 3D chip.
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