Redundant Radix-2r Number System for Accelerating Arithmetic Operations on the FPGAs

The main contribution of this paper is to present hardware algorithms for redundant radix-2^r number system in the FPGA to speed the arithmetic operations for numbers with many bits, which have applications in security systems such as RSA encryption and decryption. Our hardware algorithms accelerate arithmetic operations including addition, multiplication, and Montgomery modulo multiplication.Quite surprisingly, our hardware algorithms of the multiplication and Montgomery multiplication for two 1024-bit numbers runs only 64 clock cycles using redundant radix-2¹⁶ number system. Also, the experimental results for Xilinx Virtex-II Pro Family FPGA XC2VP100-6 show that the clock frequency of our circuit is independent of the number of bits. The speed up factors of our hardware algorithm using the redundant number system over those using the conventional number system are 8.3 for 1024-bit addition, 3.4 for 1024-bit multiplication, and 2.5 for 1024-bit Montgomery modulo multiplication. Further, for 256-bit Montgomery modulo multiplication, our hardware algorithm runs in 0.38 mus, while a previously known implementation runs in 1.22 mus. Thus, our approach using redundant number system for arithmetic operations is very efficient.