An Efficient Algorithm for Modulus Operation and Its Hardware Implementation in Prime Number Calculation

Abstract: This paper presents a novel algorithm for the modulus operation for FPGA implementation. The proposed algorithm use only addition, subtraction, logical, and bit shift operations, avoiding the complexities and hardware costs associated with multiplication and division. It demonstrates consistent performance across operand sizes ranging from 32-bit to 2048-bit, addressing scalability challenges in cryptographic applications. Implemented in Verilog HDL and tested on a Xilinx Zynq-7000 family FPGA, the algorithm shows a predictable linear scaling of cycle count with bit length difference (BLD), described by the equation $y=2x+2$, where $y$ represents the cycle count and $x$ represents the BLD. The application of this algorithm in prime number calculation up to 500,000 shows its practical utility and performance advantages. Comprehensive evaluations reveal efficient resource utilization, robust timing performance, and effective power management, making it suitable for high-performance and resource-constrained platforms. The results indicate that the proposed algorithm significantly improves the efficiency of modular arithmetic operations, with potential implications for cryptographic protocols and secure computing.