Improving Coding Performance Using Numerical Methods and an Applied Study on Elliptical Curves

Abstract

Modern digital infrastructures, including smart cities and the Internet of Things (IoT), increasingly require secure and efficient cryptographic mechanisms to ensure data confidentiality and privacy. Elliptic Curve Cryptography (ECC) has become one of the most widely utilized encryption schemes because it provides strong security with relatively small key sizes compared to traditional algorithms such as RSA. However, the practical deployment of ECC encounters several numerical and computational challenges related to efficiency and precision, especially when operating over large finite fields or within resource‑constrained devices. This research addresses these challenges by integrating advanced numerical methods to improve the performance of ECC algorithms. The study examines the core arithmetic operations in ECC—specifically point addition and point multiplication—and identifies critical computational stages that can benefit from numerical optimization. A set of numerical techniques is reviewed and applied, including the Newton–Raphson method to accelerate modular inverse calculations, numerical integration methods such as the trapezoidal rule and Simpson’s rule to approximate nonlinear elliptic‑curve functions, and fast multiplication algorithms.

To validate the proposed approach, an applied experimental study was conducted using MATLAB. Experimental codes were developed to implement ECC algorithms enhanced with the selected numerical methods. Performance evaluation was based on key metrics such as execution time, computational accuracy, and numerical error magnitude. The study also explores the potential use of artificial intelligence techniques, including machine learning algorithms, to optimize initial parameter selection and accelerate convergence in iterative processes.

The results demonstrate clear improvements in execution time and significant reductions in numerical error, confirming the effectiveness of numerical methods in enhancing ECC performance for modern resource‑limited systems.