Advances in Computational Number Theory and Its Applications in Modern Cryptography

Computational number theory, also known as algorithmic number theory, is a modern and rapidly evolving field focused on the development and analysis of efficient algorithms to solve classical and contemporary problems in number theory and arithmetic geometry. This branch of mathematics plays a pivotal role in the design and implementation of secure cryptographic systems. As computational power increases and algorithms become more sophisticated, computational number theory continues to expand its influence across mathematics, computer science, and especially cryptography. This paper explores the fundamental principles of computational number theory and its close relationship with modern cryptographic practices. It examines essential cryptographic systems such as RSA, elliptic curve cryptography (ECC), and emerging post-quantum algorithms. Core number-theoretic tools—like primality testing, integer factorization, modular arithmetic, and Diophantine equation solving—are analyzed in the context of their cryptographic applications. The historical development and theoretical underpinnings of cryptography are also traced, alongside recent advances and active research areas in the field. By showcasing the intersection of theory and application, this paper highlights the indispensable role of computational number theory in enabling secure communication in the digital age.