NUMBER THEORY APPLICATIONS IN MODERN CRYPTOGRAPHY

Abstract

This paper explores the critical role of number theory in modern cryptography, highlighting its foundational significance in securing digital communication. Number theory, the branch of mathematics concerned with the properties and relationships of numbers, underpins many cryptographic algorithms and protocols that protect sensitive information in today’s digital landscape. The discussion begins with the historical evolution of cryptography, tracing its transformation from simple ciphers to complex mathematical frameworks. Key concepts such as prime numbers, modular arithmetic, and discrete logarithms are introduced as essential elements that provide security against unauthorized access. The RSA algorithm, one of the most widely used public key cryptographic systems, exemplifies the application of number theory through its reliance on the difficulty of factoring large composite numbers. Additionally, the Diffie-Hellman key exchange illustrates how discrete logarithms enable secure key agreements over insecure channels.