ANALYSIS OF NUMBER THEORY FOR CRYPTOGRAPHY AND SECURITY APPLICATIONS

A key component of cryptography and security applications is number theory, a subfield of pure mathematics. In order to ensure the confidentiality, integrity, and validity of data in contemporary cryptographic systems, this study seeks to provide an analysis of the fundamental ideas of number theory and their applications. The essential concepts of number theory, such as prime numbers, modular arithmetic, and integer properties, are introduced at the outset of the article. In specifically, it looks at the role prime numbers play in the creation of robust cryptographic keys and the execution of key exchange protocols. Modular exponentiation, the basis for numerous encryption and decryption techniques, is also investigated in relation to the idea of modular arithmetic. In addition, the paper investigates how number theory might be used to create digital signatures that guarantee the validity and integrity of data. It examines the mathematical underpinnings of digital signature algorithms like the Elliptic Curve Digital Signature Algorithm (ECDSA) and the RSA signature scheme, which use ideas like modular arithmetic and the properties of prime numbers to confirm the authenticity and integrity of digital documents. The paper also examines the applications and constraints of number theory-based cryptography systems. It talks about how improvements in computer hardware and computational complexity affect these systems' security. An examination of the newly emerging topic of postquantum cryptography, which seeks to create cryptographic algorithms that are still secure in the presence of quantum computers, is included in the analysis.