PRIMITIVE ROOTS: THEORY AND SOME OF ITS APPLICATIONS.

ABSTRACT

The study of primitive roots is not only theoretical but also encompasses farreaching applications in modular arithmetic, number theory, and cryptography. We begin with an introduction that unveils its historical significance and key contributors. Fundamental concepts in number theory and cryptography are explored in detail. This foundational chapter highlights the importance of these foundational concepts in understanding primitive roots and their applications. Chapter 2 deals with the theoretical foundations of primitive roots in number theory. It begins by defining primitive roots and exploring their fundamental properties. The chapter also presents theorems establishing the conditions for the existence of primitive roots, along with their proofs. Overall, the chapter provides an overview of the essential concepts and properties associated with primitive roots, highlighting their significance in mathematics. Chapter 3 explores the practical applications and real-world implications of primitive roots within the domain of number theory and cryptography from solving modular congruences to their role in cryptographic protocols such as the DiffieHellman Key Exchange and the ElGamal Encryption. Finally, we explore the algorithmic dimension of primitive roots and the discrete logarithm problem, presenting various computational methods and their complexities. Each method we mention is analyzed in terms of its efficiency, complexity, and suitability for different scenarios. Through practical examples readers gain insight into the computational challenges involved in computing primitive roots and solving discrete logarithm problems. In conclusion the project synthesizes theoretical insights, practical applications and computational aspect offering a comprehensive understanding of primitive roots