Elliptic Curve Cryptography defined on Finite Fields

Abstract

This memory is a map that guides us through understanding ECC and why it is so promising , we’ve split it into three main chapters that progressively explore finite fields, elliptic curves, and their cryptographic applications. The first chapter introduces the fundamentals by examining the structure and proper- ties of finite fields. The second chapter delves into elliptic curves, their geometric and algebraic repre- sentations, and the arithmetic operations defined over finite fields. Building on these concepts, the third chapter focuses on elliptic curve cryptography, a powerful approach to secure communication and data protection based on the hard- ness of the elliptic curve discrete logarithm problem, Key exchange protocols, and other cryptographic schemes utilizing elliptic curves are covered.Through this logical progres- sion, a comprehensive understanding of these fundamental topics is developed.