This thesis explores the use of elliptic curves in cryptography, focusing on the Elliptic Curve Discrete Logarithm Problem (ECDLP). It begins with an introduction to cryptography, covering the objectives of cryptography, symmetric key algorithms, and public key algorithms. The thesis then delves into the theory of elliptic curves, including their definition, properties, and arithmetic operations over finite fields and fields of characteristic 2. The main section of the thesis discusses various attacks on the ECDLP, such as the Pohlig-Hellman and Baby-Step/Giant-Step (BSGS) algorithms, Pollard's rho and lambda methods, and parallel collision search. It also examines improvements to these algorithms, including better random walks and the use of anomalous binary curves, which offer significant performance enhancements. The thesis concludes with a discussion on the practical implications of these attacks and improvements, highlighting the security and efficiency of elliptic curve cryptosystems. It emphasizes the advantages of using elliptic curves over traditional RSA and Diffie-Hellman systems in terms of key size and security.This thesis explores the use of elliptic curves in cryptography, focusing on the Elliptic Curve Discrete Logarithm Problem (ECDLP). It begins with an introduction to cryptography, covering the objectives of cryptography, symmetric key algorithms, and public key algorithms. The thesis then delves into the theory of elliptic curves, including their definition, properties, and arithmetic operations over finite fields and fields of characteristic 2. The main section of the thesis discusses various attacks on the ECDLP, such as the Pohlig-Hellman and Baby-Step/Giant-Step (BSGS) algorithms, Pollard's rho and lambda methods, and parallel collision search. It also examines improvements to these algorithms, including better random walks and the use of anomalous binary curves, which offer significant performance enhancements. The thesis concludes with a discussion on the practical implications of these attacks and improvements, highlighting the security and efficiency of elliptic curve cryptosystems. It emphasizes the advantages of using elliptic curves over traditional RSA and Diffie-Hellman systems in terms of key size and security.