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 its objectives, symmetric and public key algorithms, and their applications. The thesis then provides an overview of elliptic curves, their definitions, properties, and the mathematical structures that underpin their use in cryptography. It discusses the discrete logarithm problem on elliptic curves and the challenges in solving it, particularly the lack of known sub-exponential time attacks on the ECDLP. The thesis analyzes various general attacks on the ECDLP, including the Pohlig-Hellman and Baby-step Giant-step (BSGS) algorithms, and Pollard's ρ and λ methods. These attacks are evaluated for their efficiency and effectiveness in solving the ECDLP. The thesis also examines specialized attacks, such as pairing-based attacks and the Smart attack on anomalous curves, which exploit specific weaknesses in certain types of elliptic curves. The thesis further explores the use of hyperelliptic curves in attacking the ECDLP, highlighting their potential as an alternative to elliptic curves. It discusses the mathematical properties of hyperelliptic curves and their application in cryptographic contexts. The thesis concludes with a summary of the key findings and recommendations for the use of elliptic curves in cryptography, emphasizing the importance of selecting curves with appropriate security properties and the need for ongoing research into more efficient and secure cryptographic methods.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 its objectives, symmetric and public key algorithms, and their applications. The thesis then provides an overview of elliptic curves, their definitions, properties, and the mathematical structures that underpin their use in cryptography. It discusses the discrete logarithm problem on elliptic curves and the challenges in solving it, particularly the lack of known sub-exponential time attacks on the ECDLP. The thesis analyzes various general attacks on the ECDLP, including the Pohlig-Hellman and Baby-step Giant-step (BSGS) algorithms, and Pollard's ρ and λ methods. These attacks are evaluated for their efficiency and effectiveness in solving the ECDLP. The thesis also examines specialized attacks, such as pairing-based attacks and the Smart attack on anomalous curves, which exploit specific weaknesses in certain types of elliptic curves. The thesis further explores the use of hyperelliptic curves in attacking the ECDLP, highlighting their potential as an alternative to elliptic curves. It discusses the mathematical properties of hyperelliptic curves and their application in cryptographic contexts. The thesis concludes with a summary of the key findings and recommendations for the use of elliptic curves in cryptography, emphasizing the importance of selecting curves with appropriate security properties and the need for ongoing research into more efficient and secure cryptographic methods.