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These curves are widely used in modern cryptography for secure communication. The Wikipedia page on elliptic curves provides a comprehensive overview of these curves, their properties, and their applications. The page begins by introducing the concept of an elliptic curve and explaining how it is defined using an equation of the form y^2 = x^3 + ax + b, where a and b are constants. It then delves into the mathematical properties of these curves, such as their symmetry and their relationship to modular arithmetic. The page also covers the group structure of elliptic curves, which is a key aspect used in cryptographic protocols. It explains how points on the curve form a finite Abelian group, and how addition and scalar multiplication can be defined on this group. This group structure enables various cryptographic operations, such as key exchange and digital signatures, to be implemented using elliptic curves. Additionally, the page describes the notion of elliptic curve cryptography (ECC) and its advantages over other traditional cryptographic systems. ECC is known for providing strong security with relatively short key lengths, making it especially suitable for resource-constrained devices like smartphones and IoT devices. Furthermore, the page explores a variety of applications of elliptic curves in cryptography, including Elliptic Curve Diffie-Hellman (ECDH) key exchange, Elliptic Curve Digital Signature Algorithm (ECDSA), and Elliptic Curve Integrated Encryption Scheme (ECIES). Finally, the page provides references to further reading, including books, research papers, and standards related to elliptic curves. It also covers some historical background on the development of elliptic curve cryptography and the pioneers in this field. Overall, the Wikipedia page on elliptic curves offers a comprehensive and accessible resource for understanding the mathematics, properties, and applications of these curves in modern cryptography.