Elliptic curve cryptography
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We will introduce the reader to elliptic curves, and explain how they can be used for cryptography and why they are more difficult to attack in comparison to systems based on finite fields, such as the famous RSA algorithm. This expository paper will discuss different algorithms to attack cryptographic systems, such as the index calculus method, Shanks’ algorithm, Pollard’s rho method and the pairing attack. Finally, we will construct the Weil pairing, prove some of its important properties and present an algorithm to compute it. The study of attacks on cryptographic systems can help to reveal that a certain cryptographic protocol is weak and therefore that it should not be used. The reader should have background in algebra, basic number theory and some cryptography, such as the RSA algorithm. No background in elliptic curves or in algebraic geometry is required, but may be helpful.