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A Course in Mathematical Cryptography by Gerhard Rosenberger, Martin Kreuzer, Benjamin Fine, Gilbert Baumslag

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8 Elliptic Curve Cryptography

8.1 The ElGamal and Elliptic Curve Encryption System

The standard public key systems that we have described so far, Diffie-Hellman, ElGamal, RSA and Rabin, require very large key spaces. In an attempt to use the same ideas but reduce the key space size it was suggested that Diffie-Hellman be applied to other abelian groups. To accomplish this, algebraic geometry was introduced into cryptography. In 1985, Neil Koblitz, and independently Victor Miller, suggested the use of elliptic curves over finite fields, and their corresponding groups, as possible cryptographic platforms. These methods have been quite successful and result, in many cases, in faster encryption and smaller key spaces than standard RSA methods. First, ...

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