Cryptography/Elliptic curve

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The set of points on an elliptic curve over a finite field can be given the structure of a finite abelian group. This group, instead of group of integers modulo a prime number, could be used to construct cryptographic primitives. The underlying hard problem associated to these groups is the Elliptic Curve Discrete Logarithm Problem (ECDLP), which is defined in an analogous way to usual finite groups. Since the points on the curve do not naturally accept a notion of smoothness, the standard index calculus methods are not applicable (see however C. Diem). Hence the key length of EC-based systems is shorter (160 bits) than those based on the integers (1024 bits). Furthermore one could define a bilinear map (pairing) on an elliptic curves, which enables one to construct new schemes such as identity based encryption.

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