Applications of Finite Fields pp 151-171 | Cite as

# Elliptic Curve Cryptosystems

## Abstract

As we have seen in Section 6.1, the elements of a finite cyclic group *G* may be used to implement several cryptographic schemes, provided that finding logarithms of elements in G is infeasible. We may take *G* to be a cyclic subgroup of *E*(*F* _{ q }), the group of *F* _{ q }-rational points of an elliptic curve defined over *F* _{ q }; this was first suggested by N. Koblitz [10] and V. Miller [17]. Since the addition in this group is relatively simple, and moreover the discrete logarithm problem in *G* is believed to be intractable, elliptic curve cryptosystems have the potential to provide security equivalent to that of existing public key schemes, but with shorter key lengths. Having short key lengths is a factor that can be crucial in some applications, for example the design of smart card systems.

## Keywords

Elliptic Curve Elliptic Curf Logarithm Problem Discrete Logarithm Discrete Logarithm Problem## Preview

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