Elliptic curves are the basis for a relative new class of public-key schemes. It is predicted that elliptic curves will replace many existing schemes in the near future. It is thus of great interest to develop algorithms which allow efficient implementations of elliptic curve crypto systems. This thesis deals with such algorithms. Efficient algorithms for elliptic curves can be classified into low-level algorithms, which deal with arithmetic in the underlying finite field and high-level

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... ms, which operate with the group operation. This thesis describes three new algorithms for efficient implementations of elliptic curve cryptosystems. The first algorithm describes the application of the Karatsuba-Ofman Algorithm to multiplication in composite fields GF ((2 n ) m ). The second algorithm deals with efficient inversion in composite Galois fields of the form GF ((2 n ) m ). The third algorithm is an entirely Efficient algorithms for elliptic curves can be classified into high-level algorithms, which operate with the group operation, and into low-level algorithms, which deal with arithmetic in the underlying finite field. For efficient implementations it is obviously the best to optimize both types of algorithms. The main part of the thesis will introduce three algorithms, one high-level algorithm for point multiplication and two low-level for finite field inversion and multiplication, respectively.