The recent developments in the study of elliptic curve public-key algorithms have shown that they could play a major factor in the design of cryptosystems of the future. This thesis describes efficient algorithms for two important aspects of such systems. The first part describes a structured approach for finding cryptographically secure curves. A comprehensive lists of elliptic curves over subfields GF (2 n ), n = 8; 9; : : : 18, was generated, which are cryptographically secure over GF ((2 n ) m ), n \Delta m = 150; : : : ; 200. The second part describes efficient algorithms for fast software implementations of elliptic curve computations which can be used in a variety of public-key protocols. These algorithms, which perform group operations over nonsupersingular elliptic curves, are optimized through the use of composite Galois fields of the form GF ((2 n ) m ). An elliptic curve key-exchange protocol over the composite field GF ((2 16 ) 11 ) was implemented using op...