Abstract: Methods and apparatus to construct finite fields over which efficient elliptic curve cryptosystems can be set up. Given a security parameter k, the said methods and apparatus consist of devices for carrying out operations in a small k0-bit field k0 and methods to successively build extension fields K1; K2, . . . , Kt, where the extension K1/K0 has degree 2 or 3 and the other extensions Ki/KI−1, are quadratic, Kt is the final field over which elliptic curves are defined, and Kt has size ko2t or 3k02t−1 just exceeding the said security parameter k.