Finite fields of low characteristic in elliptic curve cryptography

The use of finite fields of low characteristic can make the implementation of elliptic curve cryptography more efficient. There are two approaches to lower the characteristic of the finite field in ECC while maintaining the same security level: Elliptic curves over a finite field extension and hyperelliptic curves over a finite field. This thesis solves some problems in both approaches.; The group orders of elliptic curves over finite field extensions are described as polynomials. The irreducibility of these polynomials is proved, and hence the primality of the group orders can be studied. Asymptotic formulas for the number of traces of elliptic curves over field extensions with almost prime orders are given and a proof based on Bateman-Horn's conjecture is given. Hence the number of curves for cryptographic use is known. Experimental data is given. The formulas fit the actual data remarkably well.; Finally, the arithmetic of real hyperelliptic curves is studied. We study the algorithm for divisor addition on the real hyperlliptic curves and give the explicit formulas.