11 - Cryptography based on hyperelliptic curves

Summary

The widespread success of cryptography based on elliptic curves motivates the investigation of other curves for possible cryptographic uses. However, elliptic curves are the only plane curves that admit a definition of point addition in such a way that the points of the curve form a group. This does not mean that other curves cannot be used. It only means that the points of those curves must be organized to form a group in some other way. It is more complicated to find group structures based on other curves. In general, the curve X must be embedded into a larger algebraic structure on which a suitable group operation can be defined. Hyperelliptic curves are a class of curves that lead to such a group structure. A hyperelliptic curve is associated in a natural way with an abelian group called the jacobian of the hyperelliptic curve. In contrast to the curve itself, the jacobian of a hyperelliptic curve does admit a suitable group structure. Based on the group structure of its jacobian, a hyperelliptic curve can be used to construct a cryptographic system. Most of the chapter is devoted to the task of defining the jacobian of a hyperelliptic curve and its relevant computational algorithms. The usual methods of cryptography constructed on a large finite group are then immediately applicable.

Because an elliptic curve is a special case of a hyperelliptic curve, this chapter also serves to extend our understanding of elliptic curves.