Chapter I Special Hyperelliptic Curve Cryptosystems of Genus Two : Efficient Arithmetic and Fast Implementation

There exists a manifold variety of cryptographic applications: from low level embedded crypto implementations up to high end cryptographic engines for servers. The latter require a flexible implementation of a variety of cryptographic primitives in order to be capable of communicating with several clients. On the other hand, on the client it only requires an implementation of one specific algorithm with fixed parameters such as a fixed field size or fixed curve parameters if using ECC/ HECC. In particular for embedded environments like PDAs or mobile communication devices, fixing these parameters can be crucial regarding speed and power consumption. In this chapter, we propose a highly efficient algorithm for a hyperelliptic curve cryptosystem of genus two, well suited for these constrained devices. This work presents a major improvement of HECC arithmetic for certain non-supersingular curves defined over fields of characteristic two. We optimized the group doubling operation and managed to speed up the whole cryptosystem by approximately 27%. Furthermore, an actual implementation of the new formulae on an embedded processor shows its practical relevance.