| This thesis is concerned with Jacobians of hyperelliptic curves and superelliptic curves. Reasons for studying Jacobians of curves fall into two distinct categories. First, with the increase in electronic communications over the past 20 years, the demand for secure and efficient means of performing public key cryptography has grown dramatically. One scheme which meets these requirement is the Diffie-Hellman protocol which may be implemented using the Jacobians of these curves. The other reason why there is an inherent interest in these objects is purely mathematical. Namely, we will show that for the class of curves that we are dealing with, there is a connection between the Jacobian of the curve and a certain ideal class group. Thus, we will be working with objects that are of fundamental interest in two different areas of mathematics.; There are two main components to this thesis. The first addresses how hard it is to solve the discrete logarithm problem in the Jacobian of hyperelliptic curves. Adleman, DeMarrais and Huang showed that, for hyperelliptic curves over prime fields of odd characteristic with sufficiently high genus, solving the discrete logarithm problem is of subexponential complexity. We will show in this thesis that their algorithm may be adapted to work for a much larger class of curves. In particular, we may remove all restrictions on the finite field, thus limiting the genera of the hyperelliptic curves amenable for cryptographic applications.; The second part of the thesis is focussed on developing an explicit arithmetic for the Jacobian of certain cubic superelliptic curves. We restrict our attention to curves of the form y3 = f( x). Assuming that f(x) is monic with no repeated roots and that our field does not have characteristic 3, we are able to show that the Jacobian of this curve is isomorphic to the ideal class group of K[C], the ring of regular functions on C. By exploiting the structure of ideals in K[C] as K[x] modules, we are able to produce a very efficient algorithm for performing group operations in the Jacobian which heuristically should take 46g 2 operations in the finite field K. |
