Rethinking low genus hyperelliptic Jacobian arithmetic over binary fields: Interplay of field arithmetic and explicit formulæ

In this paper, we present several improvements on the best known explicit formulæ for hyperelliptic curves of genus three and four in characteristic two, including the issue of reducing memory requirements. To show the effectiveness of these improvements and to allow a fair comparison of the curves of different genera, we implement all formulæ using a highly optimized software library for arithmetic in binary fields. This library was designed to minimize the impact of a whole series of overheads which have a larger significance as the genus of the curves increases. The current state of the art in attacks against the discrete logarithm problem is taken into account for the choice of the field and group sizes. Performance tests are done on two personal computers with very different architectures. Our results can be shortly summarized as follows: Curves of genus three provide performance similar, or better, to that of curves of genus two, and these two types of curves can perform faster than elliptic curves - indeed on some processors often twice as fast. Curves of genus four attain a performance level comparable to elliptic curves. A large choice of curves is therefore available for the deployment of curve-based cryptography, with curves of genus three and four providing their own advantages as larger cofactors can be allowed for the group order.