Mathematics – Number Theory
Scientific paper
2006-10-03
LMS J. Comput. Math. 10 (2007) 307-328, may be downloaded from http://www.lms.ac.uk/jcm/10/lms2006-049/
Mathematics
Number Theory
25 pages, identical to version 2 except for a remark about the published version of the article, which includes Magma code for
Scientific paper
Let C be an arbitrary smooth algebraic curve of genus g over a large finite field K. We revisit fast addition algorithms in the Jacobian of C due to Khuri-Makdisi (math.NT/0409209, to appear in Math. Comp.). The algorithms, which reduce to linear algebra in vector spaces of dimension O(g) once |K| >> g, and which asymptotically require O(g^{2.376}) field operations using fast linear algebra, are shown to perform efficiently even for certain low genus curves. Specifically, we provide explicit formulae for performing the group law on Jacobians of C_{3,4} curves of genus 3. We show that, typically, the addition of two distinct elements in the Jacobian of a C_{3,4} curve requires 117 multiplications and 2 inversions in K, and an element can be doubled using 129 multiplications and 2 inversions in K. This represents an improvement of approximately 20% over previous methods.
Abu Salem Fatima K.
Khuri-Makdisi Kamal
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