Fast Jacobian Group Operations for C_3,4 Curves over a Large Finite Field

Abstract

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 Mathematics of Computation). 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 C3,4 curves of genus 3. We show that, typically, the addition of two distinct elements in the Jacobian of a C3,4 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.Remark (added August 22, 2007): A revised version of this article has been published as LMS J. Comput. Math. 10 (2007) 307-328 with an appendix of sample Magma code of our algorithms. The URL for the published version is:

On Jacobian group arithmetic for typical divisors on curves

“…As a result of our current investigations, we discovered along the way a nontrivial speedup of the algorithms of [1], that saves 19 multiplications in K per operation in the Jacobian, a speedup of approximately 15%. This is described in the Appendix.…”

“…In Proposition 2.1 and Eq. (2) on p. 310 of [1], we asserted that the ideal I D was "typically" generated by two elements…”

“…Let R be the coordinate ring of the affine curve C − {P ∞ }; then the group of K-rational points of the Jacobian of C can be identified with the ideal class group of the Dedekind domain R. A previous series of articles (by three different groups of authors) on certain degree 3 covers of P 1 , particularly C 3,4 curves [1][2][3][4][5]9], gives explicit formulas for group arithmetic in the Jacobian of C when K is a large finite field, under a certain genericity assumption on the divisors whose classes are being added in the Jacobian. This genericity assumption was first introduced in [3], where such divisors were called "typical"; all the articles above give fast algorithms for Jacobian group arithmetic under the hypothesis that the divisors (also, in fact, the pairs of divisors) one encounters are typical, and that the result of the group operation is typical.…”

“…In this article, we give a straightforward explicit condition for a divisor to be typical, for arbitrary C. For the C 3,4 case, we show that the algorithms in [1], which involve two inversions (and approximately 125 multiplications) in K per group operation in the Jacobian of a C 3,4 curve, give correct results and yield typical divisors as output, provided that both inversions can be carried out, i.e., that one encounters nonzero elements of K at those two moments. Our general criterion for typicality can be expressed in terms of the rank of certain matrices, or equivalently in terms of the structure of suitable Gröbner bases for the ideal I D and its first syzygies.…”

“…Our general criterion for typicality can be expressed in terms of the rank of certain matrices, or equivalently in terms of the structure of suitable Gröbner bases for the ideal I D and its first syzygies. For an arbitrary curve C with distinguished point P ∞ , we describe a modification of the algorithms from [1], that allows us to carry out Jacobian arithmetic for typical elements. To our knowledge, this is the first set of algorithms for typical divisors on curves that (1) works with a precise definition of typicality (and a weaker notion of semi-typicality), (2) gives necessary and sufficient conditions on the typicality of the input and/or output divisors to guarantee success of the algorithms, and (3) can certify that the end result of one's operations is correct upon success, or else identify non-typical divisors encountered in the computation.…”