Efficient algorithms for the Riemann-Roch problem and for addition in the Jacobian of a curve
Journal of Symbolic Computation
Computing Riemann---Roch spaces in algebraic function fields and related topics
Journal of Symbolic Computation
Computing in the jacobian of a plane algebraic curve
ANTS-I Proceedings of the First International Symposium on Algorithmic Number Theory
ANTS-I Proceedings of the First International Symposium on Algorithmic Number Theory
Fast Jacobian Group Arithmetic on CabCurves
ANTS-IV Proceedings of the 4th International Symposium on Algorithmic Number Theory
Arithmetic on superelliptic curves
Mathematics of Computation
An index calculus algorithm for plane curves of small degree
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
Group law computations on jacobians of hyperelliptic curves
SAC'11 Proceedings of the 18th international conference on Selected Areas in Cryptography
Halving for the 2-Sylow subgroup of genus 2 curves over binary fields
Finite Fields and Their Applications
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We use an embedding of the symmetric dth power of any algebraic curve C of genus g into a Grassmannian space to give algorithms for working with divisors on C, using only linear algebra in vector spaces of dimension O(g), and matrices of size O(g2) × O(g). When the base field k is finite, or if C has a rational point over k, these give algorithms for working on the Jacobian of C that require O(g4) field operations, arising from the Gaussian elimination. Our point of view is strongly geometric, and our representation of points on the Jacobian is fairly simple to deal with; in particular, none of our algorithms involves arithmetic with polynomials. We note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in Florian Hess' 1999 Ph.D. thesis, which works with function fields as extensions of k[x]. However, for special classes of curves, Hess' algorithms are asymptotically more efficient than ours, generalizing other known efficient algorithms for special classes of curves, such as hyperelliptic curves (Cantor 1987), superelliptic curves (Galbraith, Paulus, and Smart 2002), and Cab curves (Harasawa and Suzuki 2000); in all those cases, one can attain a complexity of O(g2).