Fields of definition of torsion points on the Jacobians of genus 2 hyperelliptic curves over finite fields
نویسندگان
چکیده
This paper deals with fields of definition of the l-torsion points on the Jacobians of genus 2 hyperelliptic curves over finite fields in order to speed Gaudry and Schost’s point counting algorithm for genus 2 hyperelliptic curves up. A result in this paper shows that the extension degrees of the fields of difinition of the l-torsion points can be in O(l) instead of O(l). The effects of the result on the point counting algorithm are also discussed in this paper. The discussion concludes that the result in this paper reduces the complexity of the algorithm over Fq to O((log q)) operations in Fq.
منابع مشابه
Counting Points on Hyperelliptic Curves over Finite Fields
We describe some algorithms for computing the cardinality of hyperelliptic curves and their Jacobians over finite fields. They include several methods for obtaining the result modulo small primes and prime powers, in particular an algorithm à la Schoof for genus 2 using Cantor’s division polynomials. These are combined with a birthday paradox algorithm to calculate the cardinality. Our methods ...
متن کامل0 The Number of Rational Points On Genus 4 Hyperelliptic Supersingular Curves
One of the big questions in the area of curves over finite fields concerns the distribution of the numbers of points: Which numbers occur as the number of points on a curve of genus g? The same question can be asked of various subclasses of curves. In this article we classify the possibilities for the number of points on genus 4 hyperelliptic supersingular curves over finite fields of order 2, ...
متن کاملConstructing pairing-friendly hyperelliptic curves using Weil restriction
A pairing-friendly curve is a curve over a finite field whose Jacobian has small embedding degree with respect to a large prime-order subgroup. In this paper we construct pairing-friendly genus 2 curves over finite fields Fq whose Jacobians are ordinary and simple, but not absolutely simple. We show that constructing such curves is equivalent to constructing elliptic curves over Fq that become ...
متن کاملComputing discrete logarithms in high-genus hyperelliptic Jacobians in provably subexponential time
We provide a subexponential algorithm for solving the discrete logarithm problem in Jacobians of high-genus hyperelliptic curves over finite fields. Its expected running time for instances with genus g and underlying finite field Fq satisfying g ≥ θ log q for a positive constant θ is given by
متن کاملComputing in Picard groups of projective curves over finite fields
We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that various desirable operations can be performed efficiently in this setting: decomposing divisors into prime divisors; computing pull-backs and push-forwards of divisors under finite mor...
متن کامل