585 research outputs found
A Generic Approach to Searching for Jacobians
We consider the problem of finding cryptographically suitable Jacobians. By
applying a probabilistic generic algorithm to compute the zeta functions of low
genus curves drawn from an arbitrary family, we can search for Jacobians
containing a large subgroup of prime order. For a suitable distribution of
curves, the complexity is subexponential in genus 2, and O(N^{1/12}) in genus
3. We give examples of genus 2 and genus 3 hyperelliptic curves over prime
fields with group orders over 180 bits in size, improving previous results. Our
approach is particularly effective over low-degree extension fields, where in
genus 2 we find Jacobians over F_{p^2) and trace zero varieties over F_{p^3}
with near-prime orders up to 372 bits in size. For p = 2^{61}-1, the average
time to find a group with 244-bit near-prime order is under an hour on a PC.Comment: 22 pages, to appear in Mathematics of Computatio
On Using Expansions to the Base of -2
This short note investigates the effects of using expansions to the base of
-2. The main applications we have in mind are cryptographic protocols, where
the crucial operation is computation of scalar multiples. For the recently
proposed groups arising from Picard curves this leads to a saving of at least
7% for the computation of an m-fold. For more general non-hyperelliptic genus 3
curves we expect a larger speed-up.Comment: 5 page
Pairings on hyperelliptic curves with a real model
We analyse the efficiency of pairing computations on hyperelliptic curves given by a real model using a balanced divisor at infinity. Several optimisations are proposed and analysed. Genus two curves given by a real model arise when considering pairing friendly groups of order dividing . We compare the performance of pairings on such groups in both elliptic and hyperelliptic versions. We conclude that pairings can be efficiently computable in real models of hyperelliptic curves
Computing canonical heights using arithmetic intersection theory
For several applications in the arithmetic of abelian varieties it is
important to compute canonical heights. Following Faltings and Hriljac, we show
how the canonical height on the Jacobian of a smooth projective curve can be
computed using arithmetic intersection theory on a regular model of the curve
in practice. In the case of hyperelliptic curves we present a complete
algorithm that has been implemented in Magma. Several examples are computed and
the behavior of the running time is discussed.Comment: 29 pages. Fixed typos and minor errors, restructured some sections.
Added new Example
Point counting on curves using a gonality preserving lift
We study the problem of lifting curves from finite fields to number fields in
a genus and gonality preserving way. More precisely, we sketch how this can be
done efficiently for curves of gonality at most four, with an in-depth
treatment of curves of genus at most five over finite fields of odd
characteristic, including an implementation in Magma. We then use such a lift
as input to an algorithm due to the second author for computing zeta functions
of curves over finite fields using -adic cohomology
- …