611 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
Quasi-quadratic elliptic curve point counting using rigid cohomology
We present a deterministic algorithm that computes the zeta function of a
nonsupersingular elliptic curve E over a finite field with p^n elements in time
quasi-quadratic in n. An older algorithm having the same time complexity uses
the canonical lift of E, whereas our algorithm uses rigid cohomology combined
with a deformation approach. An implementation in small odd characteristic
turns out to give very good results.Comment: 14 page
A p-adic quasi-quadratic point counting algorithm
In this article we give an algorithm for the computation of the number of
rational points on the Jacobian variety of a generic ordinary hyperelliptic
curve defined over a finite field of cardinality with time complexity
and space complexity , where . In the latter
complexity estimate the genus and the characteristic are assumed as fixed. Our
algorithm forms a generalization of both, the AGM algorithm of J.-F. Mestre and
the canonical lifting method of T. Satoh. We canonically lift a certain
arithmetic invariant of the Jacobian of the hyperelliptic curve in terms of
theta constants. The theta null values are computed with respect to a
semi-canonical theta structure of level where is an integer
and p=\mathrm{char}(\F_q)>2. The results of this paper suggest a global
positive answer to the question whether there exists a quasi-quadratic time
algorithm for the computation of the number of rational points on a generic
ordinary abelian variety defined over a finite field.Comment: 32 page
Computing Hilbert Class Polynomials
We present and analyze two algorithms for computing the Hilbert class
polynomial . The first is a p-adic lifting algorithm for inert primes p
in the order of discriminant D < 0. The second is an improved Chinese remainder
algorithm which uses the class group action on CM-curves over finite fields.
Our run time analysis gives tighter bounds for the complexity of all known
algorithms for computing , and we show that all methods have comparable
run times
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