476 research outputs found
Computing Zeta Functions of Hyperelliptic Curves over Finite Fields of Characteristic 2
We present an algorithm for computing the zeta function of an arbitrary hyperelliptic curve over a finite field Fq of characteristic 2, thereby extending the algorithm of Kedlaya for small odd characteristic. For a genus g hyperelliptic curve over n , the asymptotic running time of the algorithm is O(g ) and the space complexity is O(g )
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
Computing zeta functions of arithmetic schemes
We present new algorithms for computing zeta functions of algebraic varieties
over finite fields. In particular, let X be an arithmetic scheme (scheme of
finite type over Z), and for a prime p let zeta_{X_p}(s) be the local factor of
its zeta function. We present an algorithm that computes zeta_{X_p}(s) for a
single prime p in time p^(1/2+o(1)), and another algorithm that computes
zeta_{X_p}(s) for all primes p < N in time N (log N)^(3+o(1)). These generalise
previous results of the author from hyperelliptic curves to completely
arbitrary varieties.Comment: 23 pages, to appear in the Proceedings of the London Mathematical
Societ
An extension of Kedlaya's algorithm for hyperelliptic curves
In this paper we describe a generalisation and adaptation of Kedlaya's
algorithm for computing the zeta function of a hyperelliptic curve over a
finite field of odd characteristic that the author used for the implementation
of the algorithm in the Magma library. We generalise the algorithm to the case
of an even degree model. We also analyse the adaptation of working with the
rather than the differential basis. This basis has the
computational advantage of always leading to an integral transformation matrix
whereas the latter fails to in small genus cases. There are some theoretical
subtleties that arise in the even degree case where the two differential bases
actually lead to different redundant eigenvalues that must be discarded.Comment: v3: some minor changes and addition of a reference to a paper by Theo
van den Bogaar
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