Computing endomorphism rings of abelian varieties of dimension two

Generalizing a method of Sutherland and the author for elliptic curves, we design a subexponential algorithm for computing the endomorphism rings of ordinary abelian varieties of dimension two over finite fields. Although its correctness and complexity analysis rest on several assumptions, we report on practical computations showing that it performs very well and can easily handle previously intractable cases.Comment: 14 pages, 2 figure

Computing endomorphism rings of elliptic curves under the GRH

We design a probabilistic algorithm for computing endomorphism rings of ordinary elliptic curves defined over finite fields that we prove has a subexponential runtime in the size of the base field, assuming solely the generalized Riemann hypothesis. Additionally, we improve the asymptotic complexity of previously known, heuristic, subexponential methods by describing a faster isogeny-computing routine.Comment: 11 pages, 1 figur

Netting venture capital from a fishing village

In northern New England, equity capital has often been difficult for small- and medium-sized companies to obtain. Keith Bisson shares Coastal Enterprises, Inc.'s philosophy of community development venture capital and explains how others can get involved.Economic development - New England ; Venture capital ; Economic development - Maine ; Economic development - New Hampshire

On polarised class groups of orders in quartic CM-fields

We give an explicit necessary condition for pairs of orders in a quartic CM-field to have the same polarised class group. This generalises a simpler result for imaginary quadratic fields. We give an application of our results to computing endomorphism rings of abelian surfaces over finite fields, and we use our results to extend a completeness result of Murabayashi and Umegaki to a list of abelian surfaces over the rationals with complex multiplication by arbitrary orders.Comment: 19 pages, v2 strengthened results slightly and changed theorem numbering, v3 further strengthened results and added more details, v4 eased the presentation but changed notations and numbering, v5 updated references, v6 removes mistaken "transitivity" statemen

Constructing Permutation Rational Functions From Isogenies

A permutation rational function $f\in \mathbb{F}_q(x)$ is a rational function that induces a bijection on $\mathbb{F}_q$, that is, for all $y\in\mathbb{F}_q$ there exists exactly one $x\in\mathbb{F}_q$ such that $f(x)=y$. Permutation rational functions are intimately related to exceptional rational functions, and more generally exceptional covers of the projective line, of which they form the first important example. In this paper, we show how to efficiently generate many permutation rational functions over large finite fields using isogenies of elliptic curves, and discuss some cryptographic applications. Our algorithm is based on Fried's modular interpretation of certain dihedral exceptional covers of the projective line (Cont. Math., 1994)