We prove a formula conjectured by Ahrens, Gordon, and McMa-hon for the number of interior points for a point connguration in R d. Our method is to show that the formula can be interpreted as a sum of Euler characteristics of certain complexes associated with the point connguration, and then compute the homology of these complexes. This method extends to other examples of convex geometries. We s...

Given a finite planar point set, we consider all possible spanning cycles whose straight line realizations are crossing-free – such cycles are also called simple polygonizations – and we are interested in the number of such simple polygonizations a set of N points can have. While the minumum number over all point configurations is easy to obtain – this is 1 for points in convex position –, the ...

Given a set S of n points in the plane, we compute in time O(n) the total number of convex polygons whose vertices are a subset of S. We give an O(m n) algorithm for computing the number of convex k-gons with vertices in S, for all values k = 3; : : : ;m; previously known bounds were exponential (O(ndk=2e)). We also compute the number of empty convex polygons (resp., k-gons, k m) with vertices ...

We explain the application of harmonic analysis to count lattice points in large regions. We also present some of our recent results in the three-dimensional case.

For counting points of jacobians of genus 2 curves over a large prime field, the best known approach is essentially an extension of Schoof’s genus 1 algorithm. We propose various practical improvements to this method and illustrate them with a large scale computation: we counted hundreds of curves, until one was found that is suitable for cryptographic use, with a state-ofthe-art security level...

Let E be a hyperelliptic curve of genus g over a finite field of degree n and small characteristic. Using deformation theory we present an algorithm that computes the zeta function of E in time essentially cubic in n and quadratic memory. This improves substantially upon Kedlaya’s result which has the same time asymptotic, but requires cubic memory size. AMS (MOS) Subject Classification Codes: ...

One of the fundamental results of descriptive complexity theory, due to Immerman [12] and Vardi [17], says that a class of ordered finite structures is definable in fixed-point logic if, and only if, it is computable in polynomial time. Much effort has been spent on the problem of capturing polynomial time, that is, describing all polynomial time computable classes of not necessarily ordered fi...

We are now ready to prove Hasse's theorem.

This series of three lectures of one hour each was preceded by two introductory lectures by Henk van Tilborg about applications of discrete logarithms (in multiplicative groups as well as elliptic curves) to cryptography. Those introductory lectures should now serve as motivation for the coming three lectures. Large cyclic subgroups of prime order in elliptic curves or in Jacobians of higher ge...

I consider the problem of computing the zeta function of an algebraic variety defined over a finite field. This problem has been pushed into the limelight in recent years because of its importance in cryptography, at least in the case of curves. Wan’s excellent survey article gives an overview of what has been achieved, and what remains to be done, on the topic [15]. The purpose of this exposit...