In this paper we present an extension of Kedlaya’s algorithm for computing the zeta function of an Artin-Schreier curve over a finite field Fq of characteristic 2. The algorithm has running time O(g log q) and needs O(g log q) storage space for a genus g curve. Our first implementation in MAGMA shows that one can now generate hyperelliptic curves suitable for cryptography in reasonable time. We...

We present an algorithm which speeds scalar multiplication on a general elliptic curve by an estimated 3.8% to 8.5% over the best known general methods when using affine coordinates. This is achieved by eliminating a field multiplication when we compute 2P+Q from given points P , Q on the curve. We give applications to simultaneous multiple scalar multiplication and to the Elliptic Curve Method...

We present an algorithm that speeds exponentiation on a general elliptic curve by an estimated 3.8% to 8.5% over the best known general exponentiation methods when using affine coordinates. This is achieved by eliminating a field multiplication when we compute 2P + Q from given points P , Q on the curve. We give applications to simultaneous multiple exponentiation and to the Elliptic Curve Meth...

We consider a plane polynomial vector field P(x, y) dx +Q(x, y) dy of degree m > 1. With each algebraic invariant curve of such a field we associate a compact Riemann surface with the meromorphic differential ω = dx/P = dy/Q. The asymptotic estimate of the degree of an arbitrary algebraic invariant curve is found. In the smooth case this estimate has already been found by Cerveau and Lins Neto ...

Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, ρE : Gal(k/k) → GL2(b Z). For a fixed number field k, we describe the image of ρE for a “random” elliptic curve E over k. In particular, if k 6= Q is linearly disjoint from the cyclotomic extension of Q, then ρE will be surjective for “most” elliptic curves over k.

Let C/Q be a curve of genus three, given as a double cover of a plane conic. Such a curve is hyperelliptic over the algebraic closure of Q, but may not have a hyperelliptic model of the usual form over Q. We describe an algorithm that computes the local zeta functions of C at all odd primes of good reduction up to a prescribed bound N . The algorithm relies on an adaptation of the ‘accumulating...

We give an algorithm that constructs, on input of a prime power q and an integer t, a supersingular elliptic curve over Fq with trace of Frobenius t in case such a curve exists. If GRH holds true, the expected run time of our algorithm is e O((log q)). We illustrate the algorithm by showing how to construct supersingular curves of prime order. Such curves can readily be used for pairing based c...

Let P be a polygonal curve in R of length n, and S be a point set of size k. We consider the problem of finding a polygonal curve Q on S such that all points in S are visited and the Fréchet distance between Q and P is at most a given ε. We show that this problem is NP-complete, regardless of whether or not the points of S are allowed to be visited more than once.