Give L an algebraic extension of k(x) where x is an indeterminate, we have defined a lattice Val(L, k) which is a point-free description of the Riemann surface X associated to L/k. (The topology is such that the open correspond to the cofinite sets. Furthermore there is an extra point which corresponds to the trivial valuation ring L.) We present here a fundamental algorithm which of L enumerat...

Algebraic curves over finite fields are being extensively used in the design of public-key cryptographic schemes. This paper surveys some topics in algebraic curve cryptography, with an emphasis on recent developments in algorithms for the elliptic and hyperelliptic curve discrete logarithm problems, and computational problems in pairing-based cryptography.

We study real algebraic plane curves, at an elementary level, using as little algebra as possible. Both cases, affine and projective, are addressed. A real curve is infinite, finite or empty according to the fact that a minimal polynomial for the curve is indefinite, semi–definite nondefinite or definite. We present a discussion about isolated points. By means of the operator, these points can ...

1.1. Languages. Let Σ be an alphabet i.e. a finite set of symbols. The set of finite words on Σ is denoted Σ∗. A language is a subset of Σ∗. For a language L ⊂ Σ∗ the associated decision problem is as follows. On input a word w ∈ Σ∗ we want to decide if w belongs to L or not. Examples : even integers, prime integers, etc. For a function f : Σ∗ → Σ∗, the associated functional problem is as follo...

We present a variety of computational techniques dealing with algebraic curves both in the plane and in space. Our main results are polynomial time algorithms (1) to compute the genus of plane algebraic curves, (2) to compute the rational parametric equations for implicitly defined rational plane algebraic curves of arbitrary degree, (3) to compute birational mappings between points on irreduci...

The Bentley Ottmann sweep line algorithm is a standard tool to compute the arrangement of algebraic curves in the plane. If degenerate positions are not excluded from the input, variants of this algorithm must, among other things, handle k ≥ 2 curves intersecting simultaneously in a single intersection point. In that situation, the algorithm knows the order of the curves immediately left of the...

If algebraic varieties like curves or surfaces are to be manipulated by computers, it is essential to be able to represent these geometric objects in an appropriate way. For some applications an implicit representation by algebraic equations is desirable, whereas for others an explicit or parametric representation is more suitable. Therefore, transformation algorithms from one representation to...

Definition Let K be a field. An elliptic curve over K is a pair (X, O) where X is a smooth absolutely integral proper curve over K with genus one and O is a K-rational point on X. Terminology (see Liu [Liu] and Stichtenoth [Sti]) A curve over K is an equidimensional variety over K of dimension one. A variety over K is a separated scheme of finite type over Spec(K). Examples A triangle with equa...