We reprove the result of Stone and DeRose, which gives the geometric classification of the affine type of an untrimmed Bézier curve, using classical algebraic geometry. We show how to derive the characterization of Stone and DeRose from three classical results: Bézout theorem, polynomial parametrization criterion and classification of the singularity type of an algebraic curve given in Weierstr...

I A p.Adic Cohomological Melhod for the i Weierstrass Family and Its Zeta Invariants I I GOROKATO Dedicated 10 my parents. ABSTRACT. Arter a survey of the Weierstrass family and cohomology, we compute lhe lifted homology of lhe Weierstrass family wilh compact supports so that explicit formulae for the zeta function of each fibre of the Weierstrass family may be obtained. The (co-)homology theor...

We propose two optimal representations for the elements of trace zero subgroups of twisted Edwards curves. For both representations, we provide efficient compression and decompression algorithms. The efficiency of the algorithm is compared with the efficiency of similar algorithms on elliptic curves in Weierstrass form.

We prove that elements of the Weierstrass gap set of a pair of points may be used to define a geometric Goppa code which has minimum distance greater than the usual lower bound. We determine the Weierstrass gap set of a pair of any two Weierstrass points on a Hermitian curve and use this to increase the lower bound on the minimum distance of particular codes defined using a linear combination o...

We present an overview of the development of the irrational numbers due to Karl Weierstrass. This construction was first presented during lectures in the 1860s in Berlin. Weierstrass never published his construction. Several of his students (Kossak, Horwitz, von Dantscher and Pincherle, to name a few) gave accounts in lecture notes from the courses. However, these notes were not published under...

We study the Abel differential equations that admits either a generalized Weierstrass first integral or a generalized Weierstrass inverse integrating factor.

Let X denote a compact Riemann surface of genus g > 1. At each point P~X, there is a sequence of g integers, l=71(P)<72(P)<. . .<Tg(P)<2g , called the Weierstrass gap sequence at P. A positive integer ~/is a Weierstrass gap at P if there exists a holomorphic differential on X with a zero of order 7 1 at P. A positive integer which is not a gap at P is called a nongap at P. The point P is called...

Elliptic Curve Cryptosystems (ECCs) are utilized as an alternative to traditional publickey cryptosystems, and are more suitable for resource limited environments due to smaller parameter size. In this dissertation we carry out a thorough investigation of side-channel attack aware ECC implementations over finite fields of prime characteristic including the recently introduced Edwards formulatio...