Reduktions- und Additionsverfahren für die Jacobische Varietät von Kurvenfamilien mit kryptischen Anwendungen

Abstract

In this paper we represent a reduction and addition algorithm for non hyperelliptic curves of genus 3. Our aim is to give an explicit representation of the group law in the jacobian varieties of curves belonging to the families $y^4=p_4(x)$ and $y^4=p_3(x)$. The idea of the algorithm is an extension of the geometric addition of points on elliptic curves. Because of the complexity of the curve structure we work hier with conics instead of chords and tangents as in the genus one case. In the construction of the algorithm we use the so called coordinate form of the divisor. The coordinate form of a divisor is a unique set of three polynomials, which we use for the computations. All computations succeed only with linear algebra knowledges. At the end of the algorithm we need one factorisation so that the result can be defined over a finite extension field. The reduction and addition ist constructed iterativ and can be applied in efficient way to divisors of every degree.