Fast Arithmetic on Jacobians of Picard Curves [chapter]

Stéphane Flon, Roger Oyono
2004 Lecture Notes in Computer Science  
In this paper we present a fast addition algorithm in the Jacobian of a Picard curve over a finite field Fq of characteristic different from 3. This algorithm has a nice geometric interpretation, comparable to the classic "chord and tangent" law for the elliptic curves. Computational cost for addition is 144M +12SQ+2I and 158M +16SQ+2I for doubling. In this article, we find explicit formulae for computing in the Jacobian of a Picard curve, basing us on some geometric aspects of these curves.
more » ... check [23], Huang and Ierardi [10] already proposed general methods for computing in the Jacobians of arbitrary algebraic curves. These algorithms are not practical from a computational point of view though, and in addition they need to extend the base field. Hess' paper [9] is closer to our geometrical point of view, in such as it provides an explicit version of Riemann-Roch theorem (see also [8] ).
doi:10.1007/978-3-540-24632-9_5 fatcat:ger33m5xabblbai5pak3bg27ma