An addition algorithm in Jacobian of Cab curves

Nowadays, elliptic curve cryptosystems receive attention and much effort is being dedicated to make it more and more practical. It is worthwhile to construct discrete logarithm based cryptosystems using more general algebraic curves, because it supplies more security sources for public key cryptosystems. The presented paper introduces Cab curves. Roughly speaking, a curve is Cab if it is non-singular in its affine part and if its singularity at infinity is “nice”. Cab curves compose a large family of algebraic curves, including elliptic, hyperelliptic and superelliptic curves. The paper shows an addition algorithm in Jacobian group of Cab curves in three steps: firstly with a geometrical point of view, which is impractical, secondly by translating the algorithm in the language of ideals, and finally, the final algorithm in which some costly steps are removed. The paper also gives experiments that prove that the algorithm behaves well in practice.