J.-P. Cherdieu, J. Estrada, E. Reinaldo
Efficient Reduction on the Jacobian Variety of Picard curves
Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976)

MSC:

14H45 Special curves and curves of low genus

14H40 Jacobians, See also {32G20}

14H05 Algebraic functions; function fields, See also {11R58}

14Q05 Curves

14Q20 Effectivity

11G10 Abelian varieties of dimension $\gtr 1$, See also {14Kxx}

11T71 Algebraic coding theory; cryptography

Abstract: In this paper, a system of coordinates for the elements on the Jacobian Variety of Picard curves is presented. These coordinates possess a nice geometric interpretation and provide us with an unifying environment to obtain an explicit structure of abelian variety for the Jacobian, as well as an efficient algorithm for the reduction and addition of divisors. Exploiting the geometry of the Picard curves, a completely effective reduction algorithm is developed, which works for curves defined over any ground field $k$, with $char(k)=0$ or $char(k)\neq 3$. In the generic case, the algorithm works recursively with the system of coordinates representing the divisors, instead of solving for points in their support. Hence, only one factorization is needed (at the end of the algorithm) and the processing of the system of coordinates involves only linear algebra and evaluation of polynomials in the definition field of the divisor $D$ to be reduced. The complexity of this deterministic reduction algorithm is $O(deg(D))$ . The addition of divisors may be performed iterating the reduction algorithm.
Keywords: Picard curves, Jacobian Varieties, Addition Law, Discrete logarithm