On the Decomposition of an Element of Jacobian of a Hyperelliptic Curve

In this manuscript, if a reduced divisor D0 of hyperelliptic curve of genus g over an extension field Fqn is written by a linear sum of ng elements of Fqn -rational points of the hyperelliptic curve whose x-coordinates are in the base field Fq, D0 is noted by a decomposed divisor and the set of such Fqn -rational points is noted by the decomposed factor of D0. We propose an algorithm which checks whether a reduced divisor is decomposed or not, and compute the decomposed factor, if it is decomposed. This algorithm needs a process for solving equations system of degree 2, (n − n)g variables, and (n − n)g equations over Fq. Further, for the cases (g, n) = (1, 3), (2, 2), and (3, 2), the concrete computations of decomposed factors are done by computer experiments.