Generalized Jacobian and Discrete Logarithm Problem on Elliptic Curves

Let E be an elliptic curve over the finite field Fq, P a point in E(Fq) of order n, and Q a point in the group generated by P. The discrete logarithm problem on E is to find the number k such that Q = kP. In this paper we reduce the discrete logarithm problem on E[n] to the discrete logarithm on the group F?q, the multiplicative group of nonzero elements of Fq, in the case where n | q ? 1, using generalized jacobian of E.