Generalized Jacobian and Discrete Logarithm Problem on Elliptic Curves

Abstract

Let E be an elliptic curve over the finite field Fq, P ap oint in E(Fq )o f ordern ,a ndQ a point in the group generated by P. The discrete logarithm problem on E is to find the number k such that Q = kP .I n this paper we reduce the discrete logarithm problem on E(n) to the discrete logarithm on the group F ∗, the multiplicative group of nonzero elements of Fq ,i n the case wheren | q − 1, using generalized jacobian of E.