Elliptic Curve Lifting Problem and Its Applications
Jung Hee Cheon, Sang Geun Hahn, Hwan Joon Kim · Proceedings of the Japan Academy Series A Mathematical Sciences · 1999
In this paper, we introduce a new method to solve the elliptic curve discrete logarithm probelm (ECDLP) over a finite field by using the elliptic curve lifting problem. Moreover, we propose to find a non-trivial point in E1(Q) in order to get a lifted elliptic curve with rank smaller than the number of lifted points. By this method, we conclude that finding a non-trivial point in E1(Q) implies solving the ECDLP, the discrete logarithm problem (DLP) and the integer factorization problem (IFP). Finally, we find that the minimum of canonical height of a point in E1(Q) is almost O(|Ẽ(Fp)|), which means that it is too large to be found by the brute force search.