Elliptic curve lifting problem and its applications
This paper considers a possible way of solving the elliptic curve discrete logarithm problem, a problem upon which a number of cryptographic systems are based. The basic idea of the paper is to lift an elliptic curve over a finite field, together with some points on it, to the rationals. Then, by finding a linear combination between the lifted points, assuming the lifted points are independent in the Mordell-Weil group, one can hopefully solve the original discrete logarithm problem. This idea was independently discovered by \textit{J. H. Silverman} [Des. Codes Cryptography 20, 5-40 (2000; Zbl 0948.11051)]. Silverman's paper contains a number of refinements on the method of the current paper. However in a later paper \textit{M. J. Jacobson} et. al., Des. Codes Cryptography 20, 41-64 (2000; Zbl 0948.11049)] the method is shown not to be a threat to elliptic curve cryptosystems. However, the method (dubbed the Xedni calculus by Silverman) does contain some interesting mathematical ideas.
scientific article; zbMATH DE number 1443391
scientific article