Analysis of the Xedni calculus attack
Elliptic curves defined over finite fields provide secure cryptosystems that are currently included in several industrial products. A crucial question hence concerns the security of the elliptic curve discrete logarithm problem (ECDLP). The Xedni attack on ECDLP involves lifting points and the curve from the considered finite field to \({\mathbb Q}\). If the lifted points are linearly dependant, then the ECDLP is solved. This article describes in details the Xedni attack, and provides valuable background (Hasse-Weil \(L\)-function, Taniyama conjecture, conjecture of Birch and Swinnerton-Dyer, heights). It is shown that this attack is asymptotically unlikely succesful. Moreover, even for small base finite fields, this attack fails very often (how often is made precise in the article), so that the Xedni calculus is impractical for the fields used in elliptic curve cryptography.
scientific article; zbMATH DE number 1441591
scientific article