## Cryptology ePrint Archive: Report 2013/660

Discrete Logarithms and Mordell-Weil Groups

Abstract: Let $E_p$ be an elliptic curve over a prime finite field $\Fp$, $p\ge5$, and $P_p,Q_p\in E_p(\Fp)$. The elliptic curve discrete logarithm problem, ECDLP, on $E_p$ is to find $m_p\in\mathbb{F}_p^{\times}$ such that $Q_p=m_p P_p$ if $Q_p\in\langle P_p\rangle$. We propose an algorithm to attack the ECDLP relying on a Hasse principle detecting linear dependence in Mordell-Weil groups of elliptic curves via a finite number of reductions.