The lifting problem for the ECDLP and the Selmer rank (Q2879553)

Let \(K=\mathbb{F}_p\) be a finite prime field, \(E/K\) an elliptic curve, \(S\) and \(T\) points on \(E(K)\). The discrete logarithm problem is the problem to determine a \(d\) such that \(T=dS\), provided such a \(d\) exists. This problem can be solved efficiently if there is an elliptic curve \(\tilde{E}\) over \(\mathbb{Q}\) such that \(\tilde{E}(\mathbb{Q})\) has rank one and \(T\) and \(S\) are reductions modulo \(p\) of points on \(\tilde{E}(\mathbb{Q})\). This is the so-called rank one attack. However, constructing such a curve \(\tilde{E}\) is non-trivial. The authors argue that the current strategies to construct an \(\tilde{E}\) of rank one for given \((E,S,T)\) are not sufficiently good for the rank one attack to be efficient.