We give a deterministic polynomial-time algorithm that computes a nontrivial rational point on an elliptic curve over a finite field, given a Weierstrass equation for the curve. For this, we reduce the problem to the task of finding a rational point on a curve of genus zero.
Mathematics | Number Theory
Shallue, Andrew and van de Woestijne, Christiaan, "Construction of Rational Points on Elliptic Curves over Finite Fields" (2006). Scholarship. 72.