Constructing elliptic curves over finite fields with prescribed torsion
Andrew Sutherland · Mathematics of Computation · 2011
We present a method for constructing optimized equations for the modular curve X 1 ( N ) X_1(N) using a local search algorithm on a suitably defined graph of birationally equivalent plane curves. We then apply these equations over a finite field F q \mathbb {F}_q to efficiently generate elliptic curves with nontrivial N N -torsion by searching for affine points on X 1 ( N ) ( F q ) X_1(N)(\mathbb {F}_q) , and we give a fast method for generating curves with (or without) a point of order 4 N 4N using X 1 ( 2 N ) X_1(2N) .