Choosing the correct elliptic curve in the CM method

We give an elementary way to distinguish between the twists of an ordinary elliptic curve E over Fp in order to identify the one with p+1-2U points, when p = U2 + dV 2 with 2U, 2V Z and E is constructed using the CM method for finding elliptic curves with a prescribed number of points. Our algorithms consist in most cases of reading off simple congruence conditions on U and V modulo 4.

• Type: Journal
• Year: 2010
• Where: MOC
• Authors: Karl Rubin, Alice Silverberg
• Keywords: Elliptic Curves, MOC 2010, Ordinary Elliptic Curve, Security Privacy, Simple Congruence Conditions