Elliptic Curves in Montgomery Form with B=1 and Their Low Order Torsion

This paper proves that the Montgomery form elliptic curves that are cheaply transformable into short Weierstrass form by a simple change of variables (x, y) 7→ (x+α, y) (instead of a more general affine change of variables) are precisely the curves with B = 1. The points of order 2 and 4 on these curves are described, and it is observed that the x-coordinates of these points are consecutive field elements. Finally, it is shown that two elliptic curves specified (in short Weierstrass form) in the SECG standard can be transformed into B = 1 Montgomery form, and also into Edwards form.