Tanja Lange

Spotlight on the transformationCurve x 2 + y 2 = c 2 (1 + dx 2 y 2 ) in Edwards form is birationallyequivalent to curveE : (1/e)v 2 = u 3 + (4/e − 2)u 2 + uin Montgomery form, where e = 1 − dc 4 .Let (x 1 , y 1 ) + (x 2 , y 2 ) = (x 3 , y 3 ) on Edwards curve. PutP i = ∞ if (x i , y i ) = (0, c);P i = (0, 0) if (x i , y i ) = (0, −c);P i = (u i , v i ) if x i ≠ 0, where u i = (c + y i )/(c − y i ) andv i = 2c(c + y i )/(c − y i )x i .Then P i ∈ E(k) and P 1 + P 2 = P 3 .D. J. Bernstein & T. Lange http://hyperelliptic.org/tanja/newelliptic – p. 18