Symmetry Arguments for Edwards Elliptic Curves

We revisit Edwards normal form for elliptic curves, and show how elementary symmetry arguments lead to a more general expression for his addition formula on these curves. Using the free parameter in our formula, we describe Edwards addition via a construction like the secant and tangent method for cubics but with rectilinear hyperbolas replacing lines; consequently, addition on the real elliptic curves known as Cassinians may be interpreted via intersections with circles—as will be graphically illustrated here. Some of our results resemble those appearing in the literature on elliptic curve cryptology; the similarities will be explained in the last two sections. But our treatment differs substantially from earlier papers and sheds new light on Edwards normal form.