Hessian form of an elliptic curve

In geometry, the Hessian curve is a plane curve similar to folium of Descartes. It is named after the German mathematician Otto Hesse. This curve was suggested for application in elliptic curve cryptography, because arithmetic in this curve representation is faster and needs less memory than arithmetic in standard Weierstrass form. Let be a field and consider an elliptic curve in the following special case of Weierstrass form over : where the curve has discriminant Then the point has order 3. To prove that has order 3, note that the tangent to at is the line which intersects with multiplicity 3 at . Conversely, given a point of order 3 on an elliptic curve both defined over a field one can put the curve into Weierstrass form with so that the tangent at is the line . Then the equation of the curve is with . To obtain the Hessian curve, it is necessary to do the following transformation: First let denote a root of the polynomial Then Note that if has a finite field of order , then every element of has a unique cube root; in general, lies in an extension field of K. Now by defining the following value another curve, C, is obtained, that is birationally equivalent to E: which is called cubic Hessian form (in projective coordinates) in the affine plane (satisfying and ). Furthermore, (otherwise, the curve would be singular). Starting from the Hessian curve, a birationally equivalent Weierstrass equation is given by under the transformations: and where: and It is interesting to analyze the group law of the elliptic curve, defining the addition and doubling formulas (because the SPA and DPA attacks are based on the running time of these operations). Furthermore, in this case, we only need to use the same procedure to compute the addition, doubling or subtraction of points to get efficient results, as said above. In general, the group law is defined in the following way: if three points lie in the same line then they sum up to zero. So, by this property, the group laws are different for every curve.