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. 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 K {displaystyle K} be a field and consider an elliptic curve E {displaystyle E} in thefollowing special case of Weierstrass form over K {displaystyle K} : where the curve has discriminant Δ = ( a 3 3 ( a 1 3 − 27 a 3 ) ) = a 3 3 δ . {displaystyle Delta =(a_{3}^{3}(a_{1}^{3}-27a_{3}))=a_{3}^{3}delta .} Then the point P = ( 0 , 0 ) {displaystyle P=(0,0)} has order 3. To prove that P = ( 0 , 0 ) {displaystyle P=(0,0)} has order 3, note that the tangent to E {displaystyle E} at P {displaystyle P} is the line Y = 0 {displaystyle Y=0} which intersects E {displaystyle E} with multiplicity 3 at P {displaystyle P} . Conversely,given a point P {displaystyle P} of order 3 on an elliptic curve E {displaystyle E} both defined over a field K {displaystyle K} one can put the curve into Weierstrassform with P = ( 0 , 0 ) {displaystyle P=(0,0)} so that the tangent at P {displaystyle P} is the line Y = 0 {displaystyle Y=0} . Then the equation of the curve is Y 2 + a 1 X Y + a 3 Y = X 3 {displaystyle Y^{2}+a_{1}XY+a_{3}Y=X^{3}} with a 1 , a 3 ∈ K {displaystyle a_{1},a_{3}in K} . Now, to obtain the Hessian curve, it is necessary to do the following transformation: First let μ {displaystyle mu } denote a root of the polynomial