Hessian Form of An Elliptic Curve - Definition

Definition

Let be a field and let E denote an elliptic curve in Weierstrass form over . Then the following curve can be obtained:

where the curve has discriminant

and (0,0) has order 3. Before proving this, note that if the characteristic of, say q, is 2 modulo 3, then the curve has 3 points of order 3; and if q is 1 modulo 3, there are 8 points of order 3.

To prove that has order 3, it is sufficient to show that using the elliptic curve group law.

(i) Compute : if is the given curve and the point, then . So, in this case, since, then .

(ii) Compute : it can be done using the tangent and chord method, that is, first construct the line through the general point and find the other intersection point with the curve.

Let be the tangent to the curve at . Now, to find the points of intersection between the curve and the line, replace every by in the curve:

iff

The roots of this equation are the x-coordinates of and, so:

Then comparing the coefficients:

and ( and ).

Note that are known and by the implicit function theorem (which is the slope of the line ). Then, and P would be known. This is a general method to compute .

So applying it to P=(0,0):

since (note cannot be zero, otherwise the denominator would vanish at and the curve would be singular),

then where

Thus, , that is (0,0) has order 3 over .

Now, to obtain the Hessian curve, it is necessary to do the following transformation:

First let denote a root of the polynomial T3 − δT2 + δ2T/3 + a₃2 = 0.

where is determined from the formula:

=1/3((-27a₃2)1/3+)

Note that if has characteristic q ≡ 2 (mod 3), then every element of has a unique cube root; otherwise, it is necessary to consider 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, D3≠1 (otherwise, the curve would be singular)

Starting from the Hessian curve, a birationally equivalent Weierstrass equation is given by

under the transformations:

and

where:

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