# What is the correct elliptic curve representation?

I'm studing the basics of elliptic curve from various resource some more mathematical someone more practical.

I know that the equation where the elliptic curves come from is the Weistraß equation

$$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$

After the trasformation $$y\rightarrow y-(a_1x+a_3)/2$$ I get the isomorphic curve : $$y^2=x^3+\frac{b_2}{4}x^2+\frac{b_4}{2}x+\frac{b_6}{4}$$ (the coefficent $$b_2,b_4,b_6$$ are a combination of the previous one).

After defining the equation I've started reading the group law and proceeding further I found out other kind of curves with different group laws(the formulas for calculating the addition and the double of a point are different).

In the end now I know that there exists three kind of curves:

1. Supersingular curves

$$y^2+cy=x^3+ax+b$$

2. Non supersingular curves

$$y^2+xy=x^3+ax+b$$

3. Ordinary curves

$$y^2=x^3+ax+b$$

### Questions

• What are the difference between this curves?
• Why are EC defined only on ordinary curves? What is their advantage?
• What is the correct elliptic curve representation?

Reading online seems that supersingular curves require more bit (to avoid this) so that's why they are not used. Is this true? What about non supersingular curves?

In general case, each elliptic curve over field $K$ can be defined by the following equation: $$y^2+a_1 xy+a_3 y=x^3+a_2 x^2+a_4 x+a_6$$ If the characteristic of field be larger than $3$, then all elliptic curves over it are isomorphic to $$y^2=x^3+ax+b$$ In the case that characteristic is $3$ or $2$, we have two form of isomorphic curves which are as follows respectively. $$y^2=x^3+ax+b$$ $$y^2=x^3+ax^2+b$$ $$y^2+cy=x^3+ax+b$$ $$y^2+xy=x^3+ax^2+b$$ Edit: The representation of elliptic curve depends on your goal and the field. Elliptic curves are not defined only on ordinary curves. As an example, we have Koblitz curves (which are non-supersingular) that are a class of computationally efficient elliptic curves. Several of these curves are recommended for cryptographic use by NIST. Supersingular curves are useful for cryptographic goal, too. In the 1,2,3,4 you can find several details.
• @Ofey Well, this does seem to answer What are the difference between this curves? Commented Nov 11, 2017 at 12:07