Affine Short Weierstrass form

Let $F$ be a finite field of characteristic $q > 3$, and let $a, b ∈ F$ be two field elements such that the so-called discriminant $4a^3 + 27b^2$ is not equal to zero. Then a Short Weierstrass elliptic curve $E_{a,b}(F)$ over $F$ in its affine representation is the set of all pairs of field elements $(x, y) ∈ F × F$ that satisfy the Short Weierstrass cubic equation $y^2 = x^3 + a · x + b$, together with a distinguished symbol $O$, called the point at infinity:

$$E_{a,b}(F) = \{(x, y) ∈ F × F | y^2 = x^3 + a · x + b\} \cup \{O\}$$

The equation $4a^3 + 27b^2 \neq 0$ ensures that the curve is non-singular, which loosely means that the curve has no cusps or self-intersections in the geometric sense, if seen as an actual curve. Cusps and self-intersections would make the group law potentially ambiguous.

Isomorphic affine Short Weierstrass curves

Let $F$ be a field, and let $(a, b)$ and $(a', b')$ be two pairs of parameters such that there is an invertible field element $c ∈ F^∗$ such that $a' = a · c^4$ and $b' = b · c^6$ hold. Then the elliptic curves $E_{a,b}(F)$ and $E_{a',b'} (F)$ are isomorphic, and there is a map that maps the curve points of $E_{a,b}(F)$ onto the curve points of $E_{a',b'} (F)$:

$$I : E_{a,b}(F) \to E_{a',b'}(F): \begin{cases} (x,y) \\ O \end{cases} \to \begin{cases} (c^2 \cdot x,c^3 \cdot y) \\ O \end{cases}$$

This map is a $1:1$ correspondence, and its inverse map is given by mapping the point at infinity onto the point at infinity, and mapping each curve point $(x, y)$ onto the curve point $(c^{−2} \cdot x, c^{−4} \cdot y)$.