<aside> 💡 What is elliptic curve group law?
The properties of the points on the curve that make up the group. How you perform operations between these points are based on properties and rules defined by the elliptic curve group law.
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short Weierstrass curves can be represented in affine representation. In affine representation, the equations for a Weierstrass curve take the form y^2 = x^3 + ax + b, where a and b are constants. This equation is known as a short Weierstrass equation. The curve is defined over a finite field, and its elements can be represented in affine coordinates.
F is finite field of characteristic q.
Discriminant: $4a^3 +27b^2$
$$ E_{a,b}(F) = \{(x,y) \in F \times F | y^2 = x^3 + ax +b\} , \{O\} $$
in affine representation is the set of all pairs of field elements x,y in FxF that satisfy the short weierstrass cubic equation.
Difference is visualization between the “curve” over a field with characteristic q = 0 (unbounded) and finite field
Field unbounded looks like a curve
Finite field looks like random points
The discriminant ensures it is non-singular
Curve has no cusps or intersections