Smoothness – Elliptic Curves

8.1.2 Smoothness

Before we can start defining a group operation on the points of an elliptic curve, we need to add one requirement: The curve needs to be smooth. By this, we mean that the curve has well-defined tangent slopes everywhere. Take a look at curve E defined by y2 = x3 − 3x + 2 over real numbers (see Figure 8.2).

Figure 8.2: The singular curve y2 = x3 − 3x + 2 over real numbers

The point (1,0) of the curve does not have a unique tangent slope. Such points are called singular points. If we try to find a tangent slope by differentiating both sides of the defining equation with respect to x, we get 2yy′ = 3×2 − 3, or y′ = . The behavior of this fraction is undefined if we plug in the singular point (1,0), meaning it takes two different limit values whether we approach x = 1 from the left or from the right. This comes from the fact that the right-hand side of the curve equation has a double zero at x = 1 (which means the derivative has a zero at x = 1 as well).

More generally, if the right-hand side of a curve E in reduced Weierstrass form is given by the cubic polynomial

it can be shown that this polynomial has a double zero somewhere if and only if 16(4a3 + 27b2) = 0. The factor 16 seems to be superficial, but it does matter if the characteristic of 𝔽 is 2, because it is always zero in this case. We will not pursue this issue here, but refer you to [82], section 5.7 instead.

We therefore define an elliptic curve E in reduced Weierstrass form y2 = x3 + ax + b over a field 𝔽 of characteristic char(𝔽) > 3 to be smooth or nonsingular, if the discriminant

This ensures that the curve has well-defined tangents everywhere.