Algebraic Description of Elliptic Curve Addition

This section provides an algebraic description of the problem of calculating the addition operation defined on an elliptic curve.

In the previous chapter, we have discussed elliptic curves and an "addition" operation from a geometric point of view. In this chapter, let's look at them from an algebraic point of view. This should help us calculate the "addition" operation using algebraic formulas.

First, let's describe the "addition" operation problem in algebraic terms:

```For a given elliptic curve represented as:
y2 = x3 + ax + b

And two given points on the curve represented as:
P = (xP, yP)
Q = (xQ, yQ)

Find a third point on the curved represented as:
R = (xR, yR)

So that R is the result of the addition operation
of P and Q as defined by our geometrical algorithm:
R = P + Q
```

Here is a diagram that illustrates how to perform the point addition operation on an elliptic curve geometrically (source: stackoverflow.com):

It will be easier to find the algebraic solution, if we divide the problem into multiple cases:

• Case 1: P and Q are symmetrical points.
• Case 2: P or Q is the infinity point.
• Case 3: P and Q are the same point.
• Case 4: P and Q are two distinct points.

See next tutorials on algebraic solutions for those cases.