Elliptic Curve Operation Summary

This section provides a summary of elliptic curve operations and their properties discussed in this chapter.

Our elliptic curve operations introduced in this chapter can be summary as the following:

1. An elliptic curve is a set of points satisfy the following equation for given coefficients, a and b:

```y2 = x3 + ax + b
```

2. For any given two points, P and Q, on an elliptic curve, the addition of P and Q (or P + Q) results to a third point, R, on the same elliptic curve. R is the symmetrical point of -R, which is the third intersection of the curve and the straight line passing through P and Q.

3. The addition operation is commutative, because the following is true:

```P + Q = Q + P
```

4. The addition operation is associative, because the following is true:

```P + (Q + S) = (P + Q) + S
```

5. For any given point, P, on an elliptic curve, the negation operation of P (or -P) results to the symmetrical point of P on the curve. -P is also called the inverse point of P.

6. For any given two points, P and Q, on an elliptic curve, the subtraction operation of Q from P (or P - Q) results to P + (-P). Or:

```P - Q = P + (-Q)
```

7. The infinity point, O = (∞, ∞), is called the identity element, because the following statements are true:

```P + O = P
P - P = O
```