EC Cryptography Tutorials - Herong's Tutorial Examples - v1.00, by Dr. Herong Yang
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
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