**EC Cryptography Tutorials - Herong's Tutorial Examples** - Version 1.00, 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:

y^{2}= x^{3}+ 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

Last update: 2019.

Table of Contents

►Geometric Introduction to Elliptic Curves

Elliptic Curve Geometric Properties

Addition Operation on an Elliptic Curve

Prove of Elliptic Curve Addition Operation

Same Point Addition on an Elliptic Curve

Infinity Point on an Elliptic Curve

Negation Operation on an Elliptic Curve

Subtraction Operation on an Elliptic Curve

Identity Element on an Elliptic Curve

Commutativity of Elliptic Curve Operations

Associativity of Elliptic Curve Operations

►Elliptic Curve Operation Summary

Algebraic Introduction to Elliptic Curves

Abelian Group and Elliptic Curves

Discrete Logarithm Problem (DLP)

Generators and Cyclic Subgroups

tinyec - Python Library for ECC

ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

ECDSA (Elliptic Curve Digital Signature Algorithm)