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

Last update: 2019.

Table of Contents

 About This Book

Geometric Introduction to Elliptic Curves

 What Is an Elliptic Curve?

 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)

 Finite Fields

 Generators and Cyclic Subgroups

 Reduced Elliptic Curve Groups

 Elliptic Curve Subgroups

 tinyec - Python Library for ECC

 EC (Elliptic Curve) Key Pair

 ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

 ECDSA (Elliptic Curve Digital Signature Algorithm)

 ECES (Elliptic Curve Encryption Scheme)

 Terminology

 References

 Full Version in PDF/EPUB