Algebraic Introduction to Elliptic Curves

This chapter provides an algebraic introduction of addition operations on elliptic curves. Algebraic solutions are provided to calculate addition operations by dividing the problem into 4 cases: two symmetric points, one infinity point, two identical points, and two distinct points.

Algebraic Description of Elliptic Curve Addition

Algebraic Solution for Symmetrical Points

Algebraic Solution for the Infinity Point

Algebraic Solution for Point Doubling

Algebraic Solution for Distinct Points

Elliptic Curves with Singularities

Elliptic Curve Point Addition Example

Elliptic Curve Point Doubling Example

Conclusion:

Table of Contents

 About This Book

 Geometric Introduction to Elliptic Curves

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

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