Initial Conditions

# The Case for Curves: Unraveling Elliptic Curves

A Brief Guide to Elliptic Curves, and Some of the Foundational Concepts behind Finite Fields, and Elliptic Curve Cryptography (ECC)

This site serves as the output of my research project, supervised by Dr. Jelena Schmalz at the University of New England (UNE), specifically for the unit SCI395 - Science Report. Here, you will find information organised to facilitate understanding, and exploration of elliptic curves.

Well-posed, Ill-posed, and Chaos Theory

Differential Equations

Contents & Using This Site

What is a Set, a Group, a Ring, and a Field?

The Discrete Logarithm Problem

Elliptic Curves over ℝ

Theorem / Proof: A line connecting P and Q always intersects a third point

The Additive Inverse

Adding Points, Doubling Points: P, 2P, 3P, 4P, 8P etc.

Expanding from Weierstrass To General Form

Associativity

The Projective Plane

Homogenous Equations

Elliptic Curves In a Finite Field

The Modular Multiplicative Inverse

The Slope of an Elliptic Curve (Formal Derivatives)

Arithmetic in a Finite field

Doubling Points in A Finite Field

Acknowledgements and Bibliography