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.

Contents & Using This Site

Introduction Welcome to this guide to elliptic curves. Elliptic curves are a cornerstone of modern cryptographic systems. They have an...

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

Exploring Fundamental Abstract Algebra.

The Discrete Logarithm Problem

The Discrete Logarithm Problem is a foundational problem in the field of cryptography and number theory.

Elliptic Curves over ℝ

Elliptic curves over real numbers involve intricate & beautiful geometric properties & have applications in various mathematical fields.

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

How do we know that a line connecting P and Q will always intersect another point on the curve?

The Additive Inverse

For every point P, on an Elliptic Curve, there exists a point P' (often denoted -P) for which P + P' = P + (-P) = P - P = the point at...

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

Adding and doubling points is an essential element of the algebra of Elliptic curves.

Expanding from Weierstrass To General Form

Taking the Weierstrass form of an Elliptic Curve, and translating our x-values and y-values, with functions, we can convert the...

Associativity

Associativity on elliptic curves refers to a fundamental property of the addition operation defined on points of the curve (Washington,...

The Projective Plane

The projective plane plays a critical role in the study of elliptic curves, providing a broader framework that extends the usual...

Homogenous Equations

Homogeneous equations are mathematical expressions in which every term has the same degree. This is a key concept in algebra. It assists...

Elliptic Curves In a Finite Field

Elliptic curves over finite fields are fundamental mathematical structures with implications in various areas of cryptography and number...

The Modular Multiplicative Inverse

Calculating the multiplicative inverse of a number in a finite field efficiently is a crucial operation, especially in the fields of...

The Slope of an Elliptic Curve (Formal Derivatives)

Before we continue with our investigation of elliptic curves in a finite field, we need to clarify some important calculations, that will...

Arithmetic in a Finite field

The operations of adding and doubling points on elliptic curves defined over finite fields form the cornerstone of elliptic curve...

Doubling Points in A Finite Field

Doubling points in a finite field becomes important and central to the process of efficiently calculating multiples of a point - that is,...

Acknowledgements and Bibliography

Acknowledgements and Bibliography - Thank Yous and references.