Crypto Basics

The idea is to implement some of the building blocks of cryptography

Finite Fields

A field is an algebraic structure that consists of a set of numbers (elements) with two defined operations (addition and multiplication) that satisfy the following properties:

• Closure
• Additive identity
• Multiplicative identity
• Additive inverse
• Multiplicative inverse

A finite field is a field with a finite number of elements and looks like: F_p = {0, 1, 2, ..., p-1} where p is called the order of the field and, in cryptography, it is a prime number.

Field elements must satisfy the abovementioned properties so we cannot use primitives.

Elliptic curves

An elliptic curve is a set of points that satisfy an equation like this:

$$y^2 = x^3 + ax + b$$

Elliptic curves are an essential part of blockchain, Bitcoin for instance utilizes a curve called secp256k1 that looks like $y^2 = x^3 + 7$.

The point addition consists of performing an operation on two points of the curve and obtaining a third point in the curve. It is a nonlinear operation in which predicting the result of the addition is not trivial. Point addition satisfies certain properties such as Identity, Commutativity, Associativity, and Invertibility.

For two points A and B we can get a line that intersects the elliptic curve. Commonly, the line will intersect three points in the curve. The result of the point addition comes from reflecting the third point over the x-axis. Special cases are considered when the line intersects only at one point, or at two points.