Notes Cryptography Mathematical Foundations of Cryptography

Mathematical Foundations of Cryptography

28 Jul 2025

§ Overview

The aim of the course is to teach the mathematical foundations behind the commonly used algorithms and systems used in modern cryptography and communications.

As such, it is useful to begin with a discussion over some relevant topics in number theory and abstract algebra to develop a foundation on which we build cryptographic schemes.

§ Contents

Algebraic Structures

Basic group and field structures.

Modular Arithmetic

Addition, multiplication, and multiplicative inverses under a $\mathbb{Z}_m$ .

Greatest Common Divisor

Finding a solution to the Diophantine equation $ax + by = c$ in $\mathbb{Z}$ .

Fast Powering

Many cryptographic schemes rely on exponentiation, and we want to do it fast and secure.

Totient Function and Euler's Theorem

The totient function has some nice properties under a cyclic group which underpins the RSA cryptosystem.