Cryptography

§ Overview

Mathematical foundations of modern cryptography including \n-Diffie-Hellman key-exchange, RSA, primality tests (AKS, Baillie-PSW), factorization (Pollard rho, sieves, Lenstra), and elliptic curves.

Notes from MATH 470 — Communications and Cryptography at Texas A&M.

Based off of lectures by Dr. Josiah Park, and An Introduction to Mathematical Cryptography by J. Hoffstein, J. Pipher, and J.H. Silverman.

§ Comments

Notation

$\mathbb{Z}_n$ and $\mathbb{Z}/n\mathbb{Z}$ are used interchangably; as are $(\mathbb{Z}_n)^\times$ and $(\mathbb{Z}/n\mathbb{Z})^\times$
$\mathbb{Z}_n$ is sometimes used in place of $(\mathbb{Z}_n)^{+}$ or $(\mathbb{Z}_n)^\times$ , which is disambiguated based on context
$\mathbb{N}$ are the natural numbers, defined without 0.
$\mathbb{P}$ are the set of primes
The interval notation $[a\dots b],$ $(a\dots b),$ $(a\dots b],$ $[a\dots b)$ defines the set of integers between $a$ and $b$ with brackets for inclusivity or parentheses for exclusivity^[a].

(2025) Over the years I've come to prefer the set notation $\{a, \dots, b\} \triangleq [a\dots b],$ even if it means that I have to explicitly state $n - 1$ instead of using a parenthesis for when something is exclusive.

^a: Not everything has been touched in my update; some portions of these notes may use the interval notation, and in others the set notation.