Why Math Matters for Cryptography

Every time you send a message, make a purchase online, or connect to a website, mathematics is working behind the scenes to keep your data safe. The same equations that mathematicians explored centuries ago now protect billions of internet transactions every day.

The Big Idea: Cryptography relies on mathematical operations that are easy to do in one direction but incredibly hard to reverse.

The Magic of One-Way Functions

Imagine you have a function that takes a secret password and transforms it into a seemingly random string of characters. Anyone can run this function forward, but going backward, from the random string back to the password, is practically impossible.

Easy Direction

Multiply two large prime numbers together:

61 × 53 = 3233

This takes a fraction of a second.

Hard Direction

Given 3233, find the two primes that multiply to make it:

3233 = ? × ?

Much harder! Imagine if the number had 600 digits.

What You'll Learn

This course builds your understanding step by step, from simple concepts to the real algorithms used in modern cryptography:

Module 1: Foundations

Modular Arithmetic - The "clock math" that wraps numbers around
Prime Numbers - The indivisible building blocks

Module 2: Number Theory

Greatest Common Divisor - Finding what numbers share
Modular Inverse - "Division" in modular arithmetic
Euler's Totient - Counting coprime numbers

Module 3: Applied Number Theory

Modular Exponentiation - Fast computation of huge powers
Discrete Logarithms - The hard problem that secures the web

Module 4: Cryptographic Protocols

RSA - The most famous encryption algorithm
Diffie-Hellman - Sharing secrets over public channels
Elliptic Curves - Modern cryptography on curves

Try It: One-Way Functions in Action

Type something below and see how a hash function transforms it. Notice how even tiny changes produce completely different outputs:

Input:

Hash output will appear here...

Note: This is a simple demonstration hash, not a cryptographic one. Real cryptographic hash functions like SHA-256 are much more complex, but the principle is the same.

A Brief History

The mathematics we'll learn has deep roots:

300 BCE - Euclid describes the algorithm for finding GCD
1760s - Euler develops the totient function
1976 - Diffie and Hellman invent public-key cryptography
1977 - Rivest, Shamir, and Adleman create RSA
1985 - Elliptic curve cryptography is proposed
Today - These algorithms protect virtually all internet traffic

Ready to begin? Let's start with the foundation of everything: modular arithmetic, also known as "clock math."

Home Next: Modular Arithmetic