Foundations
Prerequisites: none — this is the place to start.
By the end you'll be able to…
- Explain what a cryptographic “one-way” function is and why it matters.
- Read the basic notation used throughout the path (mod, gcd, exponents).
- Use the on-site calculators with confidence.
Modular Arithmetic
Prerequisites: Foundations.
By the end you'll be able to…
- Compute with clock (modular) arithmetic and reduce large numbers mod n.
- Perform fast modular exponentiation by squaring.
- See why modular math gives us reversible-yet-hard operations.
Primes & RSA
Prerequisites: Modular Arithmetic.
By the end you'll be able to…
- Test whether a number is prime and explain why primes are “atoms.”
- Walk through RSA key generation, encryption and decryption.
- Articulate why factoring large semiprimes is hard.
Discrete Logarithms
Prerequisites: Primes & RSA.
By the end you'll be able to…
- Define the discrete logarithm problem and why it's one-way.
- Walk through Diffie–Hellman key exchange step by step.
Elliptic Curves
Prerequisites: Discrete Logarithms.
By the end you'll be able to…
- Add points on an elliptic curve geometrically and algebraically.
- Explain why ECC gives the same security as RSA with smaller keys.
- Connect curves to real signatures (Bitcoin, Ethereum).
Post-Quantum
Prerequisites: Elliptic Curves.
By the end you'll be able to…
- Explain how Shor's algorithm threatens RSA and ECC.
- Describe lattice-based cryptography at a high level.
- Name the NIST post-quantum standards and why they matter.
Symmetric Cryptography
Prerequisites: Post-Quantum.
By the end you'll be able to…
- Explain how one shared key both encrypts and decrypts — the fast workhorse of real data.
- Describe block ciphers (AES), stream ciphers and modes of operation (CBC, CTR, GCM).
- Say why the one-time pad is provably unbreakable yet impractical to key.
Cryptographic Protocols
Prerequisites: Symmetric Cryptography.
By the end you'll be able to…
- Explain how key exchange, authentication and encryption combine into a secure session.
- Describe MAC / HMAC, digital signatures, certificates (PKI) and forward secrecy.
- Recognise how TLS — the HTTPS padlock — puts every piece together, and what a zero-knowledge proof achieves.
Certificate of Completion
Finish all 8 stages to earn a verifiable, digitally-signed certificate — with a QR code anyone can check in their browser.
Sample lesson — Modular exponentiation
This is the lesson pattern used across the path: short steps, a worked example you can reveal, then a quick check.
The problem
We want 7^13 mod 11. Computing 7^13 directly is huge — but we never need the giant number, only its remainder.
Square and reduce
Repeatedly square, reducing mod 11 each time so numbers stay small.
7^1 = 7
7^2 = 49 ≡ 5 (mod 11)
7^4 = 5^2 = 25 ≡ 3 (mod 11)
7^8 = 3^2 = 9 (mod 11)
Now 13 = 8 + 4 + 1, so 7^13 = 7^8 · 7^4 · 7^1 ≡ 9 · 3 · 7 = 189 ≡ 2 (mod 11).
Answer: 2. Only small multiplications — that's why your phone can do this for 600-digit numbers.
3^4 mod 5?