The CryptoMath Teacher's Pack

Modular arithmetic

Learning objectives

• Calculate a mod n and explain it using a clock face.
• Recognise that modular arithmetic "wraps around" and loses information.
• Explain why that information loss makes reversal hard.

Starter (10 min)

• "It's 9 o'clock. What time is it 8 hours later?" Establish that 9 + 8 = 5 on a clock — that's 17 mod 12.
• Quick-fire: 25 mod 7, 100 mod 12, 50 mod 9.

Main activity (25 min)

• Open the Modular Arithmetic section and use the live calculator on the board.
• In pairs, students compute 7² mod 11, then 7⁴, then 7⁸ — noticing the numbers stay small.
• Build up to 7¹³ mod 11 using the worked example on the learning path.

Plenary (15 min)

• Discussion: "Given only the answer 2, could you work out the question was 7¹³?" Draw out that many inputs give the same output.
• Exit ticket: 3⁴ mod 5 (answer: 1).

Extension

• Set the Arcade Blitz as a timed challenge — 60 seconds of modular exponentiation.

Primes & RSA

Learning objectives

• Explain why primes are the "atoms" of multiplication.
• Walk through RSA key generation on small numbers.
• Articulate why factoring large semiprimes is computationally hard.

Starter (10 min)

• Race: multiply 61 × 53 (fast). Then: factor 3233 (slow). Time both — the asymmetry is the lesson.

Main activity (35 min)

• Work through the RSA guide together: p, q, n, φ(n), e, d.
• Students encrypt the letter "A" (65) with the public key (3233, 17) → ciphertext 2790.
• Then decrypt it back with d = 2753. The round trip is the "aha" moment.

Plenary (15 min)

• Students attempt the Weakened RSA boss battle in pairs — factoring, totient, private key, decrypt.

Answers

• 3233 = 61 × 53 · φ(n) = 3120 · d = 2753 · m = 65 ("A")

Hashing & blockchain

Learning objectives

• Describe a hash function as a one-way fingerprint.
• Observe the avalanche effect — one character changes everything.
• Explain how chained hashes make a ledger tamper-evident.

Starter (10 min)

• Open the Simulator, hash a student's name, then change one letter. Show the output is completely different.

Main activity (25 min)

• Students mine a block by finding a nonce that produces leading zeros — real SHA-256, in the browser.
• Then tamper with an earlier block and watch every later block turn invalid.

Plenary (15 min)

• Discuss: why does this make a distributed ledger trustworthy without a bank?
• Introduce the birthday attack: why hashes need 256 bits, not 64.

Quantum & post-quantum cryptography

Learning objectives

• Explain at a high level how Shor's algorithm threatens RSA and ECC.
• Distinguish Shor's (exponential) from Grover's (quadratic) speedup.
• Describe lattice-based cryptography and name the NIST standards.

Starter (10 min)

• Provocation: "Everything you encrypted in Lesson 2 could be broken. Not by a better computer — by a different kind of computer."

Main activity (25 min)

• Work through the post-quantum guide, focusing on the table of what breaks and what survives.
• Use the Quantum page's classical-vs-quantum comparison.
• Discuss "harvest now, decrypt later" — a genuinely current policy problem.

Plenary (15 min)

• Debate: should governments mandate post-quantum migration now, given the cost and the uncertainty about timelines?