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Lesson Plan: The Mechanics of RSA Encryption

Duration: 90 Minutes
Prerequisites: Basic understanding of prime numbers and algebra.
Objective: Students will understand the mathematical foundation of RSA (GCD and Modular Arithmetic) and observe how these concepts are implemented in code.

1. Warm-up: The "Magic" of Modular Math (15 mins)

  • Concept: Explain Modular Arithmetic (Clock Math). Focus on why it’s useful for cryptography: it’s easy to perform but hard to "undo" without specific information.
  • Exercise: Simple Modular exponentiation.
    • Question: What is $7^3 \pmod{10}$?
    • Work: $7 \times 7 = 49 \equiv 9$. $9 \times 7 = 63 \equiv 3$.
  • Online Tool: WolframAlpha (Type: 7^3 mod 10) or the Omni Calculator - Modulo.

2. The Greatest Common Divisor & Euclid (25 mins)

[Image of Euclidean Algorithm flowchart]

  • The Problem: For RSA, we need to find two numbers that are "coprime" (GCD = 1).
  • The Algorithm: Teach Euclidean Algorithm (repeated division).
    • Example: Find $GCD(1071, 462)$.
    • $1071 = 2(462) + 147$
    • $462 = 3(147) + 21$
    • $147 = 7(21) + 0 \rightarrow$ GCD is 21.
  • Guided Practice: Have students find $GCD(35, 12)$. (Result is 1, so they are coprime—essential for RSA).
  • Online Tool: Calculator.net GCD Calculator.

3. Understanding RSA (The SimpleRSA Model) (20 mins)

Explain the five steps of RSA using the logic found in the GitHub repo's TestOne:

  1. Pick two primes ($p, q$): Keep them small for class (e.g., 61 and 53).
  2. Compute $n = p \times q$: This is the modulus used in the code.
  3. Compute Totient $\phi(n) = (p-1)(q-1)$: This is where the Euclid math comes in.
  4. Choose $e$: Must be coprime with $\phi(n)$ (use Euclid here!).
  5. Determine $d$: The "modular inverse" (the secret key).

4. Code Deep-Dive: SimpleRSA (20 mins)

Open the SimpleRSA GitHub and walk through the four levels of complexity provided:

  • TestOne: Highlight how it encrypts one character at a time using MulMod(). This mirrors the math they did in step 1.
  • TestTwo/Three: Discuss the ModularPow() function. Explain that for large numbers, we can't just do $base^{exponent}$ because the number becomes too large for a computer to hold (overflow).
  • TestFour: Show how "packing" works (putting multiple characters into one BigInteger). This is how real-world encryption is efficient.

5. Hands-on Activity: The Human RSA (10 mins)

  • Break students into pairs.
  • Student A (Receiver): Picks two small primes, calculates $n$ and $e$, and gives them to Student B.
  • Student B (Sender): Uses a calculator to encrypt a single number (a "message") using $m^e \pmod{n}$.
  • Student A: Decrypts it using their private key $d$.
  • Calculator Tip: Use the Use the Google Search Bar or a Scientific Calculator (typically the % or mod button).

Resources for Students