Encrypting with matrices: when can a Hill cipher be undone?

AI HLAA HL Ambitious Also in: Matrices & Markov

Research question

Which 2 × 2 matrices can be used as keys for a Hill cipher on 26 letters, how many valid keys are there, and how many known plaintext letters does an attacker need?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

Matrices meet modular arithmetic: a key works only when its determinant has an inverse mod 26. Counting valid keys and breaking the cipher with known plaintext are both real, finishable investigations.

The mathematics you'll need

Matrix multiplication and inverses
Determinants and invertibility mod 26 (new)
Counting invertible matrices (inclusion–exclusion or direct counting)
Solving linear systems mod 26
Known-plaintext attack

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

No data needed; encrypt your own messages.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Explain encryption with a 2 × 2 key.
Show why the determinant must be coprime to 26.
Count the valid keys.
Break a message using a few known letters.
Reflect on why the cipher is not secure.

Pitfalls that cost marks

Using the ordinary inverse instead of the inverse mod 26.
Counting by computer without explaining the count.
No worked example by hand.

Showing personal engagement

Exchange encrypted messages with a friend.
Find keys that leave some messages unchanged.
Time your own attack.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Good fit	Matrix multiplication and inverses; Determinants and invertibility mod 26 (new)
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Good fit	Matrix multiplication and inverses; Determinants and invertibility mod 26 (new)

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Build and break your own small cipher or code, invent examples to test each result, and record the conjectures you made and the ones that turned out to be false.

Reflection (D)

Reflect on what each result guarantees and what it doesn't: which errors a check digit misses, which attacks a cipher survives, and how the answer depends on the size of the numbers. For this idea, start with: using the ordinary inverse instead of the inverse mod 26 — say how it affects your answer.

Use of mathematics (E)

SL: Counting principles, probability or frequency statistics used correctly; any number theory (modular arithmetic, primes) introduced with your own small worked examples and explained, not quoted.

HL: Rigorous proofs (by contradiction or induction) of the number-theory facts you rely on, counting arguments made general, or a statistical attack tested formally.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Count valid 3 × 3 keys, or compare mod 26 with mod 29 (a prime) and explain the difference.

Extending it for HL

Prove the key result in general (why the check digit catches every single-digit error, why the decryption undoes the encryption) rather than checking examples.

See a complete IA, marked

Our annotated exemplar How long does a game of Snakes and Ladders last on my grandmother's board? (AI HL) asks a different question, but shows how a complete cryptography exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).