Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

How does RSA encryption work, and how long would it take to break a small key?

AA HLAA SL Ambitious Also in: Networks & graphs, Cryptography

Research question

Why does RSA decryption recover the original message, and how does the time taken to factorise n = pq by trial division grow with the number of digits?

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Why it makes a good exploration

RSA protects online payments using modular arithmetic. Explaining why it works, encrypting a message by hand, and timing factorisation of growing keys shows exactly why large primes keep secrets safe.

The mathematics you'll need

Modular arithmetic (explain)
Euler's totient and Fermat's little theorem
Modular inverses via the Euclidean algorithm
Growth of computation time — exponential in digits

Where the data comes from

Use small primes for worked examples; time a trial-division program for increasing key sizes.

OEIS (On-Line Encyclopedia of Integer Sequences) — Check a sequence you have found and read its known formulas and references.

A possible outline

Explain modular arithmetic with examples.
Generate keys with small primes.
Encrypt and decrypt a short message.
Prove why decryption works.
Time factorisation and model how time grows with digits.

Pitfalls that cost marks

Only describing the algorithm without proof.
Arithmetic errors — check every step with technology.
Claiming real keys can be broken by your method.

Showing personal engagement

Encrypt a message to a friend and have them decrypt it.
Predict the time to factor a 50-digit n from your model.
Discuss quantum computing's threat accurately.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Fits — ambitious at SL	Modular arithmetic (explain); Euler's totient and Fermat's little theorem
AA HL	Good fit	Modular arithmetic (explain); Euler's totient and Fermat's little theorem
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	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

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

Reflection (D)

For this idea, start with the weak points: only describing the algorithm without proof; and arithmetic errors — check every step with technology. Say how each one affects your answer.

Use of mathematics (E)

SL: modular arithmetic (explain); euler's totient and Fermat's little theorem; modular inverses via the Euclidean algorithm; growth of computation time — exponential in digits, each explained in your own words and carried out correctly.

How pure maths explorations reach the top bands in general: Pure maths, number & proof IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Prove Fermat's little theorem by induction or counting.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL pure maths exploration adds.

See a complete IA, marked

What is the quickest way to deliver a leaflet to every house on my estate? (AI HL) is a complete pure maths exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA HL and AA SL. We rate it Ambitious: it suits confident students, who should expect to learn some mathematics on their own.

What mathematics does this IA use?

Modular arithmetic (explain), euler's totient and Fermat's little theorem, modular inverses via the Euclidean algorithm and growth of computation time — exponential in digits.

Where can I get data for this IA?

Use small primes for worked examples; time a trial-division program for increasing key sizes. Checked sources and tools: OEIS (On-Line Encyclopedia of Integer Sequences).

Turn this idea into your IA

The planner opens with “How does RSA encryption work, and how long would it take to break a small key?” filled in: the question, the maths and the data sources.

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