IA ideas library
Cryptography and number theory IA ideas: codes, primes and modular arithmetic
Codes and ciphers are a natural home for pure mathematics with a purpose: modular arithmetic, prime numbers, counting arguments and proof. You can break a cipher with letter frequencies, prove why a check digit catches typing mistakes, or count how many keys an attacker would have to try.
Most of this mathematics is beyond the IB syllabus, which makes it a good fit for confident AA students, especially at HL where proof is expected. Keep the scope small: one idea understood and proved is worth more than a tour of famous ciphers.
Cryptography & number theory ideas (9)
- Which typing errors does a check digit catch? ISBN-10 versus ISBN-13AA HLAA SLSolid
- Breaking a cipher with letter frequencies and chi-squaredAI SLAA SLAI HLAA HLAccessible
- Finding the key length of a Vigenère cipher with the index of coincidenceAA HLAI HLAmbitious
- How long would it take to guess your password? Counting and logarithmsAA SLAI SLAA HLAI HLAccessible
- Agreeing a secret in public: the Diffie–Hellman key exchangeAA HLAmbitious
- Which numbers are the sum of two squares?AA HLAA SLAmbitious
- Patterns in the last digits of powersAA SLAA HLSolid
- Perfect numbers and Mersenne primes: proving Euclid's ruleAA HLAmbitious
- Encrypting with matrices: when can a Hill cipher be undone?AI HLAA HLAmbitious
Also relevant (4)
- The Collatz conjecture: what can we say about stopping times?AA SLAA HLAI SLAI HLSolid
- How many primes are there below n? Testing the prime number theoremAA SLAA HLAI SLSolid
- How does RSA encryption work, and how long would it take to break a small key?AA HLAA SLAmbitious
- Spreading beats evenly: Euclid's algorithm and musical rhythmAA SLAA HLSolid
Frequently asked questions
Is RSA too hard for a Maths IA?
RSA needs modular arithmetic, Euler's totient function and modular inverses, none of which are in the syllabus. It is possible at AA HL if you build the ideas up carefully with small numbers. Smaller questions, such as check digits or affine ciphers, are easier to make rigorous.
Can AI students do a cryptography IA?
Yes, especially a statistical one: frequency analysis with a chi-squared goodness-of-fit test, or the probability of guessing a password. Make sure the mathematics you use is at the level of your course.
Free: the IA 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.
While you wait for the email: read a free annotated exemplar excerpt →
Related types of IA: Pure maths, number & proof
Other topics: Modelling · Calculus · Differential equations · Statistics · Probability · Geometry & Voronoi · Finance · Sport · Environment · Health & biology · Networks & graphs · Pure maths · Art & music · Games & puzzles · Optimisation · Simulation · Numerical methods · Fractals & chaos · Matrices & Markov · Trig & navigation · Kinematics · Voting & game theory · Surveys
By interest: Computer science · Physics and engineering · Psychology and social sciences · Geography and travel · Chemistry · Biology · Economics · Business management · Sports, exercise and health science · Environmental systems and societies · Music · Visual arts · Design technology · By course: AA SL · AA HL · AI SL · AI HL