RSA art

Modular multiplication drawn as glowing string art, and the same maths turned into RSA encryption: keys, a message, the square-and-multiply trick and an eavesdropper trying to break it.

Toy only. Never use this for real security: these keys are tiny and there is no padding.

view n200 k2.00 p q e message seed

Square and multiply

The numbers

Key generation

cycles against k

line density against kdistinct lines per point

cycle structure of encryption

Eavesdropper

Eve sees the public key (n, e). To read anything she has to split n into p × q. Here she tries every odd divisor in turn, the brute-force way. Toy only, never use for real security.

key size

n = -

time to factor, extrapolatedlog scale

Challenge: read the hidden message

An intercepted message, encrypted with the public key below. Find the private exponent d and type it in. Hint: factor n with the eavesdropper, set p and q in the controls above, and the key cards will show d.

Diffie-Hellman: the other door

The same modular powers do a second job. RSA locks with a public key; Diffie-Hellman lets two strangers agree a secret key in public. Here it runs in a group made from your key's own primes. Open the full Diffie-Hellman page for the paint-mixing version, step-by-step playback and the race. Toy only.

Gallery of patterns found

How it works

Put n points on a circle and join each point i to (i × k) mod n. With k = 2 a cardioid appears, with k = 3 a nephroid, and as k slides through fractions the lines sweep between them. Counting where the lines go, the cycles and the repeats, is modular arithmetic made visible.

RSA uses powers instead of multiples. Pick two primes p and q and publish n = p × q and an exponent e. Anyone can encrypt a number m as c = m^e mod n, but only someone who knows d can undo it with m = c^d mod n. Choosing d so that e × d leaves remainder 1 when divided by φ(n) = (p - 1)(q - 1) makes the two powers cancel, by Euler's theorem.

This is a one-way function with a trapdoor. Multiplying p and q takes a moment; getting them back from n alone is the hard direction, and finding d needs φ(n), which needs p and q. The public key locks, the private key unlocks. Square and multiply makes the big powers cheap: one squaring per bit of the exponent, plus a multiply for each 1 bit.

the same circle that draws a cardioid hides a lock: decryption draws exactly the same chords as encryption, walked the other way.

c = m^e mod n m = c^d mod nm is the message as a number below n, c the ciphertext. (n, e) is public; d is private.

e × d ≡ 1 (mod φ(n))φ(n) = (p - 1)(q - 1) counts the numbers below n that share no factor with it.

Toy only. Real RSA uses primes hundreds of digits long, random padding such as OAEP, and audited libraries. Never use this page to protect anything.