RSA Cryptography Foundation

Intro to Proofs · Axiom Academy

Every padlock in your browser bar runs on one number-theory idea — modular exponentiation. Put RSA encryption in your hands. Anyone can scramble a message to you with your public key (n = 3233, e = 17) , but only your private key (d = 2753) turns it back — and an eavesdropper who sees everything still can't read it. Three moves show why. Encrypt: scramble it with the public key Pick any message number and the public key locks it: raise it to the 17th power, fold it back down modulo 3233, and out comes ciphertext that looks like noise. That's , live. Decrypt: the private key runs it backward The ciphertext is useless to everyone but you. Raise it to the private exponent and the scramble undoes itself — hands back the exact original. Drag the message and watch the round trip close. The trapdoor: easy to lock, impossible to pick An eavesdropper sees n and e — so why can't they just compute d? Because that needs the factors of n, and factoring is the one-way street. Drag the key size and watch multiplying stay instant while factoring blows up. One relationship, three moves: lock it with , unlock it with , and stay safe because factoring is hard . The same number theory rides under every HTTPS padlock, signed software update, and encrypted message — pure modular arithmetic, quietly guarding the digital world.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.