Modular arithmetic for cryptography

In 1801 a 24-year-old Carl Friedrich Gauss published Disquisitiones Arithmeticae, and on its first pages introduced a notation for something mathematicians had handled clumsily for centuries: a ≡ b (mod n), “a is congruent to b modulo n”. It turned the arithmetic of remainders into an algebra you could calculate with.

For 175 years it was pure mathematics. Then RSA and Diffie–Hellman made it the arithmetic of every secure connection, and the questions Gauss studied, which numbers have inverses and how powers behave, became engineering questions.