Elliptic-curve cryptography
Public-key cryptography built on points on a curve rather than large primes — the same security at a tenth of the key length.
Proposed independently by Neal Koblitz and Victor Miller in 1985, it only took hold in the 2000s, partly because the patent situation stayed murky well into the 1990s. Security rests on the fact that adding a point to itself many times on the curve is easy, while working out how many times it was done is not — the discrete logarithm problem wearing a different costume.
The win is size: a 256-bit curve is roughly equivalent to 3072-bit RSA. That makes ECC the obvious choice on phones, smartcards and anything else on a battery. Curve selection is politically loaded — parts of the industry distrust NIST's P-256 for constants nobody can account for, and Curve25519 from Daniel Bernstein in 2005 won adoption precisely by justifying every design decision. ECC falls to Shor's algorithm just as RSA does, so post-quantum cryptography applies here too.