Shor's algorithm
Factors large numbers in polynomial time — and thereby renders RSA and elliptic curves unusable the day the hardware exists.
Peter Shor published the algorithm in 1994 at Bell Labs. It was the first convincing proof that quantum computers can solve a practically important problem exponentially faster than known classical methods, and it turned the field from a theoretical curiosity into something governments fund.
The core move is unexpected: factoring is recast as finding the period of a function, and periodicity is precisely what the quantum Fourier transform is good at. Breaking RSA-2048 is estimated to require thousands of error-corrected logical qubits, hence millions of physical ones — far beyond today's machines, but close enough that migration to post-quantum cryptography is already under way.