How is it possible to send encrypted information across an insecure channel (like the internet) so that only the intended recipient can decode it, without sharing the secret key in advance?

In 1976, well before this question arose, a new mathematical theory of encryption (public-key cryptography) invented by Diffie and Hellman made digital commerce and finance possible. The technology advances of the last 20 years bring new and urgent problems, including the need to compute on encrypted data in the cloud and to have cryptography that can withstand the speed-ups of quantum computers.