Loosely speaking, the most commonly-used public-key algorithms are based on three problems: factoring (RSA), finite field discrete logarithms (Diffie-Hellman), and elliptic curve discrete logarithms (ECDH and ECDSA). But even if quantum computing never becomes powerful enough to break current public keys, the fear of the quantum doomsayers is based on a completely valid observation: the internet has put nearly all of its cryptographic eggs into the single basket of the hidden subgroup problem. If somebody can efficiently solve the hidden subgroup problem, whether it’s with quantum computers or classical computers, they will be able to break the vast majority of public-key cryptography used on the internet.