With obsolete network password protocols , whoever goes first gives away the password. Worse still, across an open channel, anyone listening in can get the password as well. For two people in the same room, the Bizcard ZKPP demonstrates how to do this with pencil and paper. Our focus is on solving this for people on the Internet.
Attackers have powerful computers.
In the failed attempts to solve the problem, people tried using the password as a key. But most cryptographic methods, whether symmetric as in DES, or asymmetric as in RSA, require very large keys. Ordinary encryption methods and one-way hash functions just didn't solve the problem. Computers became more powerful, minimal safe key sizes got longer, and over time it became obvious that older methods for verifying passwords were broken. Even when the first zero-knowledge proofs of knowledge were developed, none were designed to work with a small secret.