One important achievement of complexity theory was the discovery of an elegant scheme for classifying problems according to their computational difficulty. Using this scheme, we can demonstrate a method for giving evidence that certain problems are computationally hard, even if we are unable to prove that they are.

cryptography

In most fields, easy computational problems are preferable to a hard one because the easy ones are cheaper to solve. Cryptography is unusual because it specifically requires computational problems that are hard, rather than easy.