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Concept
Computational hardness assumption
Summary
In computational complexity theory, a computational hardness assumption is the hypothesis that a particular problem cannot be solved efficiently (where efficiently typically means "in polynomial time"). It is not known how to prove (unconditional) hardness for essentially any useful problem. Instead, computer scientists rely on reductions to formally relate the hardness of a new or complicated problem to a computational hardness assumption about a problem that is better-understood.
Computational hardness assumptions are of particular importance in cryptography. A major goal in cryptography is to create cryptographic primitives with provable security. In some cases, cryptographic protocols are found to have information theoretic security; the one-time pad is a common example. However, information theoretic security cannot always be achieved; in such cases, cryptographers fall back to computational security. Roughly speaking, this means that these systems are secure assuming that any adversaries are computationally limited, as all adversaries are in practice.
Computational hardness assumptions are also useful for guiding algorithm designers: a simple algorithm is unlikely to refute a well-studied computational hardness assumption such as P ≠ NP.
Computer scientists have different ways of assessing which hardness assumptions are more reliable.
We say that assumption is stronger than assumption when implies (and the converse is false or not known).
In other words, even if assumption were false, assumption may still be true, and cryptographic protocols based on assumption may still be safe to use.
Thus when devising cryptographic protocols, one hopes to be able to prove security using the weakest possible assumptions.
Best, worst and average case
An average-case assumption says that a specific problem is hard on most instances from some explicit distribution, whereas a worst-case assumption only says that the problem is hard on some instances.
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