# If $P \neq NP$ why doesn't this prove the existence of OWF?

i know that $P = NP \Rightarrow$ non existence of OWF. but i don't understand why the claim: $P \neq NP \Rightarrow$ existence of OWF is wrong?

An intuitive answer would be enough.

• Where did you read that the claim is wrong? – mikeazo Nov 12 '16 at 13:24
• the existence of a proof that P and NP are not equal would not directly imply the existence of one-way functions. wikipedia – odu9 Nov 12 '16 at 13:27
• i would guess that : it's not sufficient for OWF to show that you can't solve for one input , but for np complete problem , its sufficient for just one input, but i want to make sure. – odu9 Nov 12 '16 at 13:29
• The wikipedia article links to Goldwasser and Bellare's lecture notes. – mikeazo Nov 12 '16 at 14:40
• I know I'm very late to the party, but aren't you asking why A => B <=/=> ~A => ~B? – J. Dionisio Jun 14 '20 at 16:09

an answer from Goldwasser and Bellare's lecture notes. P ≠ NP is not a sufficient one. P ≠ NP only implies that the encryption scheme is hard to break in the worst case. It does not rule-out the possibility that the encryption scheme is easy to break in almost all cases. In fact, one can easily construct “encryption schemes” for which the breaking problem is NP-complete and yet there exist an efficient breaking algorithm that succeeds on 99% of the cases. Hence, worst-case hardness is a poor measure of security. Security requires hardness on most cases or at least average-case hardness. Hence, a necessary condition for the existence of secure encryption schemes is the existence of languages in NP which are hard on the average. Furthermore, P≠ NP is not known to imply the existence of languages in NP which are hard on the average.

• In addition, even a hard-on-average language in NP does not necessarily imply the existence of an "easy-to-compute and hard-to-invert" function. – Geoffroy Couteau Nov 12 '16 at 20:24