NP Problems with unique solution

Is there any class of NP problems that have one unique solution? I'm asking that, because when I was studying cryptography I read about the knapsack and I found very interesting the idea.

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migrated from mathoverflow.netApr 23 '14 at 10:38

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You seem to be shaky on your terminology; NP problems can be arbitrarily easy, and they are usually decision problems (which always have a unique solution, yes or no); read more at our reference questions. I assume you want NPO-hard problems with unique solutions? – Raphael Apr 23 '14 at 14:34
Yes, I was meaning NP complete or NP hard, or whatever that is not in P... Sorry and thanks to point out – Marco Apr 23 '14 at 20:59
We don't know if NP-complete problems are not in P... – Raphael Apr 23 '14 at 21:06

UP: If $x \in L$, then there is exactly one "proof" ("witness", "certificate", "accepting path"). If $x \not\in L$, there are exactly zero "proofs".
US: If $x \in L$, then there is exactly one "proof". If $x \not\in L$, there may be zero proofs, or 2+ proofs, as long as there is not exactly one.