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You're right that my statement wasn't specific enough; I was going for brevity. The precise statement is: You need a problem with a known polynomial-time nondeterministic algorithm, but no known polynomial-time deterministic algorithm.

A very important point though: public-key cryptosystems rely on one-way functions, functions that are hard to invert in the average case, NOT just the worst case, which is not your typical complexity-theory mindset. So, I would be careful when talking about complexity classes in the context of crypto because it can be very, very easy to trip up and say something that's either not comprehensive or just downright incorrect.

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