One way functions and pseudorandom generators
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Pseudorandom generators transform in polynomial time a short random “seed” into a long “pseudorandom” string. This string cannot be random in the classical sense of , but testing that requires an unrealistic amount of time (say, exhaustive search for the seed). Such pseudorandom generators were first discovered in  assuming that the function (a x modb) is one-way, i.e., easy to compute, but hard to invert on a noticeable fraction of instances. In  this assumption was generalized to the existence of any one-way permutation. The permutation requirement is sufficient but still very strong. It is unlikely to be proven necessary, unless something crucial, like P=NP, is discovered. Below, among other observations, a weaker assumption about one-way functions is proposed, which is not only sufficient, but also necessary for the existence of pseudorandom generators.
KeywordsPolynomial Time Hamiltonian Cycle Probabilistic Algorithm Pseudorandom Generator Bell System Technical Journal
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