One way functions and pseudorandom generators

Abstract

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 [6], but testing that requires an unrealistic amount of time (say, exhaustive search for the seed). Such pseudorandom generators were first discovered in [2] 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 [12] 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.

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Supported by NSF grant #DCR-8304498, DCR-8607492.

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Levin, L.A. One way functions and pseudorandom generators. Combinatorica 7, 357–363 (1987). https://doi.org/10.1007/BF02579323

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Keywords

  • Polynomial Time
  • Hamiltonian Cycle
  • Probabilistic Algorithm
  • Pseudorandom Generator
  • Bell System Technical Journal