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ABSTRACT

We show that the existence of one-way functions is necessary and sufficient for the existence of pseudo-random generators in the following sense. Let ƒ be an easily computable function such that when x is chosen randomly: (1) from ƒ(x) it is hard to recover an x1 with ƒ(x1) = ƒ(x) by a small circuit, or; (2) ƒ has small degeneracy and from ƒ(x) it is hard to recover x by a fast algorithm. From one-way functions of type (1) or (2) we show how to construct pseudo-random generators secure against small circuits or fast algorithms, respectively, and vice-versa. Previous results show how to construct pseudo-random generators from one-way functions that have special properties ([Blum, Micali 82], [Yao 82], [Levin 85], [Goldreich, Krawczyk, Luby 88]). We use the results of [Goldreich, Levin 89] in an essential way.

REFERENCES

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INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS
      G.2.1 Combinatorics
          Subjects: Generating functions

Additional Classification:
  E. Data

  F. Theory of Computation
  F.3 LOGICS AND MEANINGS OF PROGRAMS
      F.3.3 Studies of Program Constructs
          Subjects: Control primitives

  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS
      G.2.1 Combinatorics
          Subjects: Permutations and combinations
  G.3 PROBABILITY AND STATISTICS
      Subjects: Random number generation

General Terms:
Algorithms, Design, Security, Theory