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Structural analyses on the complexity of inverting functions

  • Osamu Watanabe
  • Seinosuke Toda
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 450)

Abstract

In this paper we investigate the complexity of inverting polynomial-time computable functions by methods developed in structural complexity theory. We first analyze upper bounds of the complexity of inverse functions by using complexity classes of functions. We prove the following: (1) NP/bit (the class of functions whose each bit is NP computable) is an upper bound for inverting honest and one-to-one functions, and (2) relative to almost all oracle, the class PF tt NP (the class of functions that are polynomial time computable by asking non-adaptive queries to an NP oracle) is an upper bound for inverting honest functions. Next we investigate relative complexity of inverse functions by using polynomial-time reducibility of functions. We prove that an honest function is NP/bit invertible if the class of its inverse functions possesses the least element under polynomial-time non-adaptive one-query reducibility.

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References

  1. [Al86]
    E. Allender, Isomorphisms and 1-L reductions, in “Proc. 1st Structure in Complexity Theory Conference”, Lecture Notes in Computer Science 223, Springer-Verlag, Berlin (1986), 12–22; the final version appeared in J. Comput. Syst. Sci. 36 (1988), 336–350.Google Scholar
  2. [BDG88]
    J. Balcázar, J. Díaz, and J. Gabarró, Structural Complexity I, EATCS Monographs on Theoretical Computer Science, Springer-Verlag, Berlin (1988).Google Scholar
  3. [Be87]
    R. Beigel, Bounded queries to SAT and Boolean hierarchy, Technical Report 87-07, Dept. Computer Science, The Johns Hopkins University (1987).Google Scholar
  4. [Be88]
    R. Beigel, NP-hard sets are P-superterse unless R = NP, Technical Report 88-04, Dept. Computer Science, The Johns Hopkins University (1988).Google Scholar
  5. [BH88]
    S. Buss and L. Hay, On truth-table reducibility to SAT and the difference hierarchy over NP, in “Proc. 3rd Structure in Complexity Theory Conference”, IEEE (1988), 224–233.Google Scholar
  6. [GS84]
    J. Grollmann and A. Selman, Complexity measures for public-key cryptosystems, in “Proc. 25th IEEE Sympos. Foundation of Comput. Sci.”, IEEE (1984), 495–503; the revised version appeared in SIAM J. Comput. 17 (1988), 309–335.Google Scholar
  7. [He87]
    L. Hemachandra, The strong exponential hierarchy collapses, in “Proc. 19th Ann. ACM Sympos. on Theory of Computing”, ACM (1987), 110–122.Google Scholar
  8. [Ka88]
    J. Kadin, The polynomial time hierarchy collapses if the Boolean hierarchy collapses, in “Proc. 3rd Structure in Complexity Theory Conference”, IEEE (1988), 278–292.Google Scholar
  9. [Kr86]
    M. Krentel, The complexity of optimization problems, in “Proc. 18th Ann. ACM Sympos. on Theory of Computing”, ACM (1986), 69–76; the final version appeared in J. Comput. Syst. Sci. 36 (1988), 490–509.Google Scholar
  10. [To90]
    S. Toda, On polynomial time truth-table reducibilities of intractable sets to p-selective sets, Theoret. Comput. Sci. (1990), to appear.Google Scholar
  11. [Va76]
    L. Valiant, Relative complexity of checking and evaluating, Inform. Process. Lett. 5 (1976), 20–23.CrossRefGoogle Scholar
  12. [Wag87]
    K. Wagner, Number-of-query hierarchies, Technical Report TR-158, Institute of Mathematics, University of Augsburg (1987).Google Scholar
  13. [Wa88a]
    O. Watanabe, On hardness of one-way functions, Inform. Process. Lett. 27 (1988), 151–157.Google Scholar
  14. [Wa88b]
    O. Watanabe, On 1-tt-sparseness of nondeterministic complexity classes, in “Proc. 15th International Colloquium on Automata, Languages and Programming”, Lecture Notes in Computer Science 317, (1988), 697–709.Google Scholar
  15. [Wa89]
    O. Watanabe, On one-way functions, in “Combinatorics, Computing and Complexity”, D. Du and G. Hu eds., Kluwer Academic Pub. (1989), 98–131.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Osamu Watanabe
    • 1
  • Seinosuke Toda
    • 2
  1. 1.Department of Computer ScienceTokyo Institute of TechnologyTokyoJapan
  2. 2.Department of Computer ScienceUniversity of Electro-CommunicationsTokyoJapan

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