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On sets with easy certificates and the existence of one-way permutations

  • Lane A. Hemaspaandra
  • Jörg Rothe
  • Gerd Wechsung
Regular Presentations
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1203)

Abstract

Can easy sets only have easy certificate schemes? In this paper, we study the class of sets that, for all NP certificate schemes (i.e., NP machines), always have easy acceptance certificates (i.e., accepting paths) that can be computed in polynomial time. We also study the class of sets that, for all NP certificate schemes, infinitely often have easy acceptance certificates. We give structural conditions that control the size of these classes. We also provide negative results showing that some of our positive claims are optimal. Our negative results are proven using a novel observation: The classic “wide spacing” oracle construction technique instantly yields non-bi-immunity results.

Easy certificate classes are also a useful notion in the study of whether one-way functions exist. This is one of the most important open questions in cryptology. We extend the results of Grollmann and Selman [GS88] by obtaining a complete characterization regarding the existence of a certain type of one-way function—(partial) one-way permutations—in terms of easy certificate classes. By Grädel's recent results about one-way functions [Grä94], this also links statements about easy certificates of NP sets with statements in finite model theory. In addition, we give a condition necessary and sufficient for the existence of (total) one-way permutations.

Keywords

Polynomial Time Turing Machine Certificate Scheme Kolmogorov Complexity Satisfying Assignment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Adl79]
    L. Adleman. Time, space, and randomness. Technical Report MIT/LCS/TM-131, MIT, Cambridge, MA, April 1979.Google Scholar
  2. [All86]
    E. Allender. The complexity of sparse sets in P. In Proceedings of the 1st Structure in Complexity Theory Conference, pages 1–11. Springer-Verlag Lecture Notes in Computer Science 223, June 1986.Google Scholar
  3. [All92]
    E. Allender. Applications of time-bounded Kolmogorov complexity in complexity theory. In O. Watanabe, editor, Kolmogorov Complexity and Computational Complexity, EATCS Monographs on Theoretical Computer Science, pages 4–22. Springer-Verlag, 1992.Google Scholar
  4. [AR88]
    E. Allender and R. Rubinstein. P-printable sets. SIAM Journal on Computing, 17(6): 1193–1202, 1988.Google Scholar
  5. [BD76]
    A. Borodin and A. Demers. Some comments on functional self-reducibility and the NP hierarchy. Technical Report TR 76-284, Cornell Department of Computer Science, Ithaca, NY, July 1976.Google Scholar
  6. [BGS75]
    T. Baker, J. Gill, and R. Solovay. Relativizations of the P=?NP question. S1AM Journal on Computing, 4(4):431–442, 1975.Google Scholar
  7. [Cha66]
    G. Chaitin. On the length of programs for computing finite binary sequences. Journal of the ACM, 13:547–569, 1966.Google Scholar
  8. [FFNR96]
    S. Fenner, L. Fortnow, A. Naik, and J. Rogers. On inverting onto functions. In Proceedings of the 11th Annual IEEE Conference on Computational Complexity, pages 213–222. IEEE Computer Society Press, May 1996.Google Scholar
  9. [FR94]
    L. Fortnow and J. Rogers. Separability and one-way functions. In Proceedings of the 5th International Symposium on Algorithms and Computation, pages 396–404. Springer-Verlag Lecture Notes in Computer Science #834, August 1994.Google Scholar
  10. [GHK92]
    J. Goldsmith, L. Hemachandra, and K. Kunen. Polynomial-time compression. Computational Complexity, 2(1):18–39, 1992.Google Scholar
  11. [Grä94]
    E. Grädel. Definability on finite structures and the existence of one-way functions. Methods of Logic in Computer Science, 1:299–314, 1994.Google Scholar
  12. [GS88]
    J. Grollmann and A. Selman. Complexity measures for public-key cryptosystems. SIAM Journal on Computing, 17(2):309–335, 1988.Google Scholar
  13. [GS91]
    A. Goldberg and M. Sipser. Compression and ranking. SIAM Journal on Computing, 20(3) 524–536, 1991.Google Scholar
  14. [Har83]
    J. Hartmanis. Generalized Kolmogorov complexity and the structure of feasible computations. In Proceedings of the 24th IEEE Symposium on Foundations of Computer Science, pages 439–445. IEEE Computer Society Press, 1983.Google Scholar
  15. [HH88]
    J. Hartmanis and L. Hemachandra. Complexity classes without machines: On complete languages for UP. Theoretical Computer Science, 58:129–142, 1988.Google Scholar
  16. [HRW95]
    L. Hemaspaandra, J. Rothe, and G. Wechsung. Easy sets and hard certificate schemes. Technical Report Math/95/5, Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany, 1995.Google Scholar
  17. [HY84]
    J. Hartmanis and Y. Yesha. Computation times of NP sets of different densities. Theoretical Computer Science, 34:17–32, 1984.Google Scholar
  18. [Ko85]
    K. Ko. On some natural complete operators. Theoretical Computer Science, 37:1–30, 1985.Google Scholar
  19. [Kol65]
    A. Kolmogorov. Three approaches for defining the concept of information quantity. Prob. Inform. Trans., 1:1–7, 1965.Google Scholar
  20. [LV90]
    M. Li and P. Vitányi. Applications of Kolmogorov complexity in the theory of computation. In A. Selman, editor, Complexity Theory Retrospective, pages 147–203. Springer-Verlag, 1990.Google Scholar
  21. [Sel95]
    A. Selman. Personal Communication, May, 1995.Google Scholar
  22. [Sip83]
    M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15th ACM Symposium on Theory of Computing, pages 330–335, 1983.Google Scholar
  23. [Tra84]
    B. Trakhtenbrot. A survey of Russian approaches to perebor (brute-force search) algorithms. Annals of the History of Computing, 6(4):384–400, 1984.Google Scholar
  24. [Val76]
    L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20–23, 1976.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Lane A. Hemaspaandra
    • 1
  • Jörg Rothe
    • 2
  • Gerd Wechsung
    • 2
  1. 1.Department of Computer ScienceUniversity of RochesterRochesterUSA
  2. 2.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany

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