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

## 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## Preview

Unable to display preview. Download preview PDF.

## References

- [Adl79]L. Adleman. Time, space, and randomness. Technical Report MIT/LCS/TM-131, MIT, Cambridge, MA, April 1979.Google Scholar
- [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 - [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 - [AR88]E. Allender and R. Rubinstein. P-printable sets.
*SIAM Journal on Computing*, 17(6): 1193–1202, 1988.Google Scholar - [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
- [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 - [Cha66]G. Chaitin. On the length of programs for computing finite binary sequences.
*Journal of the ACM*, 13:547–569, 1966.Google Scholar - [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 - [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 - [GHK92]J. Goldsmith, L. Hemachandra, and K. Kunen. Polynomial-time compression.
*Computational Complexity*, 2(1):18–39, 1992.Google Scholar - [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 - [GS88]J. Grollmann and A. Selman. Complexity measures for public-key cryptosystems.
*SIAM Journal on Computing*, 17(2):309–335, 1988.Google Scholar - [GS91]A. Goldberg and M. Sipser. Compression and ranking.
*SIAM Journal on Computing*, 20(3) 524–536, 1991.Google Scholar - [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 - [HH88]J. Hartmanis and L. Hemachandra. Complexity classes without machines: On complete languages for UP.
*Theoretical Computer Science*, 58:129–142, 1988.Google Scholar - [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
- [HY84]J. Hartmanis and Y. Yesha. Computation times of NP sets of different densities.
*Theoretical Computer Science*, 34:17–32, 1984.Google Scholar - [Ko85]K. Ko. On some natural complete operators.
*Theoretical Computer Science*, 37:1–30, 1985.Google Scholar - [Kol65]A. Kolmogorov. Three approaches for defining the concept of information quantity.
*Prob. Inform. Trans.*, 1:1–7, 1965.Google Scholar - [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 - [Sel95]A. Selman. Personal Communication, May, 1995.Google Scholar
- [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 - [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 - [Val76]L. Valiant. The relative complexity of checking and evaluating.
*Information Processing Letters*, 5:20–23, 1976.Google Scholar