Advertisement

computational complexity

, Volume 5, Issue 1, pp 76–97 | Cite as

The computational complexity of recognizing permutation functions

  • Keju Ma
  • Joachim von zur Gathen
Article

Abstract

Let\(\mathbb{F}_q \) be a finite field withq elements and\(f \in \mathbb{F}_q \left( x \right)\) a rational function over\(\mathbb{F}_q \). No polynomial-time deterministic algorithm is known for the problem of deciding whetherf induces a permutation on\(\mathbb{F}_q \). The problem has been shown to be in co-R\( \subseteq \)co-NP, and in this paper we prove that it is inR\( \subseteq \)NP and hence inZPP, and it is deterministic polynomial-time reducible to the problem of factoring univariate polynomials over\(\mathbb{F}_q \). Besides the problem of recognizing prime numbers, it seems to be the only natural decision problem inZPP unknown to be inP. A deterministic test and a simple probabilistic test for permutation functions are also presented.

Subject classifications

68Q15 68Q25 11Y16 12Y05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Leonard M. Adleman and Ming-Deh Huang,Primality Testing and Abelian Varieties Over Finite Fields, vol. 1512 ofLecture Notes in Mathematics. Springer-Verlag, 1992.Google Scholar
  2. A. V. Aho, J. E. Hopcroft, andJ. D. Ullman,The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading MA, 1974.Google Scholar
  3. E. Bach, Weil bounds for singular curves.AAECC, to appear.Google Scholar
  4. S. Barnett,Polynomials and Linear Control Systems, vol. 77 ofMonographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, New York NY, 1983.Google Scholar
  5. E. Bombieri, On exponential sums in finite fields.Amer. J. Math. 88 (1966), 71–105.Google Scholar
  6. E. Bombieri andH. Davenport, On two problems of Mordell.Amer. J. Math. 88 (1966), 61–70.Google Scholar
  7. D. G. Cantor andE. Kaltofen, On fast multiplication of polynomials over arbitrary algebras.Acta. Inform. 28 (1991), 693–701.Google Scholar
  8. A. L. Chistov and D. Yu. Grigoryev, Polynomial-time factoring of the multivariable polynomials over a global field. LOMI preprint E-5-82, Leningrad, USSR, 1982.Google Scholar
  9. S. D. Cohen, The distribution of polynomials over finite fields.Acta Arith. 17 (1970), 255–271.Google Scholar
  10. H. Davenport andD. J. Lewis, Notes on congruences (I).Quart. J. Math. Oxford 14 (1963), 51–60.Google Scholar
  11. S. A. Evdokimov, Efficient factorization of polynomials over finite fields and the generalized Riemann hypothesis. Technical Report, Universität Bonn, 1993.Google Scholar
  12. M. D. Fried and M. Jarden,Field Arithmetic. Springer-Verlag, 1986.Google Scholar
  13. J. von zur Gathen, Tests for permutation polynomials.SIAM J. Comput. 20 (1991a), 591–602.Google Scholar
  14. J. von zur Gathen, Values of polynomials over finite fields.Bull. Austral. Math. Soc. 43 (1991b), 141–146.Google Scholar
  15. J. von zur Gathen andE. Kaltofen, Factorization of multivariate polynomials over finite fields.Math. Comp. 45 (1985), 251–261.Google Scholar
  16. J. von zur Gathen, M. Karpinski, and I. E. Shparlinski, Counting curves and their projections. InProc. 25th ACM Symp. Theory of Computing, 1993, 805–812.Google Scholar
  17. S. Goldwasser andJ. Kilian, Almost all primes can be quickly certified. InProc. 18th Ann. ACM Symp. Theory of Computing, Berkeley, CA, 1986, 316–329. See also: J. Kilian,Uses of randomness in algorithms and protocols, ACM Distinguished Doctoral Dissertation Series, MIT Press, Cambridge MA, 1990.Google Scholar
  18. D. R. Hayes, A geometric approach to permutation polynomials over a finite field.Duke Math. J. 34 (1967), 293–305.Google Scholar
  19. E. Kaltofen, Fast parallel absolute irreducibility testing.J. Symb. Computation 1 (1985), 57–67.Google Scholar
  20. E. Kaltofen, Deterministic irreducibility testing of polynomials over large finite fields.J. Symb. Comp. 4 (1987), 77–82.Google Scholar
  21. A. K. Lenstra, Factoring multivariate polynomials over finite fields.J. Comput. System Sci. 30 (1985), 235–248.Google Scholar
  22. R. Lidl andG.L. Mullen, When does a polynomial over a finite field permute the elements of the field?Amer. Math. Monthly 95 (1988), 243–246.Google Scholar
  23. R. Lidl andG.L. Mullen, When does a polynomial over a finite field permute the elements of the field?, II.Amer. Math. Monthly 100 (1993), 71–74.Google Scholar
  24. K. Ma and J. von zur Gathen, Counting value sets of functions and testing permutation functions. InAbstract of Int. Conf. Number Theoretic and Algebraic Methods in Computer Science, Moscow, 1993, 62–65. Finite Fields and Their Applications1 (1995), to appear.Google Scholar
  25. C. R. MacCluer, On a conjecture of Davenport and Lewis concerning exceptional polynomials.Acta Arith.12 (1967), 289–299.Google Scholar
  26. G. L. Miller, Riemann's hypothesis and tests for primality.J. Comput. System Sci. 13 (1976), 300–317.Google Scholar
  27. G. L. Mullen, Permutation polynomials over finite fields. InProc. 1992 Conf. Finite Fields, Coding Theory, and Advances in Communications and Computing, ed.G. L. Mullen and P. J.-S. Shiue, vol. 141 ofLecture Notes in Pure and Applied Mathematics. Marcel Dekker, 1993, 131–151.Google Scholar
  28. V. Pratt, Every prime has a succinct certificate.SIAM J. of Comput. (1975), 214–220.Google Scholar
  29. M. O. Rabin, Probabilistic algorithms for testing primality.J. of Number Theory 12 (1980), 128–138.Google Scholar
  30. A. Schönhage andV. Strassen, Schnelle Multiplikation großer Zahlen.Computing 7 (1971), 281–292.Google Scholar
  31. I. E. Shparlinski,Computational and algorithmic problems in finite fields, vol. 88 ofMathematics and its applications. Kluwer Academic Publishers, 1992a.Google Scholar
  32. I. E. Shparlinski, A deterministic test for permutation polynomials.Comput complexity 2 (1992b), 129–132.Google Scholar
  33. I. E. Shparlinski, On bivariate polynomial factorization over finite fields.Math. Comp. 60 (1993), 787–791.Google Scholar
  34. R. Solovay andV. Strassen, A fast Monte-Carlo test for primality.SIAM J. Comput. 6 (1977), 84–85. Erratum, in7 (1978), 118.Google Scholar
  35. D. Wan, Ap-adic lifting lemma and its applications to permutation polynomials. InProc. 1992 Conf. Finite Fields, Coding Theory, and Advances in Communications and Computing, ed.G. L. Mullen and P. J.-S. Shiue, vol. 141 ofLecture Notes in Pure and Applied Mathematics. Marcel Dekker, 1993, 209–216.Google Scholar
  36. K. S. Williams, On exceptional polynomials.Canad. Math. Bull. 11 (1968), 279–282.Google Scholar

Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • Keju Ma
    • 1
  • Joachim von zur Gathen
    • 1
  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada

Personalised recommendations