computational complexity

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

The computational complexity of recognizing permutation functions

  • Keju Ma
  • Joachim von zur Gathen


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 


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Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

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

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