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A Sublinear Time Randomized Algorithm for Coset Enumeration in the Black Box Model

  • Bin Fu
  • Zhixiang Chen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5092)

Abstract

Coset enumeration is for enumerating the cosets of a subgroup H of a finite index in a group G. We study coset enumeration algorithms by using two random sources to generate random elements in a finite group G and its subgroup H. For a finite set S and a real number c > 0, a random generator R S is a c-random source for S if c· min {Pr[a = R S ())|a ∈ S]} ≥ max {Pr[a = R S ())|a ∈ S]}. Let c be an arbitrary constant. We present an \(O({|G|\over \sqrt{|H|}}(\log |G|)^3)\)-time randomized algorithm that, given two respective c-random sources R G for a finite group G and R H for a subgroup H ⊆ G, computes the index \(t={|G|\over |H|}\) and a list of elements a 1,a 2, ⋯ , a t  ∈ G such that a i H ∩ a j H = ∅ for all \(i\not=j\), and \(\cup_{i=1}^t a_iH=G\). This algorithm is sublinear time when |H| = Ω((log|G|)6 + ε ) for some constant ε> 0.

Keywords

Finite Group Random Element Intersection Graph Algorithm Approximate Great Common Divisor 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Bin Fu
    • 1
  • Zhixiang Chen
    • 1
  1. 1.Dept. of Computer ScienceUniversity of TexasPan AmericanUSA

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