Local expansion of vertex-transitive graphs and random generation in finite groups
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Cryptography and Network Security (4th Edition)
Cryptography and Network Security (4th Edition)
Introduction to Algorithms, Third Edition
Introduction to Algorithms, Third Edition
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Coset enumeration is for enumerating the cosets of a subgroup Hof 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 Gand its subgroup H. For a finite set Sand a real number c 0, a random generator RSis a c-random source for Sif c· min {Pr[a= RS())|a茂戮驴 S]} 茂戮驴 max {Pr[a= RS())|a茂戮驴 S]}. Let cbe an arbitrary constant. We present an $O({|G|\over \sqrt{|H|}}(\log |G|)^3)$-time randomized algorithm that, given two respective c-random sources RGfor a finite group Gand RHfor a subgroup H茂戮驴 G, computes the index $t={|G|\over |H|}$ and a list of elements a1,a2, 茂戮驴 , at茂戮驴 Gsuch that aiH茂戮驴 ajH= 茂戮驴 for all $i\not=j$, and $\cup_{i=1}^t a_iH=G$. This algorithm is sublinear time when |H| = 茂戮驴((log|G|)6 + 茂戮驴) for some constant 茂戮驴 0.