Black box group

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In computational group theory, a black box group (black-box group) is a group G whose elements are encoded by bit strings of length N, and group operations are performed by an oracle (the "black box"). These operations include:

Contents

This class is defined to include both the permutation groups and the matrix groups. The upper bound on the order of G given by |G| ≤ 2N shows that G is finite.

Applications

The black box groups were introduced by Babai and Szemerédi in 1984. [1] They were used as a formalism for (constructive) group recognition and property testing . Notable algorithms include the Babai's algorithm for finding random group elements, [2] the Product Replacement Algorithm, [3] and testing group commutativity . [4]

Many early algorithms in CGT, such as the Schreier–Sims algorithm, require a permutation representation of a group and thus are not black box. Many other algorithms require finding element orders. Since there are efficient ways of finding the order of an element in a permutation group or in a matrix group (a method for the latter is described by Celler and Leedham-Green in 1997), a common recourse is to assume that the black box group is equipped with a further oracle for determining element orders. [5]

See also

Notes

  1. ↑ Babai, L.; Szemeredi, E. (1984). "On the Complexity of Matrix Group Problems I". 25th Annual Symposium on Foundations of Computer Science, 1984. pp. 229–240. doi:10.1109/SFCS.1984.715919. ISBN   0-8186-0591-X.
  2. ↑ L. Babai, Local expansion of vertex-transitive graphs and random generation in finite groups, Proc. 23rd STOC (1991), 164–174.
  3. ↑ Frank Celler; Charles R. Leedham-Green; Scott H. Murray; Alice C. Niemeyer; E.A. O'Brien (1995). "Generating random elements of a finite group". Communications in Algebra. 23 (3): 4931–4948. CiteSeerX   10.1.1.43.2250 . doi:10.1080/00927879508825509.
  4. ↑ Pak, Igor (2012). "Testing commutativity of a group and the power of randomization". LMS Journal of Computation and Mathematics. 15: 38–43. doi: 10.1112/S1461157012000046 .
  5. ↑ See Holt et al. (2005).

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