Black box group

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:

• taking a product g·h of elements g and h,
• taking an inverse g⁻¹ of element g,
• deciding whether g = 1.

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| ≤ 2^N 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

• Implicit graph
• Matroid oracle

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).

References

• Derek F. Holt, Bettina Eick, Eamonn A. O'Brien, Handbook of computational group theory, Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, Florida, 2005. ISBN 1-58488-372-3
• Ákos Seress, Permutation group algorithms, Cambridge Tracts in Mathematics, vol. 152, Cambridge University Press, Cambridge, 2003. ISBN 0-521-66103-X.
• Kantor, William M.; Seress, Ákos (2001). Black Box Classical Groups. Memoirs of the American Mathematical Society. Vol. 708. American Mathematical Society. ISBN 978-0-8218-2619-5. ISSN 0065-9266.