Ore revisited: an algorithmic investigation of the simple commutator promise problem

Abstract

Motivated by a desire to test the security of the pubic key exchange protocol of I. Anshel, M. Anshel, and D. Goldfeld, (``An Algebraic Method for Public-Key Cryptography'', Mathematical Research Letters, vol. 6, pp. 1-5, 1999), we study algorithmic approaches to the simple commutator decision and promise problems (SCDP/SCPP) for the braid groups B_n. We take as our point of departure a seminal paper of O. Ore, (``Some Remarks on Commutators'', Proceedings of the American Mathematical Society, Vol. 2, No. 2, pp.307-314, 1951), which studies the SCPP for the symmetric groups. Our results build on the work of H. Cejtin and I. Rivin, (``A Property of Alternating Groups'', arXiv:math.GR/0303036).