In this note, we describe a probabilistic attack on public key cryptosystems based on the word/conjugacy problems for finitely presented groups of the type proposed recently by Anshel, Anshel and Goldfeld. In such a scheme, one makes use of the property that in the given group the word problem has a polynomial time solution, while the conjugacy problem has no known polynomial solution. An example is the braid group from topology in which the word problem is solvable in polynomial time while the only known solutions to the conjugacy problem are exponential. The attack in this paper is based on having a canonical representative of each string relative to which a length function may be computed. Hence the term length attack. Such canonical representatives are known to exist for the braid group.

We study combinatorial properties, such as inversion table, weak order and Bruhat order, for certain infinite permutations that realize the affine Coxeter group ~ A n .

We prove a finite version of the well-known theorem that says that the number of partitions of an integer N into distinct parts is equal to the number of partitions of N into odd parts. Our version says that the number of "lecture hall partitions of length n" of N equals the number of partitions of N into small odd parts: 1; 3; 5; : : :; 2n \Gamma 1. We give two proofs: one via Bott's formula for the Poincar'e series of the affine Coxeter group e Cn , and one direct proof. 1 Introduction Let D be the set of integer partitions with distinct parts, and let O be the set of integer partitions with odd parts. It is well-known, since Euler [6], that the generating function for the elements of D is equal to the generating function for the elements of O: X ¯2D q j¯j = Y i1 (1 + q i ) = Y i1 1 \Gamma q 2i 1 \Gamma q i = Y i0 1 1 \Gamma q 2i+1 = X ¯2O q j¯j ; (1) where the weight j¯j of a partition ¯ = (¯ 1 ; : : : ; ¯m ) is ¯ 1 + \Delta \Delta \Delta + ¯m . In other ...

Let W denote a simply-laced Coxeter group with n generators. We construct an n-dimensional representation φ of W over the finite field F2 of two elements. The action of φ(W) on F n 2 by left multiplication is corresponding to a combinatorial structure extracted and generalized from Vogan diagrams. In each case W of types A, D and E, we determine the orbits of F n 2 under the action of φ(W), and find that the kernel of φ is the center Z(W) of W.

Abstract not found

We use the author’s combinatorial theory of full heaps to cate-gorify the action of a large class of Weyl groups on their root systems, and thus to give an elementary and uniform construction of a family of faithful permutation representations of Weyl groups. Examples include the standard representations of affine Weyl groups as permutations of Z and geometrical examples such as the realization of the Weyl group of type E6 as permutations of 27 lines on a cubic surface; in the latter case, we also show how to recover the incidence relations between the lines from the structure of the heap. An-other class of examples involves the action of certain Weyl groups on sets of pairs (t, f), where t ∈ Z and f is a function from a suitably chosen set to the two-element set {+, −}. Each of the permutation representations corresponds

Abstract not found

Let ˜ W be an affine Weyl group, and let C be a left, right, or twosided Kazhdan–Lusztig cell in ˜ W. Let Red(C) be the set of all reduced expressions of elements of C, regarded as a formal language in the sense of the theory of computation. We show that Red(C) is a regular language. Hence the reduced expressions of the elements in any Kazhdan–Lusztig cell can be enumerated by a finite state automaton.

Abstract not found

Abstract not found