In recent years, Artin’s braid groups have undergone intense scrutiny as a possible cryptographic platform, due mainly to the key-exchange protocols proposed in [1] and [12]. The security of both of these protocols relies on the difficulty of variants of the conjugacy search problem in braid groups, a well-studied problem (see [9], [6], [10]). More recently, Dehornoy has proposed a zero-knowled...

Given an automorphism of a free group Fn, we consider the following invariants: e is the number of exponential strata (an upper bound for the number of different exponential growth rates of conjugacy classes); d is the maximal degree of polynomial growth of conjugacy classes; R is the rank of the fixed subgroup. We determine precisely which triples (e, d, R) may be realized by an automorphism o...

Trapezoid maps are a kind of continuous and piecewise linear maps with a flat top. By the conjugacy relationship, we present a complete classification for four families of trapezoid maps. Firstly, using an extension method, we construct all homeomorphic solutions of conjugacy equation φ ◦ f = g ◦ φ for some non-monotone continuous maps f and g. Secondly, using an iterative construction method a...

Let G be a finite group and H a normal subgroup such that G/H is cyclic. Given a conjugacy class g of G we define its centralizing subgroup to be HCG(g). Let K be such that H ≤ K ≤ G. We show that the G-conjugacy classes contained in K whose centralizing subgroup is K, are equally distributed between the cosets of H in K. The proof of this result is entirely elementary. As an application we fin...

Let G be a finite group and let T1 denote the number of times a triple (x, y, z) ∈ G3 binds X, where X = {xyz, xzy, yxz, yzx, zxy, zyx}, to one conjugacy class. Let T2 denote the number of times a triple in G3 breaks X into two conjugacy classes. We have established the following results: i) the probability that a triple (x, y, z) ∈ D3 n binds X to one conjugacy class is ≥ 58 . ii) for groups s...

We introduce a generalized Anshel-Anshel-Goldfeld (AAG) key establishment protocol (KEP) for magmas. This leads to the foundation of non-associative public-key cryptography (PKC), generalizing the concept of non-commutative PKC. We show that left selfdistributive systems appear in a natural special case of a generalized AAG-KEP for magmas, and we propose, among others instances, concrete realiz...

Several decades ago, John McKay suggested a correspondence between nodes of the affine E8 Dynkin diagram and certain conjugacy classes in the Monster group. Thanks to Monstrous Moonshine, this correspondence can be recast as an assignment of discrete subgroups of PSL2(R) to nodes of the affine E8 Dynkin diagram. The purpose of this article is to give an explanation for this latter correspondenc...

Suppose $G$ is a connected complex semisimple group and $W$ its Weyl group. The lifting of an element to semisimple. This induces well-defined map from the set elliptic conjugacy classes $G$. In this paper, we give uniform algorithm compute map. We also consider twisted case.

The paper presents some open problems associated to the nonlinear conjugate gradient algorithms for unconstrained optimization. Mainly, these problems refer to the initial direction, the conjugacy condition, the step length computation, new formula for conjugate gradient parameter computation based on function’s values, the influence of accuracy of line search procedure, how we can take the pro...

A global convergence theorem for unconstrained minimization algorithms with an eecient line search is derived. The theorem applies to a new version of the conjugate gradient method derived here in terms of minimizing the eeect of zigzagging. The global convergence condition makes much weaker demands on the line search than previous methods. Local Q-linear convergence in a neighborhood of a stro...