Several key agreement protocols are based on the following "Generalized Conjugacy Search Problem": Find, given elements b_1,...,b_n and xb_1x^{-1},...,xb_nx^{-1} in a nonabelian group G, the conjugator x. In the case of subgroups of the braid group B_N, Hughes and Tannenbaum suggested a length-based approach to finding x. Since the introduction of this approach, its effectiveness and successfulness were debated.
We introduce several effective realizations of this approach. In particular, a new length function is defined on B_N which possesses significantly better properties than the natural length associated to the Garside normal form. We give experimental results concerning the success probability of this approach, which suggest that very large computational power is required for this method to successfully solve the Generalized Conjugacy Search Problem when its parameters are as in existing protocols.
We present an algorithm for solving the conjugacy search problem in the four strand braid group. The computational complexity is cubic with respect to the braid length.
In this note we solve the twisted conjugacy problem for braid groups, i.e. we propose an algorithm which, given two braids $u,v\in B_n$ and an automorphism $ϕ\in Aut (B_n)$, decides whether $v=(ϕ(x))^{-1}ux$ for some $x\in B_n$. As a corollary, we deduce that each group of the form $B_n \rtimes H$, a semidirect product of the braid group $B_n$ by a torsion-free hyperbolic group $H$, has solvable conjugacy problem.
We show that there is a family of pseudo-Anosov braids independently parameterized by the braid index and the (canonical) length whose smallest conjugacy invariant sets grow exponentially in the braid index and linearly in the length and conclude that the conjugacy problem remains exponential in the braid index under the current knowledge.
There are recent cryptographic protocols that are based on Multiple Simultaneous Conjugacy Problems in braid groups. We improve an algorithm, due to Sang Jin Lee and Eonkyung Lee, to solve these problems, by applying a method developed by the author and Nuno Franco, originally intended to solve the Conjugacy Search Problem in braid groups.
Let $G$ be a group endowed with a solution to the conjugacy problem and with an algorithm which computes the centralizer in $G$ of any element of $G$. Let $H$ be a subgroup of $G$. We give some conditions on $H$, under which we provide a solution to the conjugacy problem in $H$. We apply our results to some Artin groups and braid type groups. In particular, we give explicit solutions to the conjugacy problem in the Artin groups of type $\tilde{A}_n$ and $\tilde{C}_n$.
We present a new algorithm to solve the conjugacy problem in Artin braid groups, which is faster than the one presented by Birman, Ko and Lee. This algorithm can be applied not only to braid groups, but to all Garside groups (which include finite type Artin groups and torus knot groups among others).
We solve the simultaneous conjugacy problem in Artin's braid groups and, more generally, in Garside groups, by means of a complete, effectively computable, finite invariant. This invariant generalizes the one-dimensional notion of super summit set to arbitrary dimensions. One key ingredient in our solution is the introduction of a provable high-dimensional version of the Birman--Ko--Lee cycling theorem. The complexity of this solution is a small degree polynomial in the cardinalities of our generalized super summit sets and the input parameters. Computer experiments suggest that the cardinalit…
Random braids that are formed by multiplying randomly chosen permutation braids are studied by analyzing their behavior under Garside's weighted decomposition and cycling. Using this analysis, we propose a polynomial-time algorithm to the conjugacy problem that is successful for random braids in overwhelming probability. As either the braid index or the number of permutation-braid factors increases, the success probability converges to 1 and so, contrary to the common belief, the distribution of hard instances for the conjugacy problem is getting sparser. We also prove a conjecture by Birman a…
We prove a conjecture due to Makanin: if a and b are elements of the Artin braid group B_n such that a^k=b^k for some nonzero integer k, then a and b are conjugate. The proof involves the Nielsen-Thurston classification of braids.
In this article we describe the summit sets in B_3, the smallest element in a summit set and we compute the Hilbert series corresponding to conjugacy classes.The results will be related to Birman-Menesco classification of knots with braid index three or less than three.
Let $n\geq 3$. In this paper we deal with the conjugacy problem in the Artin braid group quotient $B_n/[P_n,P_n]$. To solve it we use systems of equations over the integers arising from the action of $B_n/[P_n,P_n]$ over the abelianization of the pure Artin braid group $P_n/[P_n,P_n]$. Using this technique we also realize explicitly infinite virtually cyclic subgroups in $B_n/[P_n,P_n]$.
An embedding of the m-times punctured disc into the n-times punctured disc, for n>m, yields an embedding of the braid group on m strands B_m into the braid group on n strands B_n, called a geometric embedding. The main example consists of adding n-m trivial strands to the right of each braid on m strands. We show that geometric embeddings do not merge conjugacy classes, meaning that if the images of two elements in B_m by a geometric embedding are conjugate in B_n, the original elements are conjugate in B_m. We also show that the result does not hold, in general, for geometric embeddings of ma…
We prove that the symplectic group $Sp(2n,\mathbb Z)$ and the mapping class group $Mod_{S}$ of a compact surface $S$ satisfy the $R_{\infty}$ property. We also show that $B_n(S)$, the full braid group on $n$-strings of a surface $S$, satisfies the $R_{\infty}$ property in the cases where $S$ is either the compact disk $D$, or the sphere $S^2$. This means that for any automorphism $ϕ$ of $G$, where $G$ is one of the above groups, the number of twisted $ϕ$-conjugacy classes is infinite.
An element in Artin's braid group B_n is said to be periodic if some power of it lies in the center of B_n. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B_n are exponential in the braid index n for the special case of periodic braids. We overcome this difficulty by putting to work several known isomorphisms between Garside structures in the braid group B_n and other Garside groups. This allows us to obtain a polynomial solution to the original problem in the spirit of the previously known algorithms. This paper is the third in a series…
Recently, there have been several progresses for the conjugacy search problem (CSP) in Garside groups, especially in braid groups. All known algorithms for solving this problem use a sort of exhaustive search in a particular finite set such as the super summit set and the ultra summit set. Their complexities are proportional to the size of the finite set, even when there exist very short conjugating elements. However, ultra summit sets are very large in some cases especially for reducible braids and periodic braids. Some possible approaches to resolve this difficulty would be either to use dif…
Algorithmic solutions to the conjugacy problem in the braid groups B_n were given by Elrifai-Morton in 1994 and by the authors in 1998. Both solutions yield two conjugacy class invariants which are known as `inf' and `sup'. A problem which was left unsolved in both papers was the number m of times one must `cycle' (resp. `decycle') in order to increase inf (resp. decrease sup) or to be sure that it is already maximal (resp. minimal) for the given conjugacy class. Our main result is to prove that m is bounded above by n-2 in the situation of the second algorithm and by ((n^2-n)/2)-1 in the situ…
In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid groups. We also prove that half-twists satisfy a special root property which allows us to reduce the solution for the conjugacy problem in half-twists into the free group. Using this algorithm one is able to check conjugacy of a given braid to one of E. Artin's generators in any power, and compute its root. Moreover, the braid element which conjugates a given half-twist to one of E. Artin's generators in any power can be restored. The result is applicable to calculations of braid monodromy of branc…
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 realizations using $f$-conjugacy in groups and shifted conjugacy in braid groups. We discuss the advantages of our schemes compared with the classical AAG-KEP based on conjugacy in braid groups.
We construct new non-associative key establishment protocols for all left self-distributive (LD), multi-LD-, and mutual LD-systems. The hardness of these protocols relies on variations of the (simultaneous) iterated LD-problem and its generalizations. We discuss instantiations of these protocols using generalized shifted conjugacy in braid groups and their quotients, LD-conjugacy and $f$-symmetric conjugacy in groups. We suggest parameter choices for instantiations in braid groups, symmetric groups and several matrix groups.
Let n\geq 3. We classify the finite groups which are realised as subgroups of the sphere braid group B_n(S^2). Such groups must be of cohomological period 2 or 4. Depending on the value of n, we show that the following are the maximal finite subgroups of B_n(S^2): Z_{2(n-1)}; the dicyclic groups of order 4n and 4(n-2); the binary tetrahedral group T_1; the binary octahedral group O_1; and the binary icosahedral group I. We give geometric as well as some explicit algebraic constructions of these groups in B_n(S^2), and determine the number of conjugacy classes of such finite subgroups. We also …
We provide new group presentations for surface braid groups which are positive. We study some properties of such presentations and we solve the conjugacy problem in a particular case.
We show thatthe double reversing algorithm proposed by dehornoy for solving the word problem in the braid group can also be used to recognize the conjugates of powers of the generators in an Artin group of spherical type. The proof uses a characterization of these powers in terms of their fractional decomposition. This algorithm could have potential applications to braid-based cryptography; it also provides a fast method for testing a necessary condition in the classification of surfaces in algebraic geometry.
This paper proposes for every $n$, linear time reductions of the word and conjugacy problems on the braid groups $B_n$ to the corresponding problems on the braid monoids $B_n^+$ and moreover only using positive words representations.
The braid group has recently attracted much attention. This is primarily based upon the discovery of its usage in various cryptosystems [AAG],[KLCHKP]. One major focus of current research has been in solving decision problems in braid groups and analyzing their complexity. Among the decision problems which have been investigated are: the word problem, the conjugacy problem and the conjugacy search problem. One decision problem which has not been thoroughly pursued is the generalized word problem. It can be shown that this problem is, in general, unsolvable for braid groups of index greater the…
In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where 'rigid' means that the left normal form changes only in the obvious way under cycling and decycling. It is also shown that, given X in a Garside group, if some power X^m is conjugate to a rigid element, then m can be bounded above by ||Δ||^3. In the particular case of braid groups, this implies that a pseudo-Anosov braid has a small power whose ultra summit set consists of rigid elements. This s…
Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid braids with a fixed number of strands, the size of this set is bounded by a polynomial in the length of the braids. In this paper we suggest a more precise bound: for rigid braids with $N$ strands and of Garside length $L$, the sliding circuit set should have at most $C\cdot L^{N-2}$ elements, for some constant $C$. We construct a family of braids which realise t…
We show that reducible braids which are, in a Garside-theoretical sense, as simple as possible within their conjugacy class, are also as simple as possible in a geometric sense. More precisely, if a braid belongs to a certain subset of its conjugacy class which we call the stabilized set of sliding circuits, and if it is reducible, then its reducibility is geometrically obvious: it has a round or almost round reducing curve. Moreover, for any given braid, an element of its stabilized set of sliding circuits can be found using the well-known cyclic sliding operation. This leads to a polynomial …
We give a complete classification of conjugacy stable parabolic subgroups of Artin-Tits groups of spherical type. This answers a question posed by Ivan Marin and generalizes a theorem obtained by Juan González-Meneses in the specific case of Artin braid groups.
The cycling operation is a special kind of conjugation that can be applied to elements in Artin's braid groups, in order to reduce their length. It is a key ingredient of the usual solutions to the conjugacy problem in braid groups. In their seminal paper on braid-cryptography, Ko, Lee et al. proposed the {\it cycling problem} as a hard problem in braid groups that could be interesting for cryptography. In this paper we give a polynomial solution to that problem, mainly by showing that cycling is surjective, and using a result by Maffre which shows that pre-images under cycling can be computed…
In [J.Birman, V.Gebhardt, J.Gonzalez-Meneses, Conjugacy in Garside groups I: cyclings, powers and rigidity] authors asked: (open question 2) is the size of USS of a rigid pseudo-Anosov braid is bounded above by some polynomial in the number of strands and the braid length? We answer this question in the negative.
In 1965, Steinberg's study of conjugacy classes in connected reductive groups led him to construct an affine subspace parametrising regular conjugacy classes, which he noticed is also a cross section for the conjugation action by the unipotent radical of a Borel subgroup on another affine subspace. Recently, generalisations of this slice and its cross section property have been obtained by Sevostyanov in the context of quantum group analogues of W-algebras and by He-Lusztig in the context of Deligne-Lusztig varieties. Such slices are often thought of as group analogues of Slodowy slices. In th…
We begin with a review of the notion of a braid group. We then discuss some known solutions to decision problems in braid groups. We then move on to proving new results in braid group algorithmics. We offer a quick solution to the generalized word problem in braid groups, in the special case of cyclic subgroups. We illustrate this solution and its complexity using a multitape Turing machine. We then turn to a discussion of decision problems in cyclic amalgamations of groups. Again using a multitape Turing machine, we solve the word problem for the cyclic amalgamation of two braid groups. We an…
Given a group $G$ and a subset $X \subset G$, an element $g \in G$ is called quasi-positive if it is equal to a product of conjugates of elements in the semigroup generated by $X$. This notion is important in the context of braid groups, where it has been shown that the closure of quasi-positive braids coincides with the geometrically defined class of $\mathbb{C}$-transverse links. We describe an algorithm that recognizes whether or not an element of a free group is quasi-positive with respect to a basis. Spherical cancellation diagrams over free groups are used to establish the validity of th…
Let $G$ be one of the Artin groups of finite type ${\mathbf B}_n={\mathbf C}_n$, and affine type $\tilde{\mathbf A}_{n-1}$ and $\tilde{\mathbf C}_{n-1}$. In this paper, we show that if $α$ and $β$ are elements of $G$ such that $α^k=β^k$ for some nonzero integer $k$, then $α$ and $β$ are conjugate in $G$. For the Artin group of type $\mathbf A_n$, this was recently proved by J. González-Meneses. In fact, we prove a stronger theorem, from which the above result follows easily by using descriptions of those Artin groups as subgroups of the braid group on $n+1$ strands. Let $P$ be a subset of $\{1…
We characterize the double centralizer of all parabolic subgroups of the braid groups. We apply this result to provide a new and potentially more efficient solution to the subgroup conjugacy problem for parabolic subgroups. In the course of the proof we also characterize the centralizer for all parabolic subgroups.
We give a new presentation of the braid group $B$ of the complex reflection group $G(e,e,r)$ which is positive and homogeneous, and for which the generators map to reflections in the corresponding complex reflection group. We show that this presentation gives rise to a Garside structure for $B$ with Garside element a kind of generalised Coxeter element, and hence obtain solutions to the word and conjugacy problems for $B$.
We construct non-associative key establishment protocols for all left self-distributive (LD), multi-LD-, and other left distributive systems. Instantiations of these protocols using generalized shifted conjugacy in braid groups lead to instances of a natural and apparently new group-theoretic problem, which we call the (subgroup) conjugacy coset problem.
In the paper we give a survey of rather new notions and results which generalize classical ones in the theory of braids. Among such notions are various inverse monoids of partial braids. We also observe presentations different from standard Artin presentation for generalizations of braids. Namely, we consider presentations with small number of generators, Sergiescu graph-presentations and Birman-Ko-Lee presentation. The work of V.V.Chaynikov on the word and conjugacy problems for the singular braid monoid in Birman-Ko-Lee generators is described as well.
We prove that the conjugacy problem in right-angled Artin groups (RAAGs), as well as in a large and natural class of subgroups of RAAGs, can be solved in linear-time. This class of subgroups contains, for instance, all graph braid groups (i.e. fundamental groups of configuration spaces of points in graphs), many hyperbolic groups, and it coincides with the class of fundamental groups of ``special cube complexes'' studied independently by Haglund and Wise.
The cycling operation endows the super summit set $S_x$ of any element $x$ of a Garside group $G$ with the structure of a directed graph $Γ_x$. We establish that the subset $U_x$ of $S_x$ consisting of the circuits of $Γ_x$ can be used instead of $S_x$ for deciding conjugacy to $x$ in $G$, yielding a faster and more practical solution to the conjugacy problem for Garside groups. Moreover, we present a probabilistic approach to the conjugacy search problem in Garside groups. The results are likely to have implications for the security of recently proposed cryptosystems based on the hardness of …
Determining the length of short conjugators in a group can be considered as an effective version of the conjugacy problem. The conjugacy length function provides a measure for these lengths. We study the behaviour of conjugacy length functions under group extensions, introducing the twisted and restricted conjugacy length functions. We apply these results to show that certain abelian-by-cyclic groups have linear conjugacy length function and certain semidirect products $\Z^d \rtimes \Z^k$ have at most exponential (if $k >1$) or linear (if $k=1$) conjugacy length functions.
We use some Lie group theory and Budney's unitarization of the Lawrence-Krammer representation, to prove that for generic parameters of definite form the image of the representation (also on certain types of subgroups) is dense in the unitary group. This implies that, except possibly for closures of full-twist braids, all links have infinitely many conjugacy classes of braid representations on any non-minimal number of (and at least 4) strands.
Given a system of equations in a "random" finitely generated subgroup of the braid group, we show how to find a small ordered list of elements in the subgroup, which contains a solution to the equations with a significant probability. Moreover, with a significant probability, the solution will be the first in the list. This gives a probabilistic solution to: The conjugacy problem, the group membership problem, the shortest representation of an element, and other combinatorial group-theoretic problems in random subgroups of the braid group. We use a memory-based extension of the standard length…
Modelled on efficient algorithms for solving the conjugacy problem in hyperbolic groups, we define and study the permutation conjugacy length function. This function estimates the length of a short conjugator between words $u$ and $v$, up to taking cyclic permutations. This function might be bounded by a constant, even in the case when the standard conjugacy length function is unbounded. We give applications to the complexity of the conjugacy problem, estimating conjugacy growth rates, and languages. Our main result states that for a relatively hyperbolic group, the permutation conjugacy lengt…
We generalize presentations of the fundamental group of discriminant complements and arrive at a class of presentations associated naturally with words in the free monoid of the alphabet $σ_1,\dots,σ_{n-1}$. Our study addresses invariance properties of these presentations and the presented groups under various operations on the words. In particular we prove that the group does only depend on the corresponding element in the positive braid monoid, and under mild hypotheses only on its conjugacy class in the braid group.
We give a unified solution to the conjugacy problem for Thompson's groups F, T, and V. The solution uses strand diagrams, which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompson's groups. Strand diagrams are closely related to piecewise-linear functions for elements of Thompson's groups, and we use this correspondence to investigate the dynamics of elements of F. Though many of the results in this paper are known, our approach is new, and it yields elegant proofs of several old results.
Let n be greater than or equal to 3. We study the quotient group B\_n/[P n,P\_n] of the Artin braid group B\_n by the commutator subgroup of its pure Artin braid group P\_n. We show that B\_n/[P n,P\_n] is a crystallographic group, and in the case n=3, we analyse explicitly some of its subgroups. We also prove that B\_n/[P n,P\_n] possesses torsion, and we show that there is a one-to-one correspondence between the conjugacy classes of the finite-order elements of B\_n/[P n,P\_n] with the conjugacy classes of the elements of odd order of the symmetric group S\_n, and that the isomorphism class …
In the paper we give a survey on braid groups and subjects connected with them. We start with the initial definition, then we give several interpretations as well as several presentations of these groups. Burau presentation for the pure braid group and the Markov normal form are given next. Garside normal form and his solution of the conjugacy problem are presented as well as more recent results on the ordering and on the linearity of braid groups. Next topics are the generalizations of braids, their homological properties and connections with the other mathematical fields, like knot theory (v…
Let $G$ be a supersolvable group and $A$ be a conjugacy class of $G$. Observe that for some integer $η(AA^{-1})>0$, $AA^{-1}=\{a b^{-1}\mid a,b\in A\}$ is the union of $η(AA^{-1})$ distinct conjugacy classes of $G$. Set ${\bf C}_G(A)=\{g\in G\mid a^g=a\text{for all} a\in A\}$. Then the derived length of $G/{\bf C}_G(A)$ is less or equal than $2η(A A^{-1})-1$.
We give a new method to compute the centralizer of an element in Artin braid groups and, more generally, in Garside groups. This method, together with the solution of the conugacy problem given by the authors in a previous paper, are two main steps for solving conjugacy systems, thus breaking recently discovered cryptosystems based in braid groups. We also present the result of our computations, where we notice that our algorithm yields surprisingly small generating sets for the centralizers.
These are Lecture Notes of a course given by the author at the French-Spanish School "Tresses in Pau", held in Pau (France) in October 2009. It is basically an introduction to distinct approaches and techniques that can be used to show results in braid groups. Using these techniques we provide several proofs of well known results in braid groups, namely the correctness of Artin's presentation, that the braid group is torsion free, or that its center is generated by the full twist. We also recall some solutions of the word and conjugacy problems, and that roots of a braid are always conjugate. …
Let W be a Coxeter group. In this paper, we establish that, up to going to some finite index normal subgroup W_0 of W, any two cyclically reduced expressions of conjugate elements of W_0 only differ by a sequence of braid relations and cyclic shifts. This thus provides a simple description of conjugacy classes in W_0. As a byproduct of our methods, we also obtain a characterisation of straight elements of W, namely of those elements w in W for which $\ell(w^n)=n\ell(w)$ for any integer n. In particular, we generalise previous characterisations of straight elements within the class of so-called…
Recently, Lusztig constructed for each reductive group a partition by unions of sheets of conjugacy classes, which is indexed by a subset of the set of conjugacy classes in the associated Weyl group. Sevostyanov subsequently used certain elements in each of these Weyl group conjugacy classes to construct strictly transverse slices to the conjugacy classes in these strata, generalising the classical Steinberg slice, and similar cross sections were built out of different Weyl group elements by He-Lusztig. In this paper we observe that He-Lusztig's and Sevostyanov's Weyl group elements share a ce…
We obtain new presentations for the imprimitive complex reflection groups of type $(de,e,r)$ and their braid groups $B(de,e,r)$ for $d,r \ge 2$. Diagrams for these presentations are proposed. The presentations have much in common with Coxeter presentations of real reflection groups. They are positive and homogeneous, and give rise to quasi-Garside structures. Diagram automorphisms correspond to group automorphisms. The new presentation shows how the braid group $B(de,e,r)$ is a semidirect product of the braid group of affine type $\widetilde A_{r-1}$ and an infinite cyclic group. Elements of $…
