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Arkadius Kalka

Publications and source records attributed to Arkadius Kalka.

13 recordsLinked to original sources
arXiv2020

The numerical statistical fan for noisy experimental designs

Identifiability of polynomial models is a key requirement for multiple regression. We consider an analogue of the so-called statistical fan, the set of all maximal identifiable hierarchical models, for cases of noisy experimental designs or measured covariate vectors with a given tolerance vector. This gives rise to the definition of the numerical statistical fan. It includes all maximal hierarchical models that avoid approximate linear dependence of the model terms. We develop an algorithm to compute the numerical statistical fan using recent results on the computation of all border bases of a design ideal from the field of algebra. The ideas are applied to data from a thermal spraying process. It turns out that the numerical statistical fan is effectively computable and much smaller than the respective statistical fan. The gained enhanced knowledge of the space of all stable identifiable hierarchical models enables improved model selection procedures.

stat.ME↗
arXiv2015

Power commutator groups

We consider the class of finitely generated groups whose relators are powers of commutators of the generators. This class contains as a small subclass graph groups (also called RAAGs), namely if all powers are one. Graph groups are the only torsionfree groups in this class. The generators are of infinite order, but we may also add torsion by assigning arbitrary orders to the generators. Then the above mentioned small subclass contains partially commutative Shephard groups. We show that these groups embed into Coxeter groups as finite index subgroups, thus establishing the linearity of these groups. The very short proof requires only elementary methods in combinatorial group theory.

math.GR↗
arXiv2015

Double Centralizers of Parabolic Subgroups of Braid Groups

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.

math.GR↗
arXiv2014

2-manifold recognition is in logspace

We prove that the homeomorphism problem for 2-manifolds can be decided in logspace. The proof relies on Reingold's logspace solution to the undirected $s,t$-connectivity problem in graphs.

math.GT↗
arXiv2014

Complete simultaneous conjugacy invariants in Artin's braid groups

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 cardinality of this invariant, for a list of order $N$ independent elements of Artin's braid group $B_N$, is generically close to~1.

math.GR↗
arXiv2014

Double coset problem for parabolic subgroups of braid groups

We solve the double coset problem for all parabolic subgroups of braid groups. The solution provides an effective reduction of this problem to the simultaneous conjugacy problem, and resolves the ambiguity introduced by the center of the group via the double centralizer theorem. We also prove that the subgroup-restricted conjugacy problem is unsolvable in braid groups of at least 5 strands.

math.GR↗
arXiv2013

Iterated LD-Problem in non-associative key establishment

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.

cs.CR↗
arXiv2013

Logspace computations for Garside groups of spindle type

M. Picantin introduced the notion of Garside groups of spindle type, generalizing the 3-strand braid group. We show that, for linear Garside groups of spindle type, a normal form and a solution to the conjugacy problem are logspace computable. For linear Garside groups of spindle type with homogenous presentation we compute a geodesic normal form in logspace.

math.GR↗
arXiv2013

Non-associative key establishment for left distributive systems

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.

cs.CR↗
arXiv2012

Non-associative public-key cryptography

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.

cs.CR↗
arXiv2012

Short expressions of permutations as products and cryptanalysis of the Algebraic Eraser

On March 2004, Anshel, Anshel, Goldfeld, and Lemieux introduced the \emph{Algebraic Eraser} scheme for key agreement over an insecure channel, using a novel hybrid of infinite and finite noncommutative groups. They also introduced the \emph{Colored Burau Key Agreement Protocol (CBKAP)}, a concrete realization of this scheme. We present general, efficient heuristic algorithms, which extract the shared key out of the public information provided by CBKAP. These algorithms are, according to heuristic reasoning and according to massive experiments, successful for all sizes of the security parameters, assuming that the keys are chosen with standard distributions. Our methods come from probabilistic group theory (permutation group actions and expander graphs). In particular, we provide a simple algorithm for finding short expressions of permutations in $S_n$, as products of given random permutations. Heuristically, our algorithm gives expressions of length $O(n^2\log n)$, in time and space $O(n^3)$. Moreover, this is provable from \emph{the Minimal Cycle Conjecture}, a simply stated hypothesis concerning the uniform distribution on $S_n$. Experiments show that the constants in these estimations are small. This is the first practical algorithm for this problem for $n\ge 256$. Remark: \emph{Algebraic Eraser} is a trademark of SecureRF. The variant of CBKAP actually implemented by SecureRF uses proprietary distributions, and thus our results do not imply its vulnerability. See also arXiv:abs/12020598

math.GR↗
arXiv2009

Complexity of relations in the braid group

We show that for any given n, there exists a sequence of words a_k in the generators sigma_1, ... sigma_{n-1} of the braid group B_n, representing the identity element of B_n, such that the number of braid relations of the form sigma_i sigma_{i+1} sigma_i = sigma_{i+1} sigma_i sigma_{i+1} needed to pass from a_k to the empty word is quadratic with respect to the length of a_k.

math.GR↗