scholarly journals A parallel evolutionary approach to solving systems of equations in polycyclic groups

2016 ◽  
Vol 8 (2) ◽  
Author(s):  
Matthew J. Craven ◽  
Daniel Robertz
Keyword(s):  
Linear Algebra ◽  
High Performance ◽  
Conjugacy Problem ◽  
Evolutionary Approach ◽  
Alternative Measure ◽  
Systems Of Equations ◽  
Key Exchange Protocol ◽  
Finitely Presented Group ◽  
Polycyclic Groups ◽  
Finitely Presented

AbstractThe Anshel–Anshel–Goldfeld (AAG) key exchange protocol is based upon the multiple conjugacy problem for a finitely-presented group. The hardness in breaking this protocol relies on the supposed difficulty in solving the corresponding equations for the conjugating element in the group. Two such protocols based on polycyclic groups as a platform were recently proposed and were shown to be resistant to length-based attack. In this article we propose a parallel evolutionary approach which runs on multicore high-performance architectures. The approach is shown to be more efficient than previous attempts to break these protocols, and also more successful. Comprehensive data of experiments run with a GAP implementation are provided and compared to the results of earlier length-based attacks. These demonstrate that the proposed platform is not as secure as first thought and also show that existing measures of cryptographic complexity are not optimal. A more accurate alternative measure is suggested. Finally, a linear algebra attack for one of the protocols is introduced.

scholarly journals Annular Dehn functions of groups

1998 ◽  
Vol 58 (3) ◽  
pp. 453-464 ◽  
Author(s):  
Stephen G. Brick ◽  
Jon M. Corson
Keyword(s):  
Conjugacy Problem ◽  
Upper Bounds ◽  
Dehn Function ◽  
Free Products ◽  
Dehn Functions ◽  
Finitely Presented Group ◽  
Hnn Extensions ◽  
Finite Subgroups ◽  
Boundary Curves ◽  
Finitely Presented

For a finite presentation of a group, or more generally, a two-complex, we define a function analogous to the Dehn function that we call the annular Dehn function. This function measures the combinatorial area of maps of annuli into the complex as a function of the lengths of the boundary curves. A finitely presented group has solvable conjugacy problem if and only if its annular Dehn function is recursive.As with standard Dehn functions, the annular Dehn function may change with change of presentation. We prove that the type of function obtained is preserved by change of presentation. Further we obtain upper bounds for the annular Dehn functions of free products and, more generally, amalgamations or HNN extensions over finite subgroups.

scholarly journals Conjugacy problem in groups with quadratic Dehn function

2020 ◽  
Vol 10 (01) ◽  
pp. 1950023 ◽  
Author(s):  
A. Yu. Olshanskii ◽  
M. V. Sapir
Keyword(s):  
Conjugacy Problem ◽  
Dehn Function ◽  
Finitely Presented Group ◽  
Finitely Presented

We construct a finitely presented group with quadratic Dehn function and undecidable conjugacy problem. This solves Rips’ problem formulated in 1994.

A FINITELY PRESENTED GROUP WITH ALMOST SOLVABLE CONJUGACY PROBLEM

2009 ◽  
Vol 02 (04) ◽  
pp. 611-635 ◽  
Author(s):  
K. Kalorkoti
Keyword(s):  
Conjugacy Problem ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Recursively Enumerable ◽  
Computation Step ◽  
Solvable Word Problem ◽  
Finitely Presented ◽  
Solvable Word ◽  
Extended Notion ◽  
Ordered Pairs

The algorithmic unsolvability of the conjugacy problem for finitely presented groups was demonstrated by Novikov in the early 1950s. Various simplifications and alternative proofs were found by later researchers and further questions raised. Recent work by Borovik, Myasnikov and Remeslennikov has considered the question of what proportion of the number of elements of a group (obtained by standard constructions) falls into the realm of unsolvability. In this paper we provide a straightforward construction, as a Britton tower, of a finitely presented group with solvable word problem but unsolvable conjugacy problem of any r.e. (recursively enumerable) Turing degree a. The question of whether two elements are conjugate is bounded truth-table reducible to the question of whether the elements are both conjugate to a single generator of the group. We also define computable normal forms, based on the method of Bokut', that are suitable for the conjugacy problem. We consider (ordered) pairs of normal words U, V for the conjugacy problem whose lengths add to l and show that the proportion of such pairs for which conjugacy is undecidable (in the case a ≠ 0) is strictly less than l2/(2λ - 1)l where λ > 4. The construction is based on modular machines, introduced by Aanderaa and Cohen. For the purposes of this construction it was helpful to extend the notion of configuration to include pairs of m-adic integers. The notion of computation step was also extended and is referred to as s-fold computation where s ∈ ℤ (the usual notion coresponds to s = 1). If gcd (m, s) = 1 then determinism is preserved, i.e., if the modular machine is deterministic then it remains so under the extended notion. Furthermore there is a simple correspondence between s-fold and standard computation in this case. Otherwise computation is non-deterministic and there does not seem to be any straightforward correspondence between s-fold and standard computation.

Unsolvable Problems in Groups With Solvable Word Problem

1970 ◽  
Vol 22 (4) ◽  
pp. 836-838 ◽  
Author(s):  
James McCool
Keyword(s):  
Word Problem ◽  
Conjugacy Problem ◽  
The Other ◽  
Decision Problems ◽  
Finitely Presented Group ◽  
Other Hand ◽  
Solvable Word Problem ◽  
Finitely Presented ◽  
Effective Procedures ◽  
Solvable Word

Let G be a finitely presented group with solvable word problem. It is of some interest to ask which other decision problems must necessarily be solvable for such a group. Thus it is easy to see that there exist effective procedures to determine whether or not such a group is trivial, or nilpotent of a given class. On the other hand, the conjugacy problem need not be solvable for such a group, for Fridman [5] has shown that the word problem is solvable for the group with unsolvable conjugacy problem given by Novikov [9].

Adaptive Precision Block-Jacobi for High Performance Preconditioning in the Ginkgo Linear Algebra Software

2021 ◽  
Vol 47 (2) ◽  
pp. 1-28
Author(s):  
Goran Flegar ◽  
Hartwig Anzt ◽  
Terry Cojean ◽  
Enrique S. Quintana-Ortí
Keyword(s):  
Linear Algebra ◽  
Graphics Processing Units ◽  
High Performance ◽  
Numerical Algorithms ◽  
Mixed Precision ◽  
Before And After ◽  
Memory Accesses ◽  
Specialized Hardware ◽  
The Individual ◽  
Graphics Processing

The use of mixed precision in numerical algorithms is a promising strategy for accelerating scientific applications. In particular, the adoption of specialized hardware and data formats for low-precision arithmetic in high-end GPUs (graphics processing units) has motivated numerous efforts aiming at carefully reducing the working precision in order to speed up the computations. For algorithms whose performance is bound by the memory bandwidth, the idea of compressing its data before (and after) memory accesses has received considerable attention. One idea is to store an approximate operator–like a preconditioner–in lower than working precision hopefully without impacting the algorithm output. We realize the first high-performance implementation of an adaptive precision block-Jacobi preconditioner which selects the precision format used to store the preconditioner data on-the-fly, taking into account the numerical properties of the individual preconditioner blocks. We implement the adaptive block-Jacobi preconditioner as production-ready functionality in the Ginkgo linear algebra library, considering not only the precision formats that are part of the IEEE standard, but also customized formats which optimize the length of the exponent and significand to the characteristics of the preconditioner blocks. Experiments run on a state-of-the-art GPU accelerator show that our implementation offers attractive runtime savings.

A survey of power and energy efficient techniques for high performance numerical linear algebra operations

2014 ◽  
Vol 40 (10) ◽  
pp. 559-573 ◽  
Author(s):  
Li Tan ◽  
Shashank Kothapalli ◽  
Longxiang Chen ◽  
Omar Hussaini ◽  
Ryan Bissiri ◽  
...  
Keyword(s):  
Linear Algebra ◽  
Energy Efficient ◽  
High Performance ◽  
Numerical Linear Algebra ◽  
Power And Energy

scholarly journals Some challenging group presentations

1999 ◽  
Vol 67 (2) ◽  
pp. 206-213 ◽  
Author(s):  
George Havas ◽  
Derek F. Holt ◽  
P. E. Kenne ◽  
Sarah Rees
Keyword(s):  
Finite Groups ◽  
Infinite Groups ◽  
Derived Length ◽  
Finitely Presented Group ◽  
Group Presentations ◽  
Coset Enumeration ◽  
Finitely Presented

AbstractWe study some challenging presentations which arise as groups of deficiency zero. In four cases we settle finiteness: we show that two presentations are for finite groups while two are for infinite groups. Thus we answer three explicit questions in the literature and we provide the first published deficiency zero presentation for a group with derived length seven. The tools we use are coset enumeration and Knuth-Bebdix rewriting, which are well-established as methods for proving finiteness or otherwise of a finitely presented group. We briefly comment on their capabilities and compare their performance.

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