4,613 research outputs found
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 -symmetric conjugacy in groups. We suggest parameter choices for
instantiations in braid groups, symmetric groups and several matrix groups.Comment: 30 pages, 5 figures. arXiv admin note: substantial text overlap with
arXiv:1305.440
Group theory in cryptography
This paper is a guide for the pure mathematician who would like to know more
about cryptography based on group theory. The paper gives a brief overview of
the subject, and provides pointers to good textbooks, key research papers and
recent survey papers in the area.Comment: 25 pages References updated, and a few extra references added. Minor
typographical changes. To appear in Proceedings of Groups St Andrews 2009 in
Bath, U
Conjugacy in Garside Groups III: Periodic braids
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 of papers by the same authors about the
conjugacy problem in Garside groups. They have a unified goal: the development
of a polynomial algorithm for the conjugacy decision and search problems in
B_n, which generalizes to other Garside groups whenever possible. It is our
hope that the methods introduced here will allow the generalization of the
results in this paper to all Artin-Tits groups of spherical type.Comment: 33 pages, 13 figures. Classical references implying Corollaries 12
and 15 have been added. To appear in Journal of Algebr
Combinatorial group theory and public key cryptography
After some excitement generated by recently suggested public key exchange
protocols due to Anshel-Anshel-Goldfeld and Ko-Lee et al., it is a prevalent
opinion now that the conjugacy search problem is unlikely to provide sufficient
level of security if a braid group is used as the platform. In this paper we
address the following questions: (1) whether choosing a different group, or a
class of groups, can remedy the situation; (2) whether some other "hard"
problem from combinatorial group theory can be used, instead of the conjugacy
search problem, in a public key exchange protocol. Another question that we
address here, although somewhat vague, is likely to become a focus of the
future research in public key cryptography based on symbolic computation: (3)
whether one can efficiently disguise an element of a given group (or a
semigroup) by using defining relations.Comment: 12 page
A New Algorithm for Solving the Word Problem in Braid Groups
One of the most interesting questions about a group is if its word problem
can be solved and how. The word problem in the braid group is of particular
interest to topologists, algebraists and geometers, and is the target of
intensive current research. We look at the braid group from a topological point
of view (rather than a geometrical one). The braid group is defined by the
action of diffeomorphisms on the fundamental group of a punctured disk. We
exploit the topological definition of the braid group in order to give a new
approach for solving its word problem. Our algorithm is faster, in comparison
with known algorithms, for short braid words with respect to the number of
generators combining the braid, and it is almost independent of the number of
strings in the braids. Moreover, the algorithm is based on a new computer
presentation of the elements of the fundamental group of a punctured disk. This
presentation can be used also for other algorithms.Comment: 24 pages, 13 figure
Quantum Knitting
We analyze the connections between the mathematical theory of knots and
quantum physics by addressing a number of algorithmic questions related to both
knots and braid groups.
Knots can be distinguished by means of `knot invariants', among which the
Jones polynomial plays a prominent role, since it can be associated with
observables in topological quantum field theory.
Although the problem of computing the Jones polynomial is intractable in the
framework of classical complexity theory, it has been recently recognized that
a quantum computer is capable of approximating it in an efficient way. The
quantum algorithms discussed here represent a breakthrough for quantum
computation, since approximating the Jones polynomial is actually a `universal
problem', namely the hardest problem that a quantum computer can efficiently
handle.Comment: 29 pages, 5 figures; to appear in Laser Journa
Magic-State Functional Units: Mapping and Scheduling Multi-Level Distillation Circuits for Fault-Tolerant Quantum Architectures
Quantum computers have recently made great strides and are on a long-term
path towards useful fault-tolerant computation. A dominant overhead in
fault-tolerant quantum computation is the production of high-fidelity encoded
qubits, called magic states, which enable reliable error-corrected computation.
We present the first detailed designs of hardware functional units that
implement space-time optimized magic-state factories for surface code
error-corrected machines. Interactions among distant qubits require surface
code braids (physical pathways on chip) which must be routed. Magic-state
factories are circuits comprised of a complex set of braids that is more
difficult to route than quantum circuits considered in previous work [1]. This
paper explores the impact of scheduling techniques, such as gate reordering and
qubit renaming, and we propose two novel mapping techniques: braid repulsion
and dipole moment braid rotation. We combine these techniques with graph
partitioning and community detection algorithms, and further introduce a
stitching algorithm for mapping subgraphs onto a physical machine. Our results
show a factor of 5.64 reduction in space-time volume compared to the best-known
previous designs for magic-state factories.Comment: 13 pages, 10 figure
Experimental approximation of the Jones polynomial with DQC1
We present experimental results approximating the Jones polynomial using 4
qubits in a liquid state nuclear magnetic resonance quantum information
processor. This is the first experimental implementation of a complete problem
for the deterministic quantum computation with one quantum bit model of quantum
computation, which uses a single qubit accompanied by a register of completely
random states. The Jones polynomial is a knot invariant that is important not
only to knot theory, but also to statistical mechanics and quantum field
theory. The implemented algorithm is a modification of the algorithm developed
by Shor and Jordan suitable for implementation in NMR. These experimental
results show that for the restricted case of knots whose braid representations
have four strands and exactly three crossings, identifying distinct knots is
possible 91% of the time.Comment: 5 figures. Version 2 changes: published version, minor errors
corrected, slight changes to improve readabilit
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