12,213 research outputs found
An Efficient Quantum Algorithm for some Instances of the Group Isomorphism Problem
In this paper we consider the problem of testing whether two finite groups
are isomorphic. Whereas the case where both groups are abelian is well
understood and can be solved efficiently, very little is known about the
complexity of isomorphism testing for nonabelian groups. Le Gall has
constructed an efficient classical algorithm for a class of groups
corresponding to one of the most natural ways of constructing nonabelian groups
from abelian groups: the groups that are extensions of an abelian group by
a cyclic group with the order of coprime with . More precisely,
the running time of that algorithm is almost linear in the order of the input
groups. In this paper we present a quantum algorithm solving the same problem
in time polynomial in the logarithm of the order of the input groups. This
algorithm works in the black-box setting and is the first quantum algorithm
solving instances of the nonabelian group isomorphism problem exponentially
faster than the best known classical algorithms.Comment: 20 pages; this is the full version of a paper that will appear in the
Proceedings of the 27th International Symposium on Theoretical Aspects of
Computer Science (STACS 2010
Recognising the small Ree groups in their natural representations
We present Las Vegas algorithms for constructive recognition and constructive
membership testing of the Ree groups 2G_2(q) = Ree(q), where q = 3^{2m + 1} for
some m > 0, in their natural representations of degree 7. The input is a
generating set X.
The constructive recognition algorithm is polynomial time given a discrete
logarithm oracle. The constructive membership testing consists of a
pre-processing step, that only needs to be executed once for a given X, and a
main step. The latter is polynomial time, and the former is polynomial time
given a discrete logarithm oracle.
Implementations of the algorithms are available for the computer algebra
system MAGMA
MOR Cryptosystem and classical Chevalley groups in odd characteristic
In this paper we study the MOR cryptosystem using finite classical Chevalley
groups over a finite field of odd characteristic. In the process we develop an
algorithm for these Chevalley groups in the same spirit as the row-column
operation for special linear group. We focus our study on orthogonal and
symplectic groups. We find the hardness of the proposed MOR cryptosystem for
these groups
Recognising the Suzuki groups in their natural representations
Under the assumption of a certain conjecture, for which there exists strong
experimental evidence, we produce an efficient algorithm for constructive
membership testing in the Suzuki groups Sz(q), where q = 2^{2m + 1} for some m
> 0, in their natural representations of degree 4. It is a Las Vegas algorithm
with running time O{log(q)} field operations, and a preprocessing step with
running time O{log(q) loglog(q)} field operations. The latter step needs an
oracle for the discrete logarithm problem in GF(q).
We also produce a recognition algorithm for Sz(q) = . This is a Las Vegas
algorithm with running time O{|X|^2} field operations.
Finally, we give a Las Vegas algorithm that, given ^h = Sz(q) for some h
in GL(4, q), finds some g such that ^g = Sz(q). The running time is O{log(q)
loglog(q) + |X|} field operations.
Implementations of the algorithms are available for the computer system
MAGMA
Manifold interpolation and model reduction
One approach to parametric and adaptive model reduction is via the
interpolation of orthogonal bases, subspaces or positive definite system
matrices. In all these cases, the sampled inputs stem from matrix sets that
feature a geometric structure and thus form so-called matrix manifolds. This
work will be featured as a chapter in the upcoming Handbook on Model Order
Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A.
Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the
numerical treatment of the most important matrix manifolds that arise in the
context of model reduction. Moreover, the principal approaches to data
interpolation and Taylor-like extrapolation on matrix manifolds are outlined
and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model
Order Reduction
A Digital Signature Scheme for Long-Term Security
In this paper we propose a signature scheme based on two intractable
problems, namely the integer factorization problem and the discrete logarithm
problem for elliptic curves. It is suitable for applications requiring
long-term security and provides a more efficient solution than the existing
ones
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