12,256 research outputs found
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
Constructive homomorphisms for classical groups
Let Omega be a quasisimple classical group in its natural representation over
a finite vector space V, and let Delta be its normaliser in the general linear
group. We construct the projection from Delta to Delta/Omega and provide fast,
polynomial-time algorithms for computing the image of an element. Given a
discrete logarithm oracle, we also represent Delta/Omega as a group with at
most 3 generators and 6 relations. We then compute canonical representatives
for the cosets of Omega. A key ingredient of our algorithms is a new,
asymptotically fast method for constructing isometries between spaces with
forms. Our results are useful for the matrix group recognition project, can be
used to solve element conjugacy problems, and can improve algorithms to
construct maximal subgroups
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
Black Box White Arrow
The present paper proposes a new and systematic approach to the so-called
black box group methods in computational group theory. Instead of a single
black box, we consider categories of black boxes and their morphisms. This
makes new classes of black box problems accessible. For example, we can enrich
black box groups by actions of outer automorphisms.
As an example of application of this technique, we construct Frobenius maps
on black box groups of untwisted Lie type in odd characteristic (Section 6) and
inverse-transpose automorphisms on black box groups encrypting .
One of the advantages of our approach is that it allows us to work in black
box groups over finite fields of big characteristic. Another advantage is
explanatory power of our methods; as an example, we explain Kantor's and
Kassabov's construction of an involution in black box groups encrypting .
Due to the nature of our work we also have to discuss a few methodological
issues of the black box group theory.
The paper is further development of our text "Fifty shades of black"
[arXiv:1308.2487], and repeats parts of it, but under a weaker axioms for black
box groups.Comment: arXiv admin note: substantial text overlap with arXiv:1308.248
Homomorphic encryption and some black box attacks
This paper is a compressed summary of some principal definitions and concepts
in the approach to the black box algebra being developed by the authors. We
suggest that black box algebra could be useful in cryptanalysis of homomorphic
encryption schemes, and that homomorphic encryption is an area of research
where cryptography and black box algebra may benefit from exchange of ideas
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