Dynamic Cryptographic Backdoors Part II Taking Control over the ...
32 pages
English

Dynamic Cryptographic Backdoors Part II Taking Control over the ...

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32 pages
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  • mémoire
Eric Filiol (speaker) - Oluwaseun Remi-Omosowon (speaker) - Leonard Mutembei , Dynamic Cryptographic Backdoors Part II Taking Control over the TOR Network ESIEA - Laval Operational Cryptology and Virology Lab (C + V )⁰ 28C3 2011 - Berlin (ESIEA - (C + V )⁰ lab) The Tor Attack 28C3 2011 1 /56
  • medium size group of bad guys
  • many encryption algorithms
  • skilled computer security specialists
  • conclusion introduction
  • cryptographic trapdoors
  • cryptgenrandom function
  • cryptographic algorithms
  • period of time
  • security

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Nombre de lectures 28
Langue English

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Hyperspace
actions
Eli Glasner
Abstract
Recovering full
informationMinimal hyperspace actions of
Uspenskij’shomeomorphism groups of theorem
h-homogeneoush-homogeneous spaces spaces and
maximal chains
Hyperspace
actions
Eli Glasner Patterns and
partitions
The dual RamseySchool of Mathematics
Theorem
Tel Aviv University
Signatures and
conclusion of the
proofDescriptive Set Theory in Paris, December 2011HyperspaceAbstract: actions
Eli Glasner
Let X be a h-homogeneous zero-dimensional compact
AbstractHausdorff space, i.e. X is a Stone dual of a
Recovering fullhomogeneous Boolean algebra. Using the dual Ramsey information
theorem and a detailed combinatorial analysis of what we Uspenskij’s
theorem
call stable collections of subsets of a finite set, we obtain
h-homogeneous
a complete list of the minimal sub-systems of the spaces and
maximal chains
compact dynamical system (Exp(Exp(X ));Homeo(X )),
Hyperspace
where Exp(X ) stands for the hyperspace comprising the actions
Patterns andclosed subsets of X equipped with the Vietoris topology.
partitions
The importance of this dynamical system stems from
The dual Ramsey
TheoremUspenskij’s characterization of the universal ambit of
Signatures andG = Homeo(X ). The results apply to X = C the Cantor
conclusion of the
proofset, the generalized Cantor sets X =f0; 1g for
non-countable cardinals, and to several other spaces. A
particular interesting case is X =! =!n!, where!
denotes the Cech-Stone compactification of the natural
numbers. This is a joint work with Yonatan Gutman.HyperspaceCan one obtain full knowledge from a given actions
Eli Glasnergroup action ?
Abstract
Recovering fullI We begin with a very general, almost philosophical
information
question. Given a dynamical system (X;G), how Uspenskij’s
theoremmuch knowledge about other G-systems can we
h-homogeneous
recover by just considering (X;G) ? spaces and
maximal chains
I In our efforts to learn more about other G-systems
Hyperspace
actionswe are allowed to consider product systems like
Patterns andXX or more generally, subsystems of arbitrary
partitions
Jproducts (i.e. closed G-invariant subsets of X ). The dual Ramsey
Theorem
I We are also permitted to consider the associated
Signatures and
conclusion of thedynamical systems induced on the compact spaces
proof
Exp(X ), of closed subsets of X equipped with the
Hausdorff topology, and M(X ), the space of
probability measures equipped with the weak
topology.HyperspaceMeasure preserving dynamical system actions
Eli Glasner
Abstract
I If we regard the measure preserving dynamical
Recovering full
system analogue of this question the answer is information
Uspenskij’sdefinitely negative. E.g. if the original system
theorem
(X;X; ; G) is of zero entropy then so will be any
h-homogeneous
spaces andsystem that is derived from it, including induced
maximal chains
systems. Similarly if the system is rigid, this property
Hyperspace
actionswill be shared by every derived system.
Patterns and
I partitionsAs for the M(X ) construction, back in the topological
The dual Ramseysetup, a theorem of Glasner and Weiss says that if
Theorem
(X;G) has zero topological entropy then the same
Signatures and
conclusion of theholds for (M(X );G). The fact that uniform rigidity is
proof
preserved in M(X ) is straightforward. Thus we can
not use the M(X ) construction as a tool in our
project.HyperspaceTopological dynamical systems actions
Eli Glasner
Abstract
I A G-dynamical system or a G-space is a pair
Recovering full
information(X;G), where G is a topological group, and X is a
Uspenskij’scompact Hausdorff space. The group G acts on X so
theorem
that the action (g;x)7! gx; GX! X is jointly
h-homogeneous
spaces andcontinuous; i.e. we are considering a representation
maximal chains
of G in the topological group Homeo(X ), equipped
Hyperspace
actionswith the compact open topology. In order to make the
Patterns andabove program a reasonable task we must assume partitions
that this representation is topologically faithful. The dual Ramsey
Theorem
I A dynamical system (X;G) is point transitive if
Signatures and
conclusion of thethere exists a point in X whose G-orbit is dense. It is
proof
minimal if every orbit is dense. A point transitive
system (X;x ;G) with a distinguished point x 2 X0 0
with dense orbit is called an ambit.Hyperspace
actions
Eli Glasner
I A homomorphism : (X;G)! (Y;G) is a
Abstractcontinuous onto map intertwining the G actions. For
Recovering fullambits we also require that the distinguished point in
information
X is mapped onto the distinguished point inY. In this
Uspenskij’s
theoremcase a homomorphism, when it exists, is unique.
h-homogeneous
I The general theory ensures the existence and spaces and
maximal chains
uniqueness of a universal G-ambit denoted
Hyperspace
actions(S(G);e;G). The universality means that for every
Patterns andambit (X;x ;G) there is a (necessarily unique)0
partitions
homomorphism of pointed dynamical systems
The dual Ramsey
Theorem : (S(G);e;G)! (X;x ;G).0
Signatures and
I The ambit (X;x )_ (Y;y ) =O(x ;y ) XY is conclusion of the0 0 0 0
proof
called the pointed product of the two ambits. More
generally one defines similarly the pointed product
of any family of ambits.HyperspaceUniversal systems actions
Eli Glasner
I By Zorn’s lemma every dynamical system has a
Abstractminimal subsystem.
Recovering full
I It then follows that any minimal subsystemM of information
S(G) is a universal G-system. The Uspenskij’s
theorem
uniqueness follows from a theorem of Ellis.
h-homogeneous
spaces andI (Ellis) There exists a universal minimal dynamical
maximal chains
system, (M(G);G); that is, for every minimal
Hyperspace
actionsdynamical system (X;G) there exists
Patterns and : (M(G);G)! (X;G). Moreover (M(G);G) is
partitions
coalescent (every endomorphism is an The dual Ramsey
Theorem
automorphism) and thereforeM(G) is unique up to
Signatures and
isomorphism. conclusion of the
proof
I (Veech) A locally compact group acts freely onS(G)
and hence also onM(G).
I (Kechris, Pestov, Todorcevic) For a locally compact,
non-compact topological group G, the spaceM(G)
is non-metrizable.HyperspaceThe enveloping semigroup of a dynamical actions
Eli Glasnersystem
Abstract
Recovering full
information
Uspenskij’s
theorem
The enveloping semigroup E = E(X;G) = E(X ) of a h-homogeneous
spaces andXdynamical system (X;G) is defined as the closure in X maximal chains
(with its compact, usually non-metrizable, pointwise Hyperspace
actions convergence topology) of the set G =fg : X! Xgg2G
Patterns andXconsidered as a subset of X . With the operation of partitions
The dual Ramseycomposition of maps this is a right topological
Theorem
semigroup (i.e. for every p2 E(X ) the map R : q7! qp,p
Signatures and
conclusion of theR : E(X )! E(X ) is continuous).p
proof

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