formulasearchengine - User contributions [en]

Kolmogorov's criterion

2014-06-08T16:03:27Z

134.76.217.21: /* Discrete-time Markov chains */ ok, sorry. I am too stupid. Being able to read properly is an advantage...never mind

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Backtracking line search

2014-06-02T08:11:24Z

134.76.223.2: Undid revision 610950116 by 24.162.254.176 (talk) did not render

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Lévy–Prokhorov metric

2014-03-24T10:45:19Z

134.76.83.135: /* Definition */ statement was false

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Q–Q plot

2014-03-05T11:01:10Z

134.76.0.127:

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Tricritical point

2014-01-07T11:00:08Z

134.76.93.247:

'''Topological data analysis''' (TDA) is a new area of study aimed at having applications in areas such as [[data mining]] and [[computer vision]].
The main problems are:
# how one infers high-dimensional structure from low-dimensional representations; and
# how one assembles discrete points into global structure.

The human brain can easily extract global structure from representations in a strictly lower dimension, i.e. we infer a 3D environment from a 2D image from each eye. The inference of global structure also occurs when converting discrete data into continuous images, e.g. [[Dot matrix printing|dot-matrix printers]] and televisions communicate images via arrays of discrete points.

The main method used by topological data analysis is:
# Replace a set of data points with a family of [[simplicial complex]]es, indexed by a proximity parameter.
# Analyse these topological complexes via [[algebraic topology]] — specifically, via the theory of '''persistent homology'''.<ref name=carlsson2009>{{cite journal|url=http://www.ams.org/bull/2009-46-02/S0273-0979-09-01249-X/S0273-0979-09-01249-X.pdf|title=Topology and data|author=Gunnar Carlsson|journal=BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY|volume=46|issue=2|date=April 2009|pages=255–308}}</ref>
# Encode the persistent homology of a data set in the form of a parameterized version of a [[Betti number]] which is called a '''barcode'''.<ref name=carlsson2009/>

==Point cloud data==
Data is often represented as points in a [[Euclidean space|Euclidean ''n''-dimensional space]] E''n''. The global ''shape'' of the data may provide information about the phenomena that the data represent.

One type of data set for which global features are certainly present is the so-called '''[[point cloud]] data''' coming from physical objects in 3D. E.g. a laser can scan an object at a set of discrete points and the cloud of such points can be used in a computer representation of the object. '''Point cloud data''' refers to any collection of points in E''n'' or a (perhaps noisy) sample of points on a lower-dimensional subset.

For point clouds in low-dimensional spaces there are numerous approaches for inferring features based on planar projections in the fields of [[computer graphics]] and [[statistics]]. Topological data analysis is needed when the spaces are high-dimensional or too twisted to allow planar projections to faithfully represent the features of the point cloud.

To convert a point cloud in a [[metric space]] into a global object, use the point cloud as the vertices of a [[Graph (mathematics)|graph]] whose edges are determined by proximity, then turn the graph into a [[simplicial complex]] and use algebraic topology to study it. An alternative approach is the [[minimum spanning tree]]-based method in the geometric data clustering.<ref>C. T. Zahn (1971): [http://web.cse.msu.edu/~cse802/Papers/zahn.pdf "Graph-theoretical methods for detecting and describing gestalt clusters"], ''IEEE Transactions on Computers'', pp. 68-86, Vol. 20, No. 1</ref> If a group of data points forms a cluster, then the geometry of this point cloud can be determined.

==Background==
# [[Topology]]
# [[Simplicial Complex]]
# [[Nerve of a covering| Nerve and Cover]]
Topological Data analysis refers to different methods and representations in order to cluster variegated data via a point cloud stated above. The following are various methods to do so.
==Combinatorial Representations==

# '''Cech Complex.''' The [[Cech cohomology|Cech complex]] <math>C_\epsilon</math> is the ''nerve'' of the ''cover'' of balls of radius <math>\epsilon</math> around each point in a set. Since balls are convex and convex sets are contractible, its nerve captures the topology of the cover. The Cech Complex is not computed in practice due to its computational complexity. The uniform ball radii imply an assumption of uniform sampling on the input, which is not valid in a real world dataset. Non-uniform radii methods can also be used, such as in the case of the ''Alpha Simplex''.
# '''Alpha Complex.''' The [[Voronoi diagram]] is the set of all Voronoi regions for the points in <math>S\subseteq Y</math>. This diagram is considered a closed cover for <math>Y</math>. The [[Delaunay triangulation|Delaunay complex]] is the nerve of the Voronoi diagram. The Voronoi cover and its nerve are fundamental geometric objects and have been extensively studied within computational geometry. Alpha complexes are constructed by first building the Delaunay complex. For each simplex of the Delaunay complex, we compute the minimum scale at which each simplex enters the alpha complex. Then the simplices are sorted by their minimum scale to get a partial order of simplices. The alpha complex is not formed with any scale <math>\epsilon</math> using this ordering. Efficient algorithms and software exist for computing Delaunay complexes, and in turn, alpha complexes in 2 and 3 dimensions. However, the construction of the Delaunay complex is difficult in higher dimensions.
# [[Vietoris-Rips complex]]

==Topological Invariants==
#'''Definition.''' [[Topological property| Topological Invariants]]
#[[Euler characteristic]]
#[[Simplicial homology]]
==Multiscale Invariants==
#'''Multifiltration Model'''. [[Morse Theory]] enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology.
#'''Persistent homology'''. ''See [[homology (mathematics)|homology]] for an introduction to the notation.''
Persistent homology essentially calculates homology groups at different spatial resolutions to see which features persist over a wide range of length scales. It is assumed that important features and structures are the ones that persist. We define persistent homology as follows:
Let <math>K^l</math> be a [[filtration (mathematics)|filtration]]. The '''p-persistent kth homology group''' of <math>K^l</math> is <math>H_k^{l,p}=Z_k^l/(B_k^{l+p}\cap Z_k^l)</math>.

Let <math>z</math> be a nonbounding <math>k</math>-cycle created at time <math>I</math> by simplex <math>\sigma</math> and let <math>z'\sim z</math> be a homologous <math>k</math>-cycle that becomes a boundary cycle at time <math>J</math> by simplex <math>\tau</math>.
Then we can define the '''persistence interval''' associated to <math>z</math> as <math>(I,J)</math>. We call <math>\sigma</math> the '''creator''' of <math>z</math> and <math>\tau</math> the '''destroyer''' of <math>z</math>. If <math>z</math> does not have a destroyer, its persistence is <math>\infty</math>.
Instead of using an index-based filtration, we can use a time-based filtration. Let <math>K</math> be a [[simplicial complex]] and <math>K^\rho=\{ \sigma^i\in K|\rho (\sigma^i)\le \rho \}</math> be a filtration defined for an associated map <math>\rho : S(K)\rightarrow \mathbb{R}</math> that maps simplices in the final complex to real numbers. Then for all real numbers <math>\pi \ge 0</math>, the '''<math>\pi</math>-persistent kth homology group''' of <math>K^\rho</math> is <math>H_k^{\rho, \pi}=Z_k^\rho /(B_k^{\rho + \pi }\cap Z_k^\rho )</math>. The persistence of a <math>k</math>-cycle created at time <math>\rho_i</math> and destroyed at <math>\rho_j</math> is <math>\rho_j - \rho_i</math>.
<ref>Afra J. Zomorodian (2005): ''Topology for Computing''. Cambridge Monographs on Applied and Computational Mathematics.</ref>

There are various software packages for computing persistence intervals of a finite filtration, such as [http://code.google.com/p/javaplex/ javaPlex], [http://www.mrzv.org/software/dionysus/ Dionysus], [http://www.sas.upenn.edu/~vnanda/perseus/index.html Perseus], and [http://phat.googlecode.com/ PHAT].

==See also==
*[[Dimensionality reduction]]
*[[Data mining]]
*[[Computer vision]]
*[[Computational topology]]
*[[Digital topology]]
*[[Digital Morse theory]]
*[[Shape analysis]]
*[[Size theory]]
*[[Structured data analysis (statistics)]]

==References==
<references />

==Further reading==
*[http://comptop.stanford.edu/ Topological Methods in Scientific Computing, Statistics and Computer Science] Stanford group
*[http://www.ams.org/bull/2008-45-01/S0273-0979-07-01191-3/S0273-0979-07-01191-3.pdf BARCODES: THE PERSISTENT TOPOLOGY OF DATA]
*[http://www.bangor.ac.uk/~mas013/TDA/TDA.html Topological Data Analysis: the algebraic topology of point data clouds?]
* {{cite book | author=Sanjay Rana | title=Topological Data Structures for Surfaces | url=http://books.google.com/books?id=bfPWmkohAQoC | publisher=John Wiley and Sons | year=2004 | isbn=978-0-470-85151-7}}
* [http://www.ams.org/bull/2009-46-02/S0273-0979-09-01249-X/S0273-0979-09-01249-X.pdf TOPOLOGY AND DATA], GUNNAR CARLSSON, BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY, Volume 46, Number 2, April 2009, Pages 255–308, Article electronically published on January 29, 2009
* [http://books.google.com/books?id=MDXa6gFRZuIC Computational Topology: An Introduction], Herbert Edelsbrunner, John L. Harer, AMS Bookstore, 2010, ISBN 978-0-8218-4925-5
*[http://books.google.com/books?id=88lCsF_dmIkC Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications], Editors Valerio Pascucci, Hans Hagen, Xavier Tricoche, Julien Tierny, Springer, 2010, ISBN 978-3-642-15013-5
*{{citation | first = Shmuel | last = Weinberger | title = What Is . . . Persistent Homology? | journal = AMS Notices | volume = 58 | issue = 01 | pages = 36–39 | year = 2011
| url = http://www.ams.org/notices/201101/rtx110100036p.pdf }}.
*[http://www.ayasdi.com/resources Ayasdi Resources on Topological Data Analysis for Big Data]
*[http://code.google.com/p/javaplex/ Software package for computing persistent homology]

{{DEFAULTSORT:Topological Data Analysis}}
[[Category:Computational topology]]
[[Category:Data analysis]]
[[Category:Homology theory]]

Natural units

2013-12-04T16:43:35Z

134.76.223.9: /* Atomic units */

The '''noisy-storage model'''<ref name="initial">{{cite journal|last=Wehner|first=S.|coauthors=C. Schaffner, B. Terhal|title=Cryptography from noisy-storage|journal=Physical Review Letters|year=2008|volume=100|pages=220502|doi=10.1103/PhysRevLett.100.220502|arxiv=0711.2895|issue=22|pmid=18643410}}</ref> refers to a cryptographic model employed in [[quantum cryptography]]. It assumes that the quantum memory device of an attacker ([[Adversary (cryptography)|adversary]]) trying to break the protocol is imperfect (noisy).
The main goal of this model is to enable the secure implementation of two-party cryptographic primitives, such as [[bit commitment]], [[oblivious transfer]] and [[Smart_Card#Applications|secure identification]].

==Motivation==
Quantum communication has proven to be extremely useful when it comes to distributing encryption keys. It allows two distant parties Alice and Bob to expand a small initial [[secret key]] into an arbitrarily long secret key by sending [[qubits]] (quantum bits) to each other. Most importantly, it can be shown that any [[eavesdropper]] trying to listen into their communication cannot intercept any information about the long key. This is known as [[quantum key distribution]] (QKD).

Yet, it has been shown that even quantum communication does not allow the secure implementation of many other two-party cryptographic tasks.<ref name="bitcom1">
{{cite journal|
last=Lo|
first=H.|
coauthors = H. Chau|
title = Is quantum bit commitment really possible?|
journal = Physical Review Letters|
volume = 78|
pages = 3410|
year = 1997|
doi=10.1103/PhysRevLett.78.3410|
issue=17
}}
</ref><ref name="lo2">
{{cite journal|
last=Lo|
first=H|
title = Insecurity of Quantum Secure Computations|
journal = Physical Review A|
volume = 56|
pages = 1154|
year = 1997|
doi=10.1103/PhysRevA.56.1154|
issue=2
}}
</ref><ref name="mayers2">
{{cite journal|
last=Mayers|
first=D.|
title = Unconditionally Secure Quantum Bit Commitment is Impossible|
journal = Physical Review Letters|
volume = 78|
pages = 3414––3417|
year = 1997|
doi=10.1103/PhysRevLett.78.3414|
issue=17
}}
</ref><ref name="qbc">
{{cite journal|
last=D'Ariano|
first=G.|
coauthors=D. Kretschmann and D. Schlingemann and R.F. Werner|
title = Quantum Bit Commitment Revisited: the Possible and the Impossible|
url = http://www.arXiv.org/abs/quant-ph/0605224v2|
journal = Physical Review A|
volume = 76|
pages = 032328|
year=2007|
doi=10.1103/PhysRevA.76.032328|
issue=3
}}
</ref> These all form instances of [[secure function evaluation]]. An example is [[oblivious transfer]]. What sets these tasks apart from key distribution is that they aim to solve problems between two parties, Alice and Bob, who do ''not'' trust each other. That is, there is no outside party like an [[eavesdropper]], only Alice and Bob. Intuitively, it is this lack of trust that makes the problem hard. Unlike in [[quantum key distribution]], Alice and Bob cannot collaborate to try and detect any eavesdropping activity. Instead, each party has to fend for himself.

Since tasks like [[Smart_Card#Applications|secure identification]] are of practical interest, one is willing to make assumptions on how powerful the [[Adversary (cryptography)|adversary]] can be. Security then holds as long as these assumptions are satisfied. In classical cryptography, i.e., without the use of quantum tools, most of these are [[Computational hardness assumption|computational assumptions]]. Such assumptions consists of two parts. First, one assumes that a particular problem is difficult to solve. For example, one might assume that it is hard to [[Integer factorization|factor]] a large [[integer]] into its [[prime]] factors (e.g. 15=5x3). Second, one assumes that the adversary has a limited amount of computing power, namely less than what is (thought to be) required to solve the chosen problem.

===Bounded storage ===
In [[information theoretic security|information theoretic cryptography]] physical assumptions appear, which do not rely on any hardness assumptions, but merely assume a limit on some other resource. In classical cryptography, the '''bounded-storage model''' introduced by [[Ueli Maurer (cryptographer)|Ueli Maurer]] assumes that the [[Adversary (cryptography)|adversary]] can only store a certain number of classical bits.<ref name="maurer92">
{{cite journal|
last=Maurer|
first=U.|
title = Conditionally-Perfect Secrecy and a Provably-Secure Randomized Cipher|
journal = Journal of Cryptology|
pages = 53––66|
volume = 5|
year = 1992|
issue = 1
}}
</ref><ref name="maurer97">
{{cite journal|
last=Cachin|
first=C.|
coauthors = U. Maurer|
title = Unconditional Security Against Memory-Bounded Adversaries|
journal=Proceedings of CRYPTO 1997|
year=1997|
pages=292–306
}}
</ref> Protocols are known that do (in principle) allow the secure implementation of any cryptographic task as long as the adversary's storage is small. Very intuitively, security becomes possible under this assumption since the adversary has to make a choice which information to keep. That is, the protocol effectively overflows his memory device leading to an inevitable lack on information for the adversary. It was later discovered that any classical [[Protocol (object-oriented programming)|protocol]] which requires the honest parties to store <math>n</math> bits in order to execute it successfully can be broken by an adversary that can store more than about <math>O(n^2)</math> bits.<ref name="maurerimposs">
{{cite journal|
last=Dziembowski|
first=S.|
coauthors = U. Maurer|
title = On Generating the Initial Key in the Bounded-Storage Model|
journal = Proceedings of EUROCRYPT|
year=2004|
pages=126–137
}}
</ref> That is, the gap between what is required to execute the protocol, and what is required to break the security is relatively small.

===Bounded quantum storage===
This gap changes dramatically when using [[quantum communication]]<ref name="bounded">
{{cite journal|
first=Damgaard|
last=I.|
coauthors= S. Fehr and L. Salvail and C. Schaffner|
title = Cryptography in the Bounded-Quantum-Storage Model|
year = 2005|
journal = Proceedings of 46th IEEE Symposium on Foundations of Computer Science|
pages = 449–458|
url = http://www.arXiv.org/abs/quant-ph/0508222v2
}}
</ref>
. That is, Alice and Bob can send [[qubit]]s to each other as part of the protocol. Likewise, one now assumes that the adversary's quantum storage is limited to a certain number of qubits. There is no restriction on how many classical bits the adversary can store. This is known as the '''bounded-''quantum''-storage model'''.<ref name=bounded/><ref name="damgardHighOrder">
{{cite journal|
last = Damgaard|
first = I.|
coauthors = S. Fehr and R. Renner and L. Salvail and C. Schaffner|
title = A Tight High-Order Entropic Quantum Uncertainty Relation With Applications|
journal = Proceedings of CRYPTO 2007|
pages = 360––378|
year = 2007|
url = http://www.arXiv.org/abs/quant-ph/0612014v2
}}
</ref> It was shown that there exist quantum protocols in which the honest parties need ''no'' quantum storage at all to execute them, but are nevertheless secure as long as Alice transmits more than twice the number of qubits than the adversary can store.

===Noisy storage===
More generally, security is possible as long as the amount of information that the adversary can store in his memory device is limited. This intuition is captured by the '''noisy-storage model''',<ref name=initial/> which includes the bounded-quantum-storage model as a special case.<ref name="unconditional">
{{cite journal|
last=Koenig|
first=Robert|
coauthors = S. Wehner and J. Wullschleger|
url = http://www.arXiv.org/abs/0906.1030v3|
title = Unconditional security from noisy quantum storage|
year = 2009}}
</ref> Such a limitation can, for example, come about if the memory device is extremely large, but very imperfect. In [[information theory]] such an imperfect memory device is also called a [[noisy channel]]. The motivation for this more general model is threefold. First, it allows one to make statements about much more general memory devices that the adversary may have available. Second, security statements could be made when the signals transmitted, or the storage device itself, uses [[Quantum key distribution|continuous variables]] whose dimension is infinite and thus cannot be captured by a bounded storage assumption without additional constraints. Third, even if the dimension of the signals itself is small, the noisy-storage analysis allows security beyond the regime where bounded-storage itself can make any security statement. For example, if the storage channel is entanglement breaking, security is possible even if the storage device is arbitrarily large (i.e., not bounded in any way).

==Assumption==

The assumption of the noisy-storage model is that during waiting times <math>\Delta t</math> introduced into the protocol, the [[Adversary (cryptography)|adversary]] can only store [[quantum information]] in his noisy memory device.<ref name=unconditional/> Such a device is simply a [[quantum channel]] <math>\mathcal{F}:\mathcal{S}(\mathcal{H}_{\rm in}) \rightarrow \mathcal{S}(\mathcal{H}_{\rm out})</math> that takes input [[quantum state|states]] <math>\rho_{\rm in} \in \mathcal{S}(\mathcal{H}_{\rm in})</math> to some noisy output states <math>\rho_{\rm out} \in \mathcal{S}(\mathcal{H}_{\rm out})</math>. Otherwise, the adversary is all powerful. For example, he can store an unlimited amount of classical information and perform any computation instantaneously.

[[File:NoisyStorageModel.png|right|thumb|During waiting times <math>\Delta t</math> the storage device has to be used.]]
The latter assumption also implies that he can perform any form of [[Error-correcting|error correcting encoding]] before and after using the noisy memory device, even if it is computationally very difficult to do (i.e., it requires a long time). In this context, this is generally referred to as an encoding attack <math>\mathcal{E}</math> and a decoding attack <math>\mathcal{D}</math>. Since the adversary's classical memory can be arbitrarily large, the encoding <math>\mathcal{E}</math> may not only generate some [[quantum state]] as input to the storage device <math>\mathcal{F}</math> but also output classical information. The adversary's decoding attack <math>\mathcal{D}</math> can make use of this extra classical information, as well as any additional information that the adversary may gain after the waiting time has passed.

In practise, one often considers storage devices that consist of <math>N</math> memory cells, each of which is subject to noise. In information-theoretic terms, this means that the device has the form <math>\mathcal{F} = \mathcal{N}^{\otimes N}</math>, where <math>\mathcal{N}: S(\Complex^d) \rightarrow S(\Complex^d)</math> is a noisy [[quantum channel]] acting on a memory cell of dimension <math>d</math>.

===Examples===

* The storage device consists of <math>N</math> [[qubit]]s, each of which is subject to [[Quantum depolarizing channel|depolarizing noise]]. That is, <math>\mathcal{F} = \mathcal{N}^{\otimes N}</math>, where <math>\mathcal{N}(\rho) = \lambda \rho + (1-\lambda) \mathsf{id}/2</math> is the 2-dimensional [[Quantum depolarizing channel|depolarizing channel]].

* The storage device consists of <math>N</math> [[qubit]]s, which are noise-free. This corresponds to the special case of '''bounded-quantum-storage'''. That is, <math>\mathcal{F} = \mathsf{id}^{\otimes N}</math>, where <math>\mathsf{id}</math> is the [[identity channel]].

==Protocols==

Most protocols proceed in two steps. First, Alice and Bob exchange <math>n</math> [[qubit]]s encoded in two or three [[mutually unbiased bases]]. These are the same encodings which are used in the [[BB84]] or six-state protocols of [[quantum key distribution]]. Typically, this takes the form of Alice sending such qubits to Bob, and Bob measuring them immediately on arrival. This has the advantage that Alice and Bob need no quantum storage to execute the protocol. It is furthermore experimentally relatively easy to create such [[qubits]], making it possible to implement such protocols using currently available technology.<ref name="curty">
{{cite journal|
last = Wehner|
first = S.|
coauthors = M. Curty and C. Schaffner and H. Lo|
journal = Physical Review A|
url = http://www.arXiv.org/abs/0911.2302v2|
pages = 052336|
title = Implementation of two-party protocols in the noisy-storage model|
volume = 81|
year = 2010|
doi = 10.1103/PhysRevA.81.052336|
issue = 5}}
</ref>

The second step is to perform classical post-processing of the measurement data obtained in step one. Techniques used depend on the protocol in question and include [[privacy amplification]], [[error-correcting codes]], min-entropy sampling, and interactive hashing.

===General===

To demonstrate that all [[secure function evaluation|two-party cryptographic tasks]] can be implemented securely, a common approach is to show that a simple cryptographic primitive can be implemented that is known to be ''universal'' for [[secure function evaluation]]. That is, once one manages to build a protocol for such a cryptographic primitive all other tasks can be implemented by using this primitive as a basic building block. One such primitive is [[oblivious transfer]]. In turn, [[oblivious transfer]] can be constructed from an even simpler building block known as [[weak string erasure]] in combination with cryptographic techniques such as [[privacy amplification]].

All protocols proposed to date allow one of the parties (Alice) to have even an unlimited amount of noise-free quantum memory. I.e., the noisy-storage assumption is applied to only one of the parties (Bob). For storage devices of the form <math>\mathcal{F} = \mathcal{N}^{\otimes N}</math> it is known that any [[secure function evaluation|two-party cryptographic task]] can be implemented securely by means of [[weak string erasure]] and [[oblivious transfer]] whenever any of the following conditions hold.

* For bounded-quantum-storage (i.e., <math>\mathcal{N} = \mathsf{id}</math>), security can be achieved using a protocol in which Alice sends <math>n > 2N</math> [[BB84]] encoded [[qubit]]s.<ref name=unconditional/> That is, security can be achieved when Alice sends more than twice the number of qubits than Bob can store. One can also look at this from Bob's perspective and say that security can be achieved when Bob can store strictly less than half of the qubits that Alice sent, i.e., <math>N < n/2</math>.

* For bounded-quantum-storage using higher dimensional memory cells (i.e., each cell is not a [[qubit]], but a [[Qudit#Variations_of_the_qubit|qudit]]), security can be achieved in a protocol in which Alice sends <math>n</math> higher dimensional qudits encoded one of the possible [[mutually unbiased bases]]. In the limit of large dimensions, security can be achieved whenever <math>n \gtrapprox N</math>. That is, security can always be achieved as long as Bob cannot store any constant fraction of the transmitted signals.<ref name="limits">
{{cite journal|
last=Mandayam|
first=P.|
coauthors = S. Wehner|
title = Achieving the physical limits of the bounded-storage model|
journal=Physical Review A|
volume=83|
pages=022329|
year=2011|
url=http://www.arXiv.org/abs/1009.1596v2|
doi=10.1103/PhysRevA.83.022329|
issue=2
}}
</ref> This is optimal for the protocols considered since for <math>n = N</math> a dishonest Bob can store all qudits sent by Alice. It is not known whether the same is possible using merely [[BB84]] encoded qubits.

* For noisy-storage and devices of the form <math>\mathcal{F} = \mathcal{N}^{\otimes N}</math> security can be achieved using a protocol in which Alice sends <math>n</math> [[BB84]] encoded [[qubits]] if

:* <math>n > 2 \cdot N \cdot C(\mathcal{N})</math>,<ref name=unconditional/> where <math>C(\mathcal{N})</math> is the [[classical capacity]] of the [[quantum channel]] <math>\mathcal{N}</math>, and <math>\mathcal{N}</math> obeys the so-called ''strong converse property'',<ref name="converse">{{cite journal|
last=Koenig|
first=R.|
coauthors=S. Wehner|
journal = Physical Review Letters|
url = http://www.arXiv.org/abs/quant-ph/0903.2838v1|
pages = 070504|
title = A Strong Converse for Classical Channel Coding Using Entangled Inputs|
volume = 103|
year = 2009|
doi=10.1103/PhysRevLett.103.070504|
pmid=19792627|
issue=7}}
</ref> or, if

:* <math>n > 2 \cdot N \cdot E_C(\mathcal{N})</math>,<ref name="entanglementcost">
{{cite journal|
first=M.|
last=Berta|
coauthors = F. Brandao and M. Christandl and S. Wehner|
title = Entanglement cost of quantum channels|
year=2011|
url = http://www.arXiv.org/abs/1108.5357
}}
</ref> where <math>E_C(\mathcal{N})</math> is the [[entanglement cost]] of the [[quantum channel]] <math>\mathcal{N}</math>. This is generally much better than the condition on the [[classical capacity]], however it is harder to evaluate <math>E_C(\mathcal{N})</math>.

* For noisy-storage and devices of the form <math>\mathcal{F} = \mathcal{N}^{\otimes N}</math> security can be achieved using a protocol in which Alice sends <math>n</math> [[qubit]]s encoded in one of the three [[mutually unbiased bases]] per qubit, if

:* <math>n > Q(\mathcal{N}) N</math>,<ref name="qcap">{{cite journal|
last=Berta|
first=M.|
coauthors = O. Fawzi, and S. Wehner|
title = Quantum to classical randomness extractors|
year=2011|
arxiv =1111.2026
}}</ref> where <math>Q</math> is the [[quantum capacity]] of <math>\mathcal{N}</math>, and the strong converse parameter of <math>\mathcal{N}</math> is not too small.

The three [[mutually unbiased bases]] are the same encodings as in the six-state protocol of [[quantum key distribution]]. The last condition does form the best known condition for most channels, yet the [[quantum capacity]] as well as the strong converse parameter are generally not easy to determine.

===Specific tasks===

Using such basic primitives as building blocks is not always the most efficient way to solve a cryptographic task. Specialized protocols targeted to solve specific problems are generally more efficient. Examples of known protocols are

* [[Bit commitment]] in the noisy-storage model,<ref name=unconditional/><ref name=limits/> and in the case of bounded-quantum-storage<ref name=damgardHighOrder/>

* [[Oblivious transfer]] in the noisy-storage model,<ref name=unconditional/> and in the case of bounded-quantum-storage<ref name=bounded/><ref name=damgardHighOrder/>

* [[Smart_Card#Applications|Secure identification]] in the bounded-quantum-storage model<ref name="secureid">
{{cite journal|
last=Damgaard|
first=I.|
coauthors = S. Fehr and L. Salvail and C. Schaffner|
title = Identification and QKD in the Bounded-Quantum-Storage Model|
journal = Proceedings of CRYPTO 2007|
pages = 342––359|
year = 2007|
url =http://www.arXiv.org/abs/0708.2557v3
}}
</ref><ref name="id2">
{{cite journal|
last=Bouman|
first=N.|
coauthors = S. Fehr, C. Gonzales-Guillen and C. Schaffner|
title = An All-But-One Entropic Uncertainty Relations, and Application to Password-based Identification|
url = http://www.arXiv.org/abs/1105.6212v1|
year=2011
}}
</ref>

==Noisy-storage and QKD==

The assumption of bounded-quantum-storage has also been applied outside the realm of [[secure function evaluation]]. In particular, it has been shown that if the eavesdropper in [[quantum key distribution]] is memory bounded, higher bit error rates can be tolerated in an experimental implementation.<ref name=damgardHighOrder/>

==References==
{{reflist}}


[[Category:Quantum cryptography]]
[[Category:Cryptography]]

Wilhelmy plate

2013-10-14T13:25:52Z

134.76.222.17: /* Advantages and Practice */

{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|24-cell honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:Icositetrachoronic tetracomb.png|220px]] A [[24-cell]] and first layer of its adjacent 4-faces.
|-
|bgcolor=#e7dcc3|Type||[[List_of_regular_polytopes#Five_Dimensions_2|Regular 4-space honeycomb]] [[Uniform_polyteron#Regular_and_uniform_honeycombs|Uniform 4-honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{3,4,3,3} r{3,3,4,3} 2r{4,3,3,4} 2r{4,3,31,1} {31,1,1,1}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|3|node|4|node|3|node|3|node}} {{CDD|node|3|node|4|node|3|node_1|3|node}} {{CDD|node|4|node|3|node_1|3|node|4|node}} {{CDD|nodes|split2|node_1|3|node|4|node}} {{CDD|nodes|split2|node_1|split1|nodes}}
|-
|bgcolor=#e7dcc3|4-face type||[[24-cell|{3,4,3}]] [[File:Schlegel wireframe 24-cell.png|40px]]
|-
|bgcolor=#e7dcc3|Cell type||[[octahedron|{3,4}]] [[File:Octahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Face type||[[triangle|{3}]]
|-
|bgcolor=#e7dcc3|[[Edge figure]]||[[tetrahedron|{3,3}]]
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[tesseract|{4,3,3}]]
|-
|bgcolor=#e7dcc3|Dual||[[demitesseractic honeycomb|{3,3,4,3}]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||<math>{\tilde{F}}_4</math>, [3,4,3,3] <math>{\tilde{C}}_4</math>, [4,3,3,4] <math>{\tilde{B}}_4</math>, [4,3,31,1] <math>{\tilde{D}}_4</math>, [31,1,1,1]
|-
|bgcolor=#e7dcc3|Properties||regular
|}
In [[Four-dimensional space|four-dimensional]] [[Euclidean geometry]], the '''24-cell honeycomb''', or '''icositetrachoric honeycomb''' is a [[regular polytope|regular]] space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) of 4-dimensional [[Euclidean space]] by regular [[24-cell]]s. It can be represented by [[Schläfli symbol]] {3,4,3,3}.

The [[dual polytope|dual]] tessellation by regular [[16-cell honeycomb]] has Schläfli symbol {3,3,4,3}. Together with the [[tesseractic honeycomb]] (or 4-cubic honeycomb) these are the only regular tessellations of Euclidean 4-space.

== Kissing number ==
If a [[3-sphere]] is [[inscribed sphere|inscribed]] in each hypercell of this tessellation, the resulting arrangement is the densest possible regular [[sphere packing]] in four dimensions, with the [[kissing number]] 24. The packing density of this arrangement is
:<math>\frac{\pi^2}{16}\cong0.61685.</math>

== Coordinates ==

The 24-cell honeycomb can be constructed as the [[Voronoi tessellation]] of the [[D4 root lattice|D4 root lattice]] or [[F4 lattice]]. Each 24-cell is then centered at a D4 lattice point, i.e. one of
:<math>\left\{(x_i)\in\mathbb Z^4 : {\textstyle\sum_i} x_i \equiv 0\;(\mbox{mod }2)\right\}.</math>
These points can also be described as [[Hurwitz quaternion]]s with even square norm.

The vertices of the honeycomb lie at the deep holes of the D4 lattice. These are the Hurwitz quaternions with odd square norm.

It can be constructed as a [[#Symmetry constructions|birectified tesseractic honeycomb]], by taking a [[tesseractic honeycomb]] and placing vertices at the centers of all the square faces. The [[24-cell]] facets exist between these vertices as ''rectified 16-cells''. If the coordinates of the tesseractic honeycomb are integers (i,j,k,l), the ''birectified tesseractic honeycomb'' vertices can be placed at all permutations of half-unit shifts in two of the four dimensions, thus: (i+½,j+½,k,l), (i+½,j,k+½,l), (i+½,j,k,l+½), (i,j+½,k+½,l), (i,j+½,k,l+½), (i,j,k+½,l+½).

== Configuration ==

Each 24-cell in the 24-cell honeycomb has 24 neighboring 24-cells. With each neighbor it shares exactly one octahedral cell.

It has 24 more neighbors such that with each of these it shares a single vertex.

It has no neighbors with which it shares only an edge or only a face.

The [[vertex figure]] of the 24-cell honeycomb is a [[tesseract]] (4-dimensional cube). So there are 16 edges, 32 triangles, 24 octahedra, and 8 24-cells meeting at every vertex. The [[edge figure]] is a [[tetrahedron]], so there are 4 triangles, 6 octahedra, and 4 24-cells surrounding every edge. Finally, the [[face figure]] is a triangle, so there are 3 octahedra and 3 24-cells meeting at every face.

== Cross-sections ==

One way to visualize 4-dimensional figures is to consider various 3-dimensional [[cross section (geometry)|cross-sections]]. Applying this technique to the 24-cell honeycomb gives rise to various 3-dimensional honeycombs with varying degrees of regularity.

{|class='wikitable' style="text-align:center; float:right; margin:0.5em; background:white;"
!colspan=2|Vertex-first sections
|-
|[[Image:Rhombic dodecahedra.png|220px]]
|[[Image:Partial cubic honeycomb.png|220px]]
|-
|[[Rhombic dodecahedral honeycomb]]
|[[Cubic honeycomb]]
|-
!colspan=2|Cell-first sections
|-
|[[Image:Rectified cubic honeycomb.png|220px]]
|[[Image:Bitruncated cubic honeycomb.png|220px]]
|-
|[[Rectified cubic honeycomb]]
|[[Bitruncated cubic honeycomb]]
|}
A ''vertex-first'' cross-section is one [[orthogonal]] to a line joining opposite vertices of one of the 24-cells. For instance, one could take any of the coordinate hyperplanes in the coordinate system given above (i.e. the planes determined by ''x''''i'' = 0). The cross-section of {3,4,3,3} by one of these hyperplanes gives a [[rhombic dodecahedral honeycomb]]. Each of the rhombic dodecahedra corresponds to a maximal cross-section of one of the 24-cells intersecting the hyperplane (the center of each such 24-cell lies in the hyperplane). Accordingly, the rhombic dodecahedral honeycomb is the [[Voronoi tessellation]] of the D3 root lattice (a [[face-centered cubic]] lattice). Shifting this hyperplane halfway to one of the vertices (e.g. ''x''''i'' = ½) gives rise to a regular [[cubic honeycomb]]. In this case the center of each 24-cell lies off the hyperplane. Shifting again, so the hyperplane intersects the vertex, gives another rhombic dodecahedral honeycomb but with new 24-cells (the former ones having shrunk to points). In general, for any integer ''n'', the cross-section through ''x''''i'' = ''n'' is a rhombic dodecahedral honeycomb, and the cross-section through ''x''''i'' = ''n'' + ½ is a cubic honeycomb. As the hyperplane moves through 4-space, the cross-section morphs between the two periodically.

A ''cell-first'' cross-section is one parallel to one of the octahedral cells of a 24-cell. Consider, for instance, the hyperplane orthogonal to (1,1,0,0). The cross-section of {3,4,3,3} by this hyperplane is a [[rectified cubic honeycomb]]. Each [[cuboctahedron]] in this honeycomb is a maximal cross-section of a 24-cell whose center lies in the plane. Meanwhile, each [[octahedron]] is a boundary cell of a 24-cell whose center lies off the plane. Shifting this hyperplane till it lies halfway between the center of a 24-cell and the boundary, one obtains a [[bitruncated cubic honeycomb]]. The cuboctahedra have shrunk, and the octahedra have grown until they are both [[truncated octahedron|truncated octahedra]]. Shifting again, so the hyperplane intersects the boundary of the central 24-cell gives a rectified cubic honeycomb again, the cuboctahedra and octahedra having swapped positions. As the hyperplane sweeps through 4-space, the cross-section morphs between these two honeycombs periodically. 

== Symmetry constructions ==

There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored 24-cell facets. In all cases, eight 24-cells meet at each vertex, but the vertex figures have different symmetry generators.

{| class="wikitable"
![[Coxeter group]]
![[Coxeter diagram]]
![[Facet (geometry)|Facets]] ([[24-cell]]s)
![[Vertex figure]] ([[8-cell]])
!Vertex figure symmetry order
|- align=center
!<math>{\tilde{F}}_4</math>
|{{CDD|node_1|3|node|4|node|3|node|3|node}}
|'''8:''' {{CDD|node_1|3|node|4|node|3|node}}
|{{CDD|node_1|4|node|3|node|3|node}}
|384
|- align=center
!<math>{\tilde{F}}_4</math>
|{{CDD|node|3|node_1|3|node|4|node|3|node}}
|'''6:''' {{CDD|node|3|node|4|node|3|node_1}} '''2:''' {{CDD|node|4|node|3|node_1|3|node}}
|{{CDD|node|3|node|4|node_1|2|node_1}}
|96
|- align=center
!<math>{\tilde{C}}_4</math>
|{{CDD|node|4|node|3|node_1|3|node|4|node}}
|'''4,4:''' {{CDD|node|3|node_1|3|node|4|node}}
|{{CDD|node|4|node_1|2|node_1|4|node}}
|64
|- align=center
!<math>{\tilde{B}}_4</math>
|{{CDD|nodes|split2|node_1|3|node|4|node}}
|'''2,2:''' {{CDD|node|3|node_1|3|node|4|node}} '''4:''' {{CDD|nodes|split2|node_1|3|node}}
|{{CDD|node_1|2|node_1|2|node_1|4|node}}
|32
|- align=center
!<math>{\tilde{D}}_4</math>
|{{CDD|nodes|split2|node_1|split1|nodes}}
|'''2,2,2,2:''' {{CDD|nodes|split2|node_1|3|node}}
|{{CDD|node_1|2|node_1|2|node_1|2|node_1}}
|16
|}

== Related honeycombs==
{{F4 honeycombs}}

{{C4_honeycombs}}

{{B4_honeycombs}}

{{D4 honeycombs}}

== See also ==
Other uniform honeycombs in 4-space:
*[[Truncated 5-cell honeycomb]]
*[[Omnitruncated 5-cell honeycomb]]
*[[Truncated 24-cell honeycomb]]
*[[Rectified 24-cell honeycomb]]
*[[Snub 24-cell honeycomb]]

== References ==
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
* '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* [[George Olshevsky]], ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)'' - Model 88
* {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations}} o4o3x3o4o, o3x3o *b3o4o, o3x3o *b3o4o, o3x3o4o3o, o3o3o4o3x - icot - O88
{{Honeycombs}}

[[Category:5-polytopes]]
[[Category:Honeycombs (geometry)]]

Small-world network

2012-08-13T08:42:01Z

134.76.222.216: improved layout of math formula

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2012-02-23T13:51:18Z

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2011-05-25T18:28:11Z

134.76.83.150: /* The tautological one-form */

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