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2014-08-11T16:01:10Z

CierraMorford:

In [[mathematics]], the '''Pontryagin classes''', named for [[Lev Pontryagin]], are certain [[characteristic class]]es. The Pontryagin class lies in [[cohomology group]]s with degree a multiple of four. It applies to real [[vector bundle]]s.

== Definition ==
Given a real vector bundle ''E'' over ''M'', its ''k''-th Pontryagin class ''pk''(''E'') is defined as
:''pk''(''E'') = ''pk''(''E'', '''Z''') = (−1)''k'' ''c''2''k''(''E'' ⊗ '''C''') ∈ ''H''4''k''(''M'', '''Z'''),
where:
*''c''2''k''(''E'' ⊗ '''C''') denotes the 2''k''-th [[Chern class]] of the [[complexification]] ''E'' ⊗ '''C''' = ''E'' ⊕ ''iE'' of ''E'',
*''H''4''k''(''M'', '''Z''') is the 4''k''-[[cohomology]] group of ''M'' with [[integer]] coefficients.

The rational Pontryagin class ''pk''(''E'', '''Q''') is defined to be the image of ''pk''(''E'') in ''H''4''k''(''M'', '''Q'''), the 4''k''-[[cohomology]] group of ''M'' with [[Rational number|rational]] coefficients.

== Properties ==
The '''total Pontryagin class'''
:<math>p(E)=1+p_1(E)+p_2(E)+\cdots\in H^*(M,\mathbf{Z}),</math>
is (modulo 2-torsion) multiplicative with respect to
[[Glossary of differential geometry and topology#W|Whitney sum]] of vector bundles, i.e.,
:<math>2p(E\oplus F)=2p(E)\smile p(F)</math>
for two vector bundles ''E'' and ''F'' over ''M''. In terms of the individual Pontryagin classes ''pk'',
:<math>2p_1(E\oplus F)=2p_1(E)+2p_1(F),</math>
:<math>2p_2(E\oplus F)=2p_2(E)+2p_1(E)\smile p_1(F)+2p_2(F)</math>
and so on.

The vanishing of the Pontryagin classes and [[Stiefel-Whitney class]]es of a vector bundle does not guarantee that the vector bundle is trivial. For example, up to [[Vector bundle#Vector bundle morphisms|vector bundle isomorphism]], there is a unique nontrivial rank 10 vector bundle ''E''10 over the [[N-sphere|9-sphere]]. (The [[clutching function]] for ''E''10 arises from the [[Orthogonal group#Homotopy groups|stable homotopy group]] π8(O(10)) = '''Z'''/2'''Z'''.) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the [[Stiefel-Whitney class]] ''w''9 of ''E''10 vanishes by the [[Stiefel-Whitney class#Relations over the Steenrod algebra|Wu formula]] ''w''9 = ''w''1''w''8 + Sq1(''w''8). Moreover, this vector bundle is stably nontrivial, i.e. the [[Glossary of differential geometry and topology#W|Whitney sum]] of ''E''10 with any trivial bundle remains nontrivial. {{Harv|Hatcher|2009|p=76}}

Given a 2''k''-dimensional vector bundle ''E'' we have
:<math>p_k(E)=e(E)\smile e(E),</math>
where ''e''(''E'') denotes the [[Euler class]] of ''E'', and <math>\smile</math> denotes the [[cup product]] of cohomology classes.

=== Pontryagin classes and curvature ===
As was shown by [[Shiing-Shen Chern]] and [[André Weil]] around 1948, the rational Pontryagin classes
:<math>p_k(E,\mathbf{Q})\in H^{4k}(M,\mathbf{Q})</math>
can be presented as differential forms which depend polynomially on the [[curvature form]] of a vector bundle. This [[Chern–Weil theory]] revealed a major connection between algebraic topology and global differential geometry.

For a [[vector bundle]] ''E'' over a ''n''-dimensional [[differentiable manifold]] ''M'' equipped with a [[connection form|connection]], the total Pontryagin class is expressed as
:<math>p=\left[1-\frac{{\rm Tr}(\Omega ^2)}{8 \pi ^2}+\frac{{\rm Tr}(\Omega ^2)^2-2 {\rm Tr}(\Omega ^4)}{128 \pi ^4}-\frac{{\rm Tr}(\Omega ^2)^3-6 {\rm Tr}(\Omega ^2) {\rm Tr}(\Omega ^4)+8 {\rm Tr}(\Omega ^6)}{3072 \pi ^6}+\cdots\right]\in H^*_{dR}(M),</math>

where Ω denotes the [[curvature form]], and ''H*''dR(''M'') denotes the [[de Rham cohomology]] groups.{{Citation needed|date=July 2009}}

=== Pontryagin classes of a manifold ===
The '''Pontryagin classes of a smooth manifold''' are defined to be the Pontryagin classes of its [[tangent bundle]].

[[Sergei Novikov (mathematician)|Novikov]] proved in 1966 that if manifolds are [[homeomorphism|homeomorphic]] then their rational Pontryagin classes ''pk''(''M'', '''Q''') in ''H''4''k''(''M'', '''Q''') are the same.

If the dimension is at least five, there are at most finitely many different smooth manifolds with given [[Homotopy#Homotopy equivalence of spaces|homotopy type]] and Pontryagin classes.

== Pontryagin numbers ==
'''Pontryagin numbers''' are certain [[topological invariant]]s of a smooth [[manifold]]. The Pontryagin number vanishes if the dimension of manifold is not divisible by 4. It is defined in terms of the Pontryagin classes of a [[manifold]] as follows:

Given a smooth 4''n''-dimensional manifold ''M'' and a collection of natural numbers
:''k''1, ''k''2, ..., ''km'' such that ''k''1+''k''2+...+''km'' =''n''.
the Pontryagin number <math>P_{k_1,k_2,\dots,k_m}</math> is defined by
:<math>P_{k_1,k_2,\dots, k_m}=p_{k_1}\smile p_{k_2}\smile \cdots\smile p_{k_m}([M])</math>
where ''pk'' denotes the ''k''-th Pontryagin class and [''M''] the [[fundamental class]] of ''M''.

=== Properties ===
#Pontryagin numbers are oriented [[cobordism]] invariant; and together with [[Stiefel-Whitney number]]s they determine an oriented manifold's oriented cobordism class.
#Pontryagin numbers of closed Riemannian manifold (as well as Pontryagin classes) can be calculated as integrals of certain polynomial from curvature tensor of Riemannian manifold.
#Such invariants as [[Signature (topology)|signature]] and [[Â genus|<math>\hat A</math>-genus]] can be expressed through Pontryagin numbers.

== Generalizations ==
There is also a ''quaternionic'' Pontryagin class, for vector bundles with [[quaternion]] structure.

== See also ==
*[[Chern–Simons form]]

== References ==
*{{cite book
|author= [[John Milnor|Milnor John W.]]
|author2=Stasheff, James D. |authorlink2=Jim Stasheff
|title= Characteristic classes
|work= Annals of Mathematics Studies
|issue=76
|publisher=Princeton University Press / University of Tokyo Press
|location=Princeton, New Jersey; Tokyo
|year= 1974
|isbn= 0-691-08122-0}}
* {{Cite journal | last=Hatcher | first=Allen | author-link=Allen Hatcher | title=Vector Bundles & K-Theory | edition=2.1 | year=2009 | ref=harv | postscript= | url=http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html}}

==External links==
* {{springer|title=Pontryagin class|id=p/p073750}}

[[Category:Characteristic classes]]
[[Category:Differential topology]]

Main Page

2014-08-11T16:00:48Z

CierraMorford:

'''ID-based encryption''' (or '''identity-based encryption (IBE)''') is an important primitive of [[ID-based cryptography]]. As such it is a type of [[public-key encryption]] in which the [[public key]] of a user is some unique information about the identity of the user (e.g. a user's email address). This can use the text-value of the name or domain name as a key or the physical IP address it translates to.

The first implementation of an email-address based PKI was developed by [[Adi Shamir]] in 1984,<ref name="iseca.org">Adi Shamir, [http://www.iseca.org/modules/mydownloads/visit.php?cid=56&lid=33 Identity-Based Cryptosystems and Signature Schemes]. ''Advances in Cryptology: Proceedings of CRYPTO 84, Lecture Notes in Computer Science'', 7:47--53, 1984</ref> which allowed users to verify [[digital signature]]s using only public information such as the user's identifier.

ID-based encryption was proposed by [[Adi Shamir]] in 1984.<ref name="iseca.org"/> He was however only able to give an instantiation of [[ID-based cryptography|identity-based signatures]]. Identity-based encryption remained an open problem for many years. One example of the research leading up to identity-based encryption is provided in Maurer.<ref>Ueli M. Maurer: Protocols for Secret Key Agreement by Public Discussion Based on Common Information. CRYPTO 1992: 461-470</ref>

The [[pairing-based cryptography|pairing]]-based [[Boneh–Franklin scheme]]<ref>Dan Boneh, Matthew K. Franklin, Identity-Based Encryption from the Weil Pairing ''Advances in Cryptology - Proceedings of CRYPTO 2001'' (2001)</ref> and [[Cocks IBE scheme|Cocks's encryption scheme]]<ref>Clifford Cocks, [http://groups.csail.mit.edu/cis/crypto/classes/6.876/papers/cocks-IBE.pdf An Identity Based Encryption Scheme Based on Quadratic Residues], ''Proceedings of the 8th IMA International Conference on Cryptography and Coding'', 2001</ref> based on [[quadratic residue]]s both solved the IBE problem in 2001.

==Usage==
Identity-based systems allow any party to generate a public key from a known identity value such as an ASCII string. A trusted third party, called the [[Private Key Generator]] (PKG), generates the corresponding private keys. To operate, the PKG first publishes a master public key, and retains the corresponding '''master private key''' (referred to as ''master key''). Given the master public key, any party can compute a public key corresponding to the identity ''ID'' by combining the master public key with the identity value. To obtain a corresponding private key, the party authorized to use the identity ''ID'' contacts the PKG, which uses the master private key to generate the private key for identity ''ID''.

As a result, parties may encrypt messages (or verify signatures) with no prior distribution of keys between individual participants. This is extremely useful in cases where pre-distribution of authenticated keys is inconvenient or infeasible due to technical restraints. However, to decrypt or sign messages, the authorized user must obtain the appropriate private key from the PKG. A caveat of this approach is that the PKG must be highly trusted, as it is capable of generating any user's private key and may therefore decrypt (or sign) messages without authorization. Because any user's private key can be generated through the use of the third party's secret, this system has inherent [[key escrow]]. A number of variant systems have been proposed which remove the escrow including [[certificate-based encryption]],<ref>Craig Gentry [http://eprint.iacr.org/2003/183.pdf Certificate-Based Encryption and the Certificate Revocation Problem] ''Advances in Cryptology - Proceedings of EUROCRYPT 2003'' (2003)</ref> [[secure key issuing cryptography]]<ref>{{cite conference | first1 = Byoungcheon | last1 = Lee | first2 = Colin | last2 = Boyd | first2 = Ed | last2 = Dawson | first3 = Kwangjo | last3 = Kim | first4 = Jeongmo | last4 = Yang | first5 = Seungjae | last5 = Yoo | id = {{citeseerx|10.1.1.6.337}} | title = Secure Key Issuing in ID-based Cryptography | copnference = ACS Conferences in Research and Practice in Information Technology - Proceedings of the Second Australian Information Security Workshop-AISW 2004 | year = 2004 }}</ref> and [[certificateless cryptography]].<ref>SS Al-Riyami, KG Paterson [http://www.springerlink.com/index/4WC47ELK7FP8XWTY.pdf Certificateless Public Key Cryptography] ''Advances in Cryptology - Proceedings of ASIACRYPT 2003'' (2003)</ref>
The steps involved are depicted in this diagram:[[Image:ID Based Encryption.png|center|thumb|600px|ID Based Encryption: Offline and Online Steps]]

==Protocol framework==
[[Dan Boneh]] and [[Matthew K. Franklin]] defined a set of four algorithms that form a complete IBE system:
* '''Setup''': This algorithm is run by the PKG one time for creating the whole IBE environment. The master key is kept secret and used to derive users' private keys, while the system parameters are made public. It accepts a [[security parameter]] <math>\textstyle k</math> (i.e. binary length of key material) and outputs:
# A set <math>\textstyle \mathcal{P}</math> of system parameters, including the [[message space]] and [[ciphertext space]] <math>\textstyle \mathcal{M}</math> and <math>\textstyle \mathcal{C}</math>,
# a master key <math>\textstyle K_m</math>.

* '''Extract''': This algorithm is run by the PKG when a user requests his private key. Note that the verification of the [[Authentication|authenticity]] of the requestor and the secure transport of <math>\textstyle d</math> are problems with which IBE protocols do not try to deal. It takes as input <math>\textstyle \mathcal{P}</math>, <math>\textstyle K_m</math> and an identifier <math>\textstyle ID \in \left\{0,1\right\}^*</math> and returns the private key <math>\textstyle d</math> for user <math>\textstyle ID</math>.

* '''Encrypt''': Takes <math>\textstyle \mathcal{P}</math>, a message <math>\textstyle m \in \mathcal{M}</math> and <math>\textstyle ID \in \left\{0,1\right\}^*</math> and outputs the encryption <math>\textstyle c \in \mathcal{C}</math>.

* '''Decrypt''': Accepts <math>\textstyle d</math>, <math>\textstyle \mathcal{P}</math> and <math>\textstyle c \in \mathcal{C}</math> and returns <math>\textstyle m \in \mathcal{M}</math>.

===Correctness constraint===
In order for the whole system to work, one has to postulate that:

<math> \forall m \in \mathcal{M}, ID \in \left\{0,1\right\}^*: Decrypt\left(Extract\left(\mathcal{P}, K_m, ID\right), \mathcal{P}, Encrypt\left(\mathcal{P}, m, ID \right) \right) = m </math>

==Encryption schemes==
The most efficient identity-based encryption schemes are currently based on [[Pairing|bilinear pairings]] on [[elliptic curves]], such as the [[weil pairing|Weil]] or [[Tate pairing|Tate]] pairings. The first of these schemes was developed by [[Dan Boneh]] and [[Matthew K. Franklin]] (2001), and performs [[probabilistic encryption]] of arbitrary ciphertexts using an [[ElGamal encryption|Elgamal]]-like approach. Though the [[BonehFranklinScheme|Boneh-Franklin scheme]] is [[Provable security|provably secure]], the security proof rests on relatively new assumptions about the hardness of problems in certain elliptic curve groups.

Another approach to identity-based encryption was proposed by [[Clifford Cocks]] in 2001. The [[Cocks IBE scheme]] is based on well-studied assumptions (the [[quadratic residuosity problem|quadratic residuosity assumption]]) but encrypts messages one bit at a time with a high degree of [[ciphertext expansion]]. Thus it is highly inefficient and impractical for sending all but the shortest messages, such as a session key for use with a [[symmetric cipher]].

== Advantages ==
One of the major advantages of any identity-based encryption scheme is that if there are only a finite number of users, after all users have been issued with keys the third party's secret can be destroyed. This can take place because this system assumes that, once issued, keys are always valid (as this basic system lacks a method of [[key revocation]]). The majority of derivatives of this system which have key revocation lose this advantage.

Moreover, as public keys are derived from identifiers, IBE eliminates the need for a public key distribution infrastructure. The [[Authentication|authenticity]] of the public keys is guaranteed implicitly as long as the transport of the private keys to the corresponding user is kept secure ([[Authentication#Computer_security|Authenticity]], [[Data integrity|Integrity]], [[Confidentiality]]).

Apart from these aspects, IBE offers interesting features emanating from the possibility to encode additional information into the identifier. For instance, a sender might specify an expiration date for a message. He appends this timestamp to the actual recipient's identity (possibly using some binary format like X.509). When the receiver contacts the PKG to retrieve the private key for this public key, the PKG can evaluate the identifier and decline the extraction if the expiration date has passed. Generally, embedding data in the ID corresponds to opening an additional channel between sender and PKG with authenticity guaranteed through the dependency of the private key on the identifier.

== Drawbacks ==
* If a Private Key Generator (PKG) is compromised, all messages protected over the entire lifetime of the public-private key pair used by that server are also compromised. This makes the PKG a high value target to adversaries. To limit the exposure due to a compromised server, the master private-public key pair could be updated with a new independent key pair. However, this introduces a key-management problem where all users must have the most recent public key for the server.
* Because the Private Key Generator (PKG) generates private keys for users, it may decrypt and/or sign any message without authorisation. This implies that IBE systems cannot be used for [[non-repudiation]]. This may not be an issue for organizations that host their own PKG and are willing to trust their system administrators and do not require non-repudiation.
* The issue of implicit key escrow does not exist with the current [[Public key infrastructure|PKI]] system wherein private keys are usually generated on the user's computer. Depending on the context key escrow can be seen as a positive feature (e.g., within Enterprises). A number of variant systems have been proposed which remove the escrow including [[certificate-based encryption]], [[secret sharing]], [[secure key issuing cryptography]] and [[certificateless cryptography]].
* A secure channel between a user and the Private Key Generator (PKG) is required for transmitting the private key on joining the system. Here, a [[Secure Sockets Layer|SSL]]-like connection is a common solution for a large-scale system. It is important to observe that users that hold accounts with the PKG must be able to authenticate themselves. In principle, this may be achieved through username,password or through public key pairs managed on smart cards.
* IBE solutions may rely on cryptographic techniques that are insecure against code breaking [[quantum computer]] attacks (see [[Shor's algorithm]])

==See also==
*[[ID-based cryptography]]

==References==
{{Reflist}}

==External links==
* [http://www.crypto.rub.de/its_seminar_ws0708.html Seminar 'Cryptography and Security in Banking'/'Alternative Cryptology', Ruhr University Bochum]
* [http://www.ietf.org/rfc/rfc5091.txt RFC 5091 - the IETF RFC defining two common IBE algorithms]
* [http://www.hpl.hp.com/techreports/2003/HPL-2003-21.pdf HP Role-Based Encryption]
* [http://www.larc.usp.br/~pbarreto/pblounge.html The Pairing-Based Crypto Lounge]
* [http://www.voltage.com/vsn The Voltage Security Network - IBE encryption web service]
* [http://vsn.visus-it.com VSN Fully Managed Email Encryption Service - UK based IBE encryption web service]
* [http://www.ferris.com/2006/05/30/the-total-cost-of-ownership-for-voltage-identity-based-encryption-solutions/ Analyst report on the cost of IBE versus PKI]

{{DEFAULTSORT:Id-Based Encryption}}
[[Category:Public-key cryptography]]
[[Category:Identity-based cryptography]]

[[fr:Schéma basé sur l'identité]]
[[ko:신원 기반 암호]]
[[ja:IDベース暗号]]

Main Page

2014-08-11T12:29:58Z

CierraMorford:

[[image:systolic array.jpg|thumb|240px|A systolic array network in which each data processing unit (DPU) receives data from one or more input streams and/or other DPUs and sends data to one or more output streams and/or other DPUs]]

In [[computer architecture]], a '''systolic array''' is a pipe network arrangement of [[Data processing system|processing units]] called cells. It is a specialized form of [[parallel computing]], where cells (i.e. processors), compute data and store it independently of each other.

==Description==
A systolic array is composed of matrix-like rows of data processing units called cells. Data processing units ([[Data processing system|DPU]]s) are similar to [[central processing unit]]s ([[CPU]])s, (except for the usual lack of a [[program counter]],<ref>The Paracel GeneMatcher series of systolic array processors do have a program counter. More complicated algorithms are implemented as a series of simple steps, with shifts specified in the instructions.</ref> since operation is [[transport triggered architecture|transport-triggered]], i.e., by the arrival of a data object). Each cell shares the information with its neighbours immediately after processing. The systolic array is often rectangular where data flows across the array between neighbour DPUs, often with different data flowing in different directions. The data streams entering and leaving the ports of the array are generated by [[auto-sequencing memory]] units, ASMs. Each ASM includes a [[data counter]]. In [[embedded system]]s a data stream may also be input from and/or output to an external source.

An example of a systolic [[algorithm]] might be designed for [[matrix multiplication]]. One [[matrix (math)|matrix]] is fed in a row at a time from the top of the array and is passed down the array, the other matrix is fed in a column at a time from the left hand side of the array and passes from left to right. Dummy values are then passed in until each processor has seen one whole row and one whole column. At this point, the result of the multiplication is stored in the array and can now be output a row or a column at a time, flowing down or across the array.<ref>[http://web.cecs.pdx.edu/~mperkows/temp/May22/0020.Matrix-multiplication-systolic.pdf Systolic Array Matrix Multiplication]</ref>

Systolic arrays are arrays of DPUs which are connected to a small number of nearest neighbour DPUs in a mesh-like topology. DPUs perform a sequence of operations on data that flows between them. Because the traditional systolic array synthesis methods have been practiced by algebraic algorithms, only uniform arrays with only linear pipes can be obtained, so that the architectures are the same in all DPUs. The consequence is, that only applications with regular data dependencies can be implemented on classical systolic arrays. Like [[SIMD]] machines, clocked systolic arrays compute in "lock-step" with each processor undertaking alternate compute | communicate
phases. But systolic arrays with asynchronous handshake between DPUs are called ''wavefront arrays''.
One well-known systolic array is Carnegie Mellon University's [[iWarp]] processor, which has been manufactured by Intel. An iWarp system has a linear array processor connected by data buses going in both directions.

==History==
The systolic array paradigm, data-stream-driven by data counters, is the counterpart of the [[von Neumann architecture|von Neumann paradigm]], instruction-stream-driven by a program counter. Because a systolic array usually sends and receives multiple data streams, and multiple data counters are needed to generate these data streams, it supports [[data parallelism]]. [[Systole (medicine)|The name]] derives from analogy with the regular pumping of blood by the heart.

[[H. T. Kung]] and [[Charles E. Leiserson]] published the first paper describing systolic arrays in 1978; however, the first machine known to have used a similar technique was the [[Colossus computer|Colossus Mark II]] in 1944.

==Applications==
''An application Example - Polynomial Evaluation''

[[Horner's rule]] for evaluating a polynomial is:

<math>
y = ( ... ( ( (a_n*x + a_{n-1})*x + a_{n-2})*x + a_{n-3})*x + ... + a_1)*x + a_0
</math>

A linear systolic array in which the processors are arranged in pairs:
one multiplies its input by <math>x</math> and passes the result to the right,
the next adds <math>a_j</math> and passes the result to the right:

==Advantages and Disadvantages==
Pros
*Faster
*Scalable
Cons
*Expensive
*Highly specialized for particular applications
*Difficult to build

==Implementations==
[[Cisco]] PXF network processor is internally organized as systolic array.<ref>http://www.cisco.com/en/US/prod/collateral/routers/ps133/prod_white_paper09186a008008902a.html</ref>

==See also==
*[[iWarp]] - Systolic Array Computer, VLSI, Intel/CMU
*[[WARP (systolic array)]] - Systolic Array Computer, GE/CMU

==Notes==
<references/>

==References==
{{More footnotes|date=April 2011}}
*H. T. Kung, C. E. Leiserson: Algorithms for VLSI processor arrays; in: C. Mead, L. Conway (eds.): Introduction to VLSI Systems; Addison-Wesley, 1979
*S. Y. Kung: VLSI Array Processors; Prentice-Hall, Inc., 1988
*N. Petkov: Systolic Parallel Processing; North Holland Publishing Co, 1992

==External links==
*[http://www.iti.fh-flensburg.de/lang/papers/isa/index.htm ''Instruction Systolic Array (ISA)'']
* [http://ieeexplore.ieee.org/iel5/92/4292150/04292156.pdf 'A VLSI Architecture for Image Registration in Real Time' (Based on systolic array), Vol. 15, September 2007]

{{DEFAULTSORT:Systolic Array}}
[[Category:Parallel computing]]
[[Category:Reconfigurable computing]]