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'''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.  
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.


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


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


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.
== 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 ''p<sub>k</sub>'',  
:<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.


==Usage==
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''<sub>10</sub> over the [[N-sphere|9-sphere]].  (The [[clutching function]] for ''E''<sub>10</sub> arises from the [[Orthogonal group#Homotopy groups|stable homotopy group]] π<sub>8</sub>(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''<sub>9</sub> of ''E''<sub>10</sub> vanishes by the [[Stiefel-Whitney class#Relations over the Steenrod algebra|Wu formula]] ''w''<sub>9</sub> = ''w''<sub>1</sub>''w''<sub>8</sub> + Sq<sup>1</sup>(''w''<sub>8</sub>).  Moreover, this vector bundle is stably nontrivial, i.e. the [[Glossary of differential geometry and topology#W|Whitney sum]] of ''E''<sub>10</sub> with any trivial bundle remains nontrivial. {{Harv|Hatcher|2009|p=76}}
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>
Given a 2''k''-dimensional vector bundle ''E'' we have  
The steps involved are depicted in this diagram:[[Image:ID Based Encryption.png|center|thumb|600px|ID Based Encryption: Offline and Online Steps]]
:<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.


==Protocol framework==
=== Pontryagin classes and curvature ===
[[Dan Boneh]] and [[Matthew K. Franklin]] defined a set of four algorithms that form a complete IBE system:
As was shown by [[Shiing-Shen Chern]] and [[André Weil]] around 1948, the rational Pontryagin classes
* '''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:
:<math>p_k(E,\mathbf{Q})\in H^{4k}(M,\mathbf{Q})</math>
# 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>,
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.
# 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>.
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>


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


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


===Correctness constraint===
[[Sergei Novikov (mathematician)|Novikov]] proved in 1966 that if manifolds are [[homeomorphism|homeomorphic]] then their rational Pontryagin classes ''p<sub>k</sub>''(''M'', '''Q''') in ''H''<sup>4''k''</sup>(''M'', '''Q''') are the same.
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>
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.


==Encryption schemes==
== Pontryagin numbers ==
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. 
'''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:


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]].
Given a smooth 4''n''-dimensional manifold ''M'' and a collection of natural numbers
:''k''<sub>1</sub>, ''k''<sub>2</sub>, ..., ''k<sub>m</sub>'' such that ''k''<sub>1</sub>+''k''<sub>2</sub>+...+''k<sub>m</sub>'' =''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 ''p<sub>k</sub>'' denotes the ''k''-th Pontryagin class and [''M''] the [[fundamental class]] of ''M''.


== Advantages ==
=== Properties ===
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.
#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.


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


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.
== See also ==
*[[Chern–Simons form]]


== Drawbacks ==
== References ==
* 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.
*{{cite book
* 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.
  |author= [[John Milnor|Milnor John W.]]
* 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]].
  |author2=Stasheff, James D. |authorlink2=Jim Stasheff
* 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.
  |title= Characteristic classes
* IBE solutions may rely on cryptographic techniques that are insecure against code breaking [[quantum computer]] attacks (see [[Shor's algorithm]])
  |work= Annals of Mathematics Studies
 
  |issue=76
==See also==
  |publisher=Princeton University Press / University of Tokyo Press
*[[ID-based cryptography]]
  |location=Princeton, New Jersey; Tokyo
 
  |year= 1974
==References==
  |isbn= 0-691-08122-0}}
{{Reflist}}
* {{Cite journal | last=Hatcher | first=Allen | author-link=Allen Hatcher  | title=Vector Bundles & K-Theory | edition=2.1 | year=2009 | ref=harv | postscript=<!--None--> | url=http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html}}


==External links==
==External links==
* [http://www.crypto.rub.de/its_seminar_ws0708.html Seminar 'Cryptography and Security in Banking'/'Alternative Cryptology', Ruhr University Bochum]
* {{springer|title=Pontryagin class|id=p/p073750}}
* [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]]
[[Category:Characteristic classes]]
[[ko:신원 기반 암호]]
[[Category:Differential topology]]
[[ja:IDベース暗号]]

Revision as of 17:01, 11 August 2014

In mathematics, the Pontryagin classes, named for Lev Pontryagin, are certain characteristic classes. The Pontryagin class lies in cohomology groups with degree a multiple of four. It applies to real vector bundles.

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 c2k(E ⊗ C) ∈ H4k(M, Z),

where:

The rational Pontryagin class pk(E, Q) is defined to be the image of pk(E) in H4k(M, Q), the 4k-cohomology group of M with rational coefficients.

Properties

The total Pontryagin class

p(E)=1+p1(E)+p2(E)+⋯∈H*(M,Z),

is (modulo 2-torsion) multiplicative with respect to Whitney sum of vector bundles, i.e.,

2p(E⊕F)=2p(E)⌣p(F)

for two vector bundles E and F over M. In terms of the individual Pontryagin classes pk,

2p1(E⊕F)=2p1(E)+2p1(F),
2p2(E⊕F)=2p2(E)+2p1(E)⌣p1(F)+2p2(F)

and so on.

The vanishing of the Pontryagin classes and Stiefel-Whitney classes of a vector bundle does not guarantee that the vector bundle is trivial. For example, up to vector bundle isomorphism, there is a unique nontrivial rank 10 vector bundle E10 over the 9-sphere. (The clutching function for E10 arises from the stable homotopy group π8(O(10)) = Z/2Z.) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel-Whitney class w9 of E10 vanishes by the Wu formula w9 = w1w8 + Sq1(w8). Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of E10 with any trivial bundle remains nontrivial. Template:Harv

Given a 2k-dimensional vector bundle E we have

pk(E)=e(E)⌣e(E),

where e(E) denotes the Euler class of E, and ⌣ 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

pk(E,Q)∈H4k(M,Q)

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, the total Pontryagin class is expressed as

p=[1−Tr(Ω2)8π2+Tr(Ω2)2−2Tr(Ω4)128π4−Tr(Ω2)3−6Tr(Ω2)Tr(Ω4)+8Tr(Ω6)3072π6+⋯]∈HdR*(M),

where Ω denotes the curvature form, and H*dR(M) denotes the de Rham cohomology groups.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

Pontryagin classes of a manifold

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

Novikov proved in 1966 that if manifolds are homeomorphic then their rational Pontryagin classes pk(M, Q) in H4k(M, Q) are the same.

If the dimension is at least five, there are at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.

Pontryagin numbers

Pontryagin numbers are certain topological invariants 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 4n-dimensional manifold M and a collection of natural numbers

k1, k2, ..., km such that k1+k2+...+km =n.

the Pontryagin number Pk1,k2,…,km is defined by

Pk1,k2,…,km=pk1⌣pk2⌣⋯⌣pkm([M])

where pk denotes the k-th Pontryagin class and [M] the fundamental class of M.

Properties

  1. Pontryagin numbers are oriented cobordism invariant; and together with Stiefel-Whitney numbers they determine an oriented manifold's oriented cobordism class.
  2. 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.
  3. Such invariants as signature and A^-genus can be expressed through Pontryagin numbers.

Generalizations

There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure.

See also

References

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