Small-gain theorem: Difference between revisions

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In the theorem, I changed "Assume two systems" to "Assume two stable systems"
 
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In [[cryptography]], '''PKCS #1''' is the first of a family of standards called [[PKCS|Public-Key Cryptography Standards (PKCS)]], published by [[RSA Laboratories]]. It provides the basic definitions of and recommendations for implementing the [[RSA (algorithm)|RSA]] algorithm for [[public-key cryptography]]. It defines the mathematical properties of public and private keys, primitive operations for encryption and signatures, secure cryptographic schemes, and related [[Abstract Syntax Notation One|ASN.1]] syntax representations.
 
The current version, 2.1, was published in June 2002, and was also republished as RFC 3447 in February 2003.
 
==Keys==
 
The PKCS #1 standard defines the mathematical definitions and properties that RSA public and private keys must have. The traditional key pair is based on a modulus, <math>n</math>, that is the product of two distinct large [[prime number]]s, <math>p</math> and <math>q</math>, such that <math>n = pq</math>.
 
Starting with version 2.1, this definition was generalized to allow for multi-prime keys, where the number of distinct primes may be two or more. When dealing with multi-prime keys, the prime factors are all generally labeled as <math>r_i</math> for some <math>i</math>, such that:
 
: <math>n = r_1 \cdot r_2 \cdot ... \cdot r_u,</math> for <math>u \ge 2</math>
 
As a notational convenience, <math>p = r_1</math> and <math>q = r_2</math>.
 
The RSA public key is represented as the tuple <math>(n, e)</math>, where the integer <math>e</math> is the public exponent.
 
The RSA private key may have two representations. The first compact form is the tuple <math>(n, d)</math>, where <math>d</math> is the private exponent. The second form has at least five terms, or more for multi-prime keys. Although mathematically redundant to the compact form, the additional terms allow for certain computational optimizations when using the key.
 
==Primitives==
 
The standard defines several basic primitives. The primitive operations provide the fundamental instructions for turning the raw mathematical formulas into computable algorithms.
 
* I2OSP, OS2IP: conversion between the potentially large nonnegative integers encountered in the mathematical formulas and their computer data representation as a sequence of bytes (an [[Octet (computing)|octet string]]).
* RSAEP, RSADP: basic encryption and decryption algorithms.
* RSASP1, RSAVP1: algorithms for producing and verifying signatures.
 
==Schemes==
 
By themselves the primitive operations do not necessarily provide any security.  The concept of a cryptographic scheme is to define higher level algorithms or uses of the primitives so they achieve certain security goals.
 
There are two schemes for encryption and decryption:
* RSAES-OAEP: improved encryption/decryption scheme; based on the [[Optimal Asymmetric Encryption Padding]] scheme proposed by [[Mihir Bellare]] and [[Phillip Rogaway]].
* RSAES-PKCS1-v1_5: older encryption/decryption scheme as first standardized in version 1.5 of PKCS #1.
 
There are also two schemes for dealing with signatures:
* RSASSA-PSS: improved probabilistic signature scheme with appendix; based on the Probabilistic Signature Scheme originally invented by Bellare and Rogaway.
* RSASSA-PKCS1-v1_5: old signature scheme with appendix as first standardized in version 1.5 of PKCS #1.
 
The two signature schemes make use of separately defined encoding methods:
* EMSA-PSS: encoding method for signature appendix, probabilistic signature scheme.
* EMSA-PKCS1-v1_5: encoding method for signature appendix as first standardized in version 1.5 of PKCS #1.
 
The signature schemes are actually signatures ''with appendix'', which means that rather than signing some input data
directly a [[hash function]] is used first to produce an intermediary representation of the data and then the result of the hash is signed. This technique is almost always used with RSA because the amount of data that can be directly signed is proportional to the size of the keys; which is almost always much smaller than the amount of data an application may wish to sign.
 
==Version history==
 
* Versions 1.1–1.3, February through March 1991, privately distributed.
* Version 1.4, June 1991, published for NIST/OSI Implementors' Workshop.
* Version 1.5, November 1993. First public publication.  Republished as RFC 2313.
* Version 2.0, September 1998. Republished as RFC 2437.
* Version 2.1, June 2002. Republished as RFC 3447.
* Version 2.2, October 2012.
 
==External links==
* [http://www.emc.com/collateral/white-papers/h11300-pkcs-1v2-2-rsa-cryptography-standard-wp.pdf PKCS #1: RSA Cryptography Standard], RSA Laboratories
* RFC 3447
* [http://www.emc.com/emc-plus/rsa-labs/historical/raising-standard-rsa-signatures-rsa-pss.htm Raising the Standard for RSA Signatures: RSA-PSS], Burt Kaliski, RSA Laboratories February 26, 2003.
 
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{{Cryptography navbox}}
 
[[Category:Cryptography standards]]

Latest revision as of 06:12, 18 November 2013

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In cryptography, PKCS #1 is the first of a family of standards called Public-Key Cryptography Standards (PKCS), published by RSA Laboratories. It provides the basic definitions of and recommendations for implementing the RSA algorithm for public-key cryptography. It defines the mathematical properties of public and private keys, primitive operations for encryption and signatures, secure cryptographic schemes, and related ASN.1 syntax representations.

The current version, 2.1, was published in June 2002, and was also republished as RFC 3447 in February 2003.

Keys

The PKCS #1 standard defines the mathematical definitions and properties that RSA public and private keys must have. The traditional key pair is based on a modulus, n, that is the product of two distinct large prime numbers, p and q, such that n=pq.

Starting with version 2.1, this definition was generalized to allow for multi-prime keys, where the number of distinct primes may be two or more. When dealing with multi-prime keys, the prime factors are all generally labeled as ri for some i, such that:

n=r1⋅r2⋅...⋅ru, for u≥2

As a notational convenience, p=r1 and q=r2.

The RSA public key is represented as the tuple (n,e), where the integer e is the public exponent.

The RSA private key may have two representations. The first compact form is the tuple (n,d), where d is the private exponent. The second form has at least five terms, or more for multi-prime keys. Although mathematically redundant to the compact form, the additional terms allow for certain computational optimizations when using the key.

Primitives

The standard defines several basic primitives. The primitive operations provide the fundamental instructions for turning the raw mathematical formulas into computable algorithms.

  • I2OSP, OS2IP: conversion between the potentially large nonnegative integers encountered in the mathematical formulas and their computer data representation as a sequence of bytes (an octet string).
  • RSAEP, RSADP: basic encryption and decryption algorithms.
  • RSASP1, RSAVP1: algorithms for producing and verifying signatures.

Schemes

By themselves the primitive operations do not necessarily provide any security. The concept of a cryptographic scheme is to define higher level algorithms or uses of the primitives so they achieve certain security goals.

There are two schemes for encryption and decryption:

There are also two schemes for dealing with signatures:

  • RSASSA-PSS: improved probabilistic signature scheme with appendix; based on the Probabilistic Signature Scheme originally invented by Bellare and Rogaway.
  • RSASSA-PKCS1-v1_5: old signature scheme with appendix as first standardized in version 1.5 of PKCS #1.

The two signature schemes make use of separately defined encoding methods:

  • EMSA-PSS: encoding method for signature appendix, probabilistic signature scheme.
  • EMSA-PKCS1-v1_5: encoding method for signature appendix as first standardized in version 1.5 of PKCS #1.

The signature schemes are actually signatures with appendix, which means that rather than signing some input data directly a hash function is used first to produce an intermediary representation of the data and then the result of the hash is signed. This technique is almost always used with RSA because the amount of data that can be directly signed is proportional to the size of the keys; which is almost always much smaller than the amount of data an application may wish to sign.

Version history

  • Versions 1.1–1.3, February through March 1991, privately distributed.
  • Version 1.4, June 1991, published for NIST/OSI Implementors' Workshop.
  • Version 1.5, November 1993. First public publication. Republished as RFC 2313.
  • Version 2.0, September 1998. Republished as RFC 2437.
  • Version 2.1, June 2002. Republished as RFC 3447.
  • Version 2.2, October 2012.

External links

Template:PKCS navbox Template:Cryptography navbox

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