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'''Pollard's ''p'' − 1 algorithm''' is a [[number theory|number theoretic]] [[integer factorization]] [[algorithm]], invented by [[John Pollard (mathematician)|John Pollard]] in 1974. It is a special-purpose algorithm, meaning that it is only suitable for [[integer]]s with specific types of factors; it is the simplest example of an [[algebraic-group factorisation algorithm]].
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The factors it finds are ones for which the number preceding the factor, ''p'' − 1, is [[smooth number#Powersmooth numbers|powersmooth]]; the essential observation is that, by working in the multiplicative group [[Modular arithmetic|modulo]] a composite number ''N'', we are also working in the multiplicative groups modulo all of ''N'''s factors.
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
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The existence of this algorithm leads to the concept of [[strong prime]]s, being primes for which ''p'' − 1 has at least one large prime factor.  Almost all sufficiently large primes are strong; if a prime used for cryptographic purposes turns out to be non-strong, it is much more likely to be through malice than through an accident of random number generation.
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==Base concepts==
'''MathML'''
Let ''n'' be a composite integer with prime factor ''p''.  By [[Fermat's little theorem]], we know that for all integers ''a'' coprime to ''p'' and for all positive integers ''K'':
:<math forcemathmode="mathml">E=mc^2</math>


:<math>a^{K(p-1)} \equiv 1\pmod{p}</math>
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


If a number ''x'' is congruent to 1 [[Modular arithmetic|modulo]] a factor of ''n'', then the [[Greatest common divisor|gcd]] (''x''&nbsp;&minus;&nbsp;1,&nbsp;''n'') will be divisible by that factor.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


The idea is to make the exponent a large multiple of ''p''&nbsp;&minus;&nbsp;1 by making it a number with very many prime factors; generally, we take the product of all prime powers less than some limit ''B''. Start with a random ''x'', and repeatedly replace it by <math>x^w \mod n</math> as ''w'' runs through those prime powers. Check at each stage, or once at the end if you prefer, whether [[Greatest common divisor|gcd]] (''x''&nbsp;&minus;&nbsp;1,&nbsp;''n'') is not equal to&nbsp;1.
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


==Multiple factors==
==Demos==


It is possible that for all the prime factors ''p'' of ''n'', ''p''&nbsp;&minus;&nbsp;1 is divisible by small primes, at which point the Pollard ''p''&nbsp;&minus;&nbsp;1 algorithm gives you ''n'' again.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


==Algorithm and running time==
The basic algorithm can be written as follows:


:'''Inputs''': ''n'': a composite integer
* accessibility:
:'''Output''': a non-trivial factor of ''n'' or <u>failure</u>
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


:# select a smoothness bound ''B''
==Test pages ==
:#<math>M \gets \prod_{\text{primes}~q \le B} q^{ \lfloor \log_q{B} \rfloor }</math>
:# randomly pick ''a'' coprime to ''n'' (note: we can actually fix ''a'', random selection here is not imperative)
:#<math>g \gets \gcd(a^M - 1, n)</math> (note: the powering can be done mod ''n'')
:# if 1 < ''g'' < ''n'' then return ''g''
:# if ''g'' = 1 then select a higher ''B'' and go to step 2 or return <u>failure</u>
:# if ''g'' = ''n'' then go to step 2 or return <u>failure</u>


If ''g'' = 1 in step 6, this indicates that for all ''p'' &minus; 1 that none were ''B''-powersmooth.  If ''g'' = ''n'' in step 7, this usually indicates that all factors were ''B''-powersmooth, but in rare cases it could indicate that ''a'' had a small order modulo ''n''.
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
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The running time of this algorithm is O(''B''&nbsp;&times;&nbsp;log&nbsp;''B''&nbsp;&times;&nbsp;log<sup>2</sup>''n''); larger values of ''B'' make it run more slowly, but are more likely to produce a factor.
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
==How to choose ''B''?==
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
Since the algorithm is incremental, it can just keep running with the bound constantly increasing.
 
Assume that ''p''&nbsp;&minus;&nbsp;1, where ''p'' is the smallest prime factor of ''n'', can be modelled as a random number of size less than&nbsp;&radic;''n''. By [[Dixon's theorem]],  the probability that the largest factor of such a number is less than (''p''&nbsp;&minus;&nbsp;1)<sup>''&epsilon;''</sub> is roughly ''&epsilon;''<sup>&minus;''&epsilon;''</sub>; so there is a probability of about 3<sup>&minus;3</sup>&nbsp;=&nbsp;1/27 that a ''B'' value of ''n''<sup>1/6</sup> will yield a factorisation.
 
In practice, the [[elliptic curve method]] is faster than the Pollard ''p''&nbsp;&minus;&nbsp;1 method once the factors are at all large; running the ''p''&nbsp;&minus;&nbsp;1 method up to ''B''&nbsp;=&nbsp;10<sup>6</sup> will find a quarter of all twelve-digit factors and 1/27 of all eighteen-digit factors, before proceeding to another method.
 
==Large prime variant==
{{incoherent}}
A variant of the basic algorithm is sometimes used; instead of requiring that ''p''&nbsp;&minus;&nbsp;1 has all its factors less than ''B'', we can require it to have all but one of its factors less than some ''B''<sub>1</sub>, and the remaining factor less than some ''B''<sub>2</sub>.  Let ''p''<sub>1</sub> be the smallest prime greater than ''B''<sub>1</sub>, ''p''<sub>2</sub> the next-largest, and so on; let ''d''<sub>''n''</sub>&nbsp;=&nbsp;''p''<sub>''n''</sub>&nbsp;&minus;&nbsp;''p''<sub>''n''&minus;1</sub>. The distribution of prime numbers is such that the ''d''<sub>''n''</sub> will all be fairly small.
 
Having computed <math>c = a^M \mod n</math>, we can easily compute once and for all <math>E_r = c^r \mod n</math> for all <math>r</math> which appear as a value of <math>d_n</math>.  Compute <math>t_1 = c^{p_1} \mod n</math>. We can then stop doing exponentiation, and compute
 
: <math>t_2 (= c^{p_2} \mod n) = t_1 E_{d_2} \mod n</math>, <math>t_3 = t_2 E_{d_3} \mod n, \dots</math>
 
with one multiplication rather than one exponentiation at each step; this is quicker by roughly a factor log&nbsp;''B'' than doing the exponentiations.  It can also be accelerated significantly using [[Fast Fourier transform]]s.
 
==Implementations==
 
* The [http://gforge.inria.fr/projects/ecm/ GMP-ECM] package includes an efficient implementation of the ''p''&nbsp;&minus;&nbsp;1 method.
* [[Prime95]] and [[MPrime]], the official clients of the [[Great Internet Mersenne Prime Search]], use p - 1 to eliminate potential candidates.
 
==References==
*{{Citation |last=Pollard |first=J. M. |year=1974 |title=Theorems of Factorization and Primality Testing |journal=Proceedings of the Cambridge Philosophical Society |volume=76 |issue=3 |pages=521–528 |doi=10.1017/S0305004100049252 |issn= }}
 
==See also==
* [[Williams' p + 1 algorithm]]
 
==External links==
*[http://modular.math.washington.edu/edu/2007/spring/ent/ent-html/node81.html Pollard's ''p''&nbsp;&minus;&nbsp;1 Method]
 
{{Number theoretic algorithms}}
 
[[Category:Integer factorization algorithms]]
 
[[de:Pollard-p-1-Methode]]
[[es:Algoritmo p − 1 de Pollard]]
[[fr:Algorithme p-1 de Pollard]]
[[nl:Pollards p-1-methode]]
[[ru:P-1 метод Полларда]]
[[th:ขั้นตอนวิธีพีลบหนึ่งของพอลลาร์ด]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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