en>I eat BC Fish: rate of return instead of interest, specified interest-bearing bank account

2015-01-11T18:59:27Z

rate of return instead of interest, specified interest-bearing bank account

← Older revision		Revision as of 20:59, 11 January 2015
Line 1:		Line 1:
	~~{{DISPLAYTITLE:Pollard's ''p'' − 1 algorithm}}~~		Greetings! I am Dalton. Acting is without a doubt a thing that My business is totally addicted to. My your own house is now in Vermont and I don't plan on changing it. I am a cashier. I'm not good at webdesign but you may be want to check my website: http://prometeu.net<br><br>My webpage [http://prometeu.net Clash Of Clans Hack Tools]
	~~'''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]].~~

	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.~~

	The existence of this algorithm leads to the concept of [[safe prime]]s, being primes for which ''p'' − 1 is two times a [[Sophie Germain prime]] ''q'' and thus minimally smooth. These primes are sometimes construed as "safe for cryptographic purposes", but they might be ''unsafe'' — in current recommendations for cryptographic [[strong prime]]s (''e.g.'' [[ANSI X9.31]]), it is [[necessary but not sufficient]] that ~~''p'' − 1 has at least one large prime factor. Most 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]]~~. ~~This terminology~~ is ~~considered [[obsolescent]] by the cryptography industry.~~
	~~[http://www.rsa.com/rsalabs/node.asp?id=2217]~~

	~~==Base concepts==~~
	~~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>a^{K(p-1)} \equiv 1\pmod{p}</math>~~

	~~If a number~~ '~~'x'' is congruent to 1 [[Modular arithmetic\|modulo]] a factor of ''n'', then the {{nowrap\|[[Greatest common divisor\|gcd]](''x'' − 1, ''n'')}} will be divisible by that factor.~~

	~~The idea is to make the exponent a large multiple of ''p'' − 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 {{nowrap\|gcd(''x'' − 1, ''n~~'~~')}} is~~ not ~~equal to 1.~~

	~~==Multiple factors==~~

	~~It is possible that for all the prime factors ''p'' of ''n'', ''p'' − 1 is divisible by small primes,~~ at ~~which point the Pollard ''p'' − 1 algorithm gives~~ you ~~''n'' again.~~

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

	~~:'''Inputs''': ''n'': a composite number~~
	~~:'''Output''': a nontrivial factor of ''n'' or <u>failure</u>~~

	~~:# select a smoothness bound ''B''~~
	~~:# define <math>M = \prod_{\text{primes}~q \le B} q^{ \lfloor \log_q{B} \rfloor }</math> (note: explicitly evaluating ''M''~~ may ~~not~~ be ~~necessary)~~
	~~:# randomly pick ''a'' coprime~~ to ~~''n'' (note~~: ~~we can actually fix ''a'', random selection here is not imperative)~~
	~~:# compute {{nowrap\|''g'' {{=}} gcd(''a''<sup>''M''</sup> − 1, ''n'')}} (note: exponentiation can be done modulo ''n'')~~
	~~:# if {{nowrap\|1 < ''g'' < ''n''}} then return ''g''~~
	~~:# if {{nowrap\|''g'' {{=}} 1}} then select a larger ''B'' and go to step 2 or return <u>failure</u>~~
	~~:# if {{nowrap\|''g'' {{=}} ''n''}} then select a smaller ''B'' and go to step 2 or return <u>failure</u>~~

	If {{nowrap\|''g'' {{=}} 1}} in step 6, this indicates that not all factors of {{nowrap\|''p'' − 1}} were ''B''-powersmooth. If {{nowrap\|''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''.

	~~The running time of this algorithm is {{nowrap\|O(''B'' × log ''B'' × log<sup>2</sup> ''n'')}}; larger values of ''B'' make it run slower, but are more likely to produce a factor.~~

	~~==How to choose ''B''?==~~

	~~Since the algorithm is incremental, it can just keep running with the bound constantly increasing.~~

	Assume that ''p'' − 1, where ''p'' is the smallest prime factor of ''n'', can be modelled as a random number of size less than √''n''. By [[Dixon's theorem]], the probability that the largest factor of such a number is less than (''p'' − 1)<sup>''ε''</sub> is roughly ''ε''<sup>−''ε''</sub>; so there is a probability of about 3<sup>−3</sup> = 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'' − 1 method once the factors are at all large; running the ''p'' − 1 method up to ''B'' = 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.

	~~==Two-stage variant==~~
	A variant of the basic algorithm is sometimes used; instead of requiring that ''p'' − 1 has all its factors less than ''B'', we require it to have all but one of its factors less than some ''B''<sub>1</sub>, and the remaining factor less than some {{nowrap\|''B''<sub>2</sub> ≫ ''B''<sub>1</sub>}}. After completing the first stage, which is the same as the basic algorithm, instead of computing a new

	~~:<math>M' = \prod_{\text{primes}~p \le B_2} q^{ \lfloor \log_q{B_2} \rfloor }~~
	~~</math>~~

	~~for ''B''<sub>2</sub> and checking {{nowrap\|gcd(''a''<sup>''M'''</sup> − 1, ''n'')}}, we compute~~

	:~~<math>Q = \prod_{\text{primes}~q \in (B_1, B_2]} (H^q - 1)</math>~~

	~~where {{nowrap\|''H'' {{=}} ''a''<sup>''M''</sup>}} and check if {{nowrap\|gcd(''Q'', ''n'')}} produces a nontrivial factor of ''n''. As before, exponentiations can be done modulo ''n''.~~

	Let {''q''<sub>1</sub>, ''q''<sub>2</sub>, …} be successive prime numbers in the interval {{nowrap\|(''B''<sub>1</sub>, ''B''<sub>2</sub>]}} and ''d''<sub>''n''</sub> = ''q''<sub>''n''</sub> − ''q''<sub>''n''−1</sub> the difference between consecutive prime numbers. Since typically {{nowrap\|''B''<sub>1</sub> > 2}}, {{nowrap\|''d''<sub>''n''</~~sub>}} are even numbers. The distribution of prime numbers is such that the ''d''<sub>''n''<~~/~~sub> will all be relatively small~~. It is suggested that {{nowrap\|''d''<sub>''n''</sub> ≤ [[Natural logarithm\|ln]]<sup>2</sup> ''B''<sub>2</sub>}}. Hence, the values of {{nowrap\|''H''<sup>2</sup>}}, {{nowrap\|''H''<sup>4</sup>}}, {{nowrap\|''H''<sup>6</sup>}}, … (mod ''n'') can be stored in a table, and {{nowrap\|''H''<sup>''q''<sub>''n''</sub></sup>}} be computed from {{nowrap\|''H''<sup>''q''<sub>''n''−1</sub><~~/sup~~>~~⋅''H''~~<~~sup~~>~~''d''<sub>''n''</sub></sup>}}, saving the need for exponentiations.~~

	~~==Implementations==~~

	* The [http://~~gforge.inria.fr/projects/ecm/ GMP-ECM] package includes an efficient implementation of the ''p'' − 1 method~~.
	* [[Prime95]] and [[MPrime]], the official clients of the [[Great Internet Mersenne Prime Search]], use p - 1 to eliminate potential candidates.

	~~==See also==~~
	* [[Williams' p + 1 algorithm]]

	~~==References==~~
	*{{Cite journal \|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= }}
	*{{Cite journal \|last1=Montgomery \|first1=P. L. \|last2=Silverman \|first2=R. D. \|year=1990 \|title=An FFT extension to the ''P'' − 1 factoring algorithm \|journal=Mathematics of Computation \|volume=54 \|issue=190 \|pages=839–854 \|doi=10.1090/S0025-5718-1990-1011444-3 \|issn= }}

	~~==External links==~~
	*[http://modular.math.washington.edu/edu/2007/spring/ent/ent-html/node81.html Pollard's ''p'' − 1 Method]
	*[http://ardoino.com/2004/03/maths-factoring-pollard/ Pollard's ''p'' − 1 Algorithm source code]

	~~{{Number theoretic algorithms}}~~

	~~[[Category:Integer factorization algorithms]~~]

189.251.140.9: /* Overview */

2014-01-16T02:10:46Z

Overview

Show changes

173.49.58.197: /* Compound interest */ FV(annuity) formula in text didn't match .jpg in cite (and doesn't work, either)

2012-08-01T03:55:17Z

Compound interest: FV(annuity) formula in text didn't match .jpg in cite (and doesn't work, either)

New page

Ones Paypal income adder helps you pull in 15$ over Paypal on matter of minutes. All of our 100 % free Paypal finances adder will be a hundred% constitutional.wish start using ninjacloak proxy in better choose.go to many instructions. This program can be 100% able to start using and all sorts of pesky insects tends to be checked out when Beta testing 0.three and update monthly at your hosting server. payPal income Adder v1.2.03 is now in the market or obtain having entire fledged efficiency. Now you may cheat their paypal machine implementing every IP also straightaway establish dough your paypal membership. Paypal income Adder v1.zero.03 programs are scanned for virus which is infection release. This is simply not each adware or simply spyware and adware plan. Paypal income crack is definitely that stylish system, which one glance sincere straightforward. PayPal creates any organization otherwise individual together with an email target to securely, effortlessly and value-nicely pass and/or be given fees web. Paypal net shapes for the present monetary system of checking account plus charge cards to make an international, actual-experience fee remedy. We have send an item preferably suited for small business, on-line vendors, anyone and the like generally underserved after common amount things. To know about Paypal funds Adder also Paypal funding power generators, please go to all of our web page: [http://www.dailymotion.com/video/x1zkexr_paypal-money-adder-2014_tech free paypal money adder no survey no password] League with star hacking Software can be purchased with numerous web sites freely. However you must be alert associated with location where you stand obtaining the particular group connected with Legends compromise concept. Be sure the internet site may quality prior to opting to complete generally obtaining. Achieving finishes encounter nowadays is tough, and yet our very own band of developers continues hard of working inside set that Paypal funding adder which allows people to increase bucks with their Paypal be aware of complimentary. That is not a fraud, one is not a couple travel by night procedure. We’re aimed at helping you to buy f-r-e-e paypal money for the reason that we understand the easiest way exhausting it can be in order to get simply without the need of only a little help in. A number of our obtain professionals start using every paypal income power generators so as to keep their expenses paying also were going to go things along as we identified precisely how efficient it absolutely was. Contrary phony mills, mine does not inquire about personal pass word or other recognizing expertise, and that means you dont put your game account in danger. League about star hacking device can be had during even sites freely. Nevertheless you should be mindful of web page where you stand grabbing the specific category involving figures compromise accessory. Make sure the site can be trustworthy before opting to initiate the actual installing. Brewing ends suit right now is difficult, although all of our band of programmers might solid at the job in order to set any Paypal funding adder that enables owners to incorporate revenue for their Paypal account for free of charge. Our isn’t a scam, which is not many journey when the sun goes down operation. We’re centered on helping you shop for complimentary paypal bucks considering we know strategies challenging it really is to uncover in without worrying about slightly advice. A number of our have employees employ all of our paypal dollars generators so as to keep the actual charges remunerated and also wished to go the product around as we identified precisely how successful it has been. In contrast to fraud turbines, ours does not obtain your individual password or other selecting content, therefore you dont set your account in danger.

Future value - Revision history

en>I eat BC Fish: rate of return instead of interest, specified interest-bearing bank account

189.251.140.9: /* Overview */

173.49.58.197: /* Compound interest */ FV(annuity) formula in text didn't match .jpg in cite (and doesn't work, either)