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Topic: Quadratic sieve

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	Quadratic sieve - Wikipedia, the free encyclopedia
		The quadratic sieve algorithm (QS) is a modern integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve).
		The quadratic sieve is a modification of Dixon's factorization method.
		This approach (called MPQS, Multiple Polynomial Quadratic Sieve) is ideally suited for parallelization, since each processor involved in the factorization can be given n, the factor base and a collection of polynomials, and it will have no need to communicate with the central processor until it is finished with its polynomials.
	en.wikipedia.org /wiki/Quadratic_sieve (2183 words)

	PlanetMath: quadratic sieve (Site not responding. Last check: 2007-09-17)
		The quadratic sieve method of factoring depends upon being able to create a set of numbers whose factorization can be expressed as a product of pre-chosen primes.
		To accomplish this, the quadratic sieve method uses a set of prime numbers called a factor base.
		This is version 6 of quadratic sieve, born on 2002-06-19, modified 2003-03-24.
	planetmath.org /encyclopedia/QuadraticSieve.html (382 words)

	PlanetMath: sieve of Eratosthenes
		The sieve of Eratosthenes is a simple algorithm for generating a list of the prime numbers between
		Today, there are faster methods, such as a quadratic sieve.
		This is version 3 of sieve of Eratosthenes, born on 2002-05-22, modified 2004-11-22.
	planetmath.org /encyclopedia/SieveOfEratosthenes.html (111 words)

	Re: Quadratic Sieve
		I'm interested to read that, as it seemed to me that with the standard quadratic sieve (using Montgomery quadratics) there was a limitation on the size of the number one could factorise.
		As I explained it, one used quadratics Q(x) = ax^2 + 2bx + c with ac - b^2 = n so that aQ(x) = (ax + b)^2 - n, and one sieved for smooth numbers Q(j) on a sieve centred at the positive root of Q(x).
		But with a sieve size of say 2 million, and a number with 100 digits, one would only get an occasional smooth number from each quadratic and as far as I could see would hardly ever if ever get two.
	www.usenet.com /newsgroups/sci.math/msg11856.html (310 words)

	Citations: The quadratic sieve factoring algorithm - Pomerance (ResearchIndex) (Site not responding. Last check: 2007-09-17)
		4.1 The quadratic sieve The first of these algorithms to be presented was the Quadratic Sieve, which was developed in the early 1980 s [Po84] It works as follows: The basic flavor of the quadratic sieve is similar to what is known as the continued fraction algorithm.
		Carl Pomerance, The quadratic sieve factoring algorithm, in [1], 169-182.
		Carl Pomerance, The quadratic sieve factoring algorithm, in [25] (1985), 169-182.
	citeseer.ist.psu.edu /context/499628/0 (2688 words)

	Factorization of RSA-130 (12 Apr 1996) (Site not responding. Last check: 2007-09-17)
		This program uses `lattice sieving with sieving by vectors' as introduced by Pollard in [P], and is based on the implementation described in [GLM].
		This was exploited in the Web-based sieving effort, which used a collection of CGI scripts ("FAFNER", from Cooperating Systems Corporation) to automate and coordinate the flow of tasks and relations within the globally distributed network of anonymous sieving clients.
		The changes also made it hard to estimate how much time was spent on the sieving stage, because the performance of the siever strongly depends on the amount of memory it gets.
	www.utm.edu /research/primes/notes/rsa130.html (944 words)

	Citations: A pipeline architecture for factoring large integers with the quadratic sieve algorithm - Pomerance, Smith, ... (Site not responding. Last check: 2007-09-17)
		describes a concept for a dedicated factoring machine, implementing a variation on the quadratic sieve algorithm, which the author s conjecture could have been built in 1988 for 50,000 to be able to factor a 100 digit number in two weeks.
		In quadratic sieve algorithms the numbers w i are the values of one (or more) quadratic polynomials with integer coe#cients.
		This was later published in [1] and is known as the self initializing quadratic sieve (siqs) 2.1 The Self Initializing Quadratic Sieve The self....
	citeseer.lcs.mit.edu /context/270985/0 (2244 words)

	[No title]
		In 1992 they would have told you that it was practical, but only faster than the quadratic sieve for numbers greater than 130-150 digits or so.
		A related algorithm, the special number field sieve, can already factor numbers of a certain specialized form--numbers not generally used for cryptography--must faster than the general number field sieve can factor general numbers of the same size.
		High estimates assume a budget of $25 billion, a general quadratic sieve algorithm running at the speed of the special number field sieve, and a technology advance of 45% per year.
	www.imada.sdu.dk /~joan/crypt/predictions.html (1586 words)

	Quadratic Sieve
		The sieving step is the time consuming part of the algorithm where we try to find values of f(x) that factors completely over the factor base.
		It is very hard to give an exact running time for the quadratic sieve, but it depends mostly on the length of the sieving interval, the size of the factor base and the threshold value that decides when to do trial division on a number.
		Check the literature section for a paper that gives a detailed analysis of the running time of the quadratic sieve since that is a whole project in itself.
	www.carceri.dk /qs/qs.html (2455 words)

	Crunch big numbers with GT3 using a quadratic sieve
		Among the many algorithms designed for this task, the quadratic sieve is one of the fastest.
		Although the number field sieve (NFS) is heuristically faster than QS, QS is still the algorithm of choice for numbers between 50 and 110 digits.
		The quadratic sieve algorithm is based in code from Dario Alejandro Alpern; on this site he talks about factoring with the sieve and presents a nifty online application for factoring with the elliptic curve method.
	www-128.ibm.com /developerworks/grid/library/gr-factor?ca=degr-L0597GT3quad (1745 words)

	The Number Field Sieve
		The number field sieve is a new factoring algorithm with a much lower asymptotic running time estimate than previous algorithms.
		Although asymptotically this is still far better than other algorithms, the point at which this method would be faster than algorithms such as the quadratic sieve appears to be in the vicinity of 200 decimal digits.
		On the other hand, the number field sieve is a very recent invention, and so it is likely that substantial improvements might occur which would make it practical.
	www.farcaster.com /papers/crypto-field/node8.html (368 words)

	Talk:Quadratic sieve - Wikipedia, the free encyclopedia
		I really believe a less contrived example would be welcome, as it would decrease the probability of such errors and would help readers get a better grasp of the concept.
		OK, I wrote a new example, with a bigger n and more strictly following the procedure outlined in the article (this example uses sieving).
		I tried to add a more approachable introduction to the ideas behind the algorithm, based roughly on the presentation from Prime Numbers: A Computational Perspective.
	en.wikipedia.org /wiki/Talk:Quadratic_sieve (212 words)

	[No title] (Site not responding. Last check: 2007-09-17)
		This was done with the self initializing quadratic sieve (siqs) [5,1,4].
		However, due to the poor factor base, we had to use a multiplier of 29, which means that we were actually sieving upon a 118 digit number.
		Perhaps a significant increase in speed could have been obtained by only sieving on these residues (ie not sieving on those which are not divisible by 2).
	www.crypto-world.com /announcements/siqs116.text (716 words)

	Q48: What are the Best Factoring Methods in Use Today?
		This was due to initial inefficiencies that made the number field sieve less efficient than the quadratic sieve.
		It is now estimated that if the number field sieve had been used, it would have taken one quarter of the time.
		Clearly, the number field sieve will overtake the mpqs as the most widely used factoring algorithm, as the size of the numbers being factored increases from about 130 digits, which is the current threshold of general numbers which can be factored, to 140 or 150 digits.
	www.x5.net /faqs/crypto/q48.html (501 words)

	The Prime Page's Links++: programs/sieves/binary_quadratic
		This sieve can be generalized by viewing it as sieving with the reducible binary quadratic form xy.
		AB99] show that such seives compete favorably with the Sieve of Eratosthenes both theoretically and in practice.
		Sieve of Atkins - This version generates the 50847534 primes up to 1000000000 in just 8 seconds on a Pentium II-350; it prints them in decimal in just 35 seconds.
	primes.utm.edu /links/programs/sieves/binary_quadratic (179 words)

	Quadratic Sieve Factoring Algorithm (Parallel version) (Site not responding. Last check: 2007-09-17)
		The quadratic sieve is a method of factoring large numbers, specifically large numbers that are a product of two large primes; the output of the program is the two primes that when multiplied, produce the number in question.
		The only parallel portion of the program is the sieving phase, which also takes a large majority of the computation time (generally around 90%).
		MUCH theory has been left out of this description, as the mathematics behind the sieve are very difficult to grasp, much less explain
	www.hcs.ufl.edu /~murphy/cluster/quadsieve.htm (634 words)

	[No title]
		To reduce the overhead of using a large factor base, cache blocking is used for the sieve interval and the factor base; this requires that sieving takes place for several polynomials simultaneously.
		This allows the sieve to be over both positive and negative values.
		Every relation found by the sieve must be associated with (i.e.
	www.sonic.net /~someone/files/c/notmine/msieve/msieve.h (527 words)

	Topas GmbH: Analysis (Site not responding. Last check: 2007-09-17)
		By means of an electromagnetic sieve vibrator EMS horizontal vibrations are applied to the microsieve which is held in sieve clamp.
		Performed by feeding material through a vibrating sieve clamped in an electromagnetic sieve vibrator EMS into the bulk-density sample cylinder (volume 100cm³) and determine net weight of material.
		The sieve frames (Ø 75mm) are made of stainless steel and fits into the sieve clamp of the electromagnetic sieve vibrator series EMS.
	www.topas-gmbh.de /analysis.htm (951 words)

	☞ factoring - quadratic factoring - quadratic factoring online guide (Site not responding. Last check: 2007-09-17)
		In 1990 my own quadratic sieve factoring algorithm had doubled the length of the numbers that could be factored, the record having 116 digits.
		Quadratic Polynomials: Factoring by Guessing There are three methods to factor a quadratic polynomial: Factoring by guessing, "completing the square", and the quadratic formula.
		There are three methods to factor a quadratic polynomial: Factoring by guessing, "completing the square", and the quadratic formula.
	www.4-factoring.info /factoring/quadratic-factoring.html (638 words)

	[No title] (Site not responding. Last check: 2007-09-17)
		It is conjectured that the 3-part of the class number of the quadratic field Q(sqrt{D}) may be bounded above by an arbitrarily small power of D.
		We use a variant of the square sieve and the q-analogue of van der Corput's method to count the number of squares of the form 4x^3 - dz^2, where d is a square-free positive integer and x and z lie in the ranges x << d^{1/2}, z<< d^{1/4}.
		As a result, we show that the 3-part of the class number of the quadratic field Q(sqrt{D}) may be bounded by O(D^{27/56 + epsilon}).
	www.math.ucla.edu /%7Entg/Local/af04/pierce.html (191 words)

	Factorization (Site not responding. Last check: 2007-09-17)
		A combination of algorithms (trial division, Pollard rho, elliptic curve method, and quadratic sieve) is used to attempt to find the complete factorization of n, where n is a non-zero integer.
		By setting ECMOnly true the use of the quadratic sieve is avoided (which is important if heavy use of temporary file space is to be avoided).
		By default, after the initial factorization attempts (trial division, Pollard) have not succeeded entirely, ECM is invoked with small bound and few curves (10), then the quadratic sieve.
	www.math.uga.edu /~matthews/DOCS/MAGMA/text342.html (1511 words)

	numlib::mpqs -- Multi-polynomial Quadratic Sieve (Site not responding. Last check: 2007-09-17)
		The number of polynomials the values of which are tested for smoothness.
		to be the bound below which every factor of a given value must be to make that value pass the trial-division part of the sieve step and become a sieve report.
		The multi-polynomial quadratic sieve is an algorithm to factor large integers without small prime factors.
	www.sciface.com /STATIC/DOC25/eng/numlib/mpqs.shtml (305 words)

	Citations: Parallel implementation of the quadratic sieve - Caron, Silverman (ResearchIndex) (Site not responding. Last check: 2007-09-17)
		Caron and R. Silverman, "Parallel implementation of the quadratic sieve", J. Supercomputing 1 (1988), 273-290.
		Caron and R. Silverman, Parallel implementation of the quadratic sieve, J. Supercomputing 1 (1988), 273--290.
		....5.2.1 Factoring Integers by Means of a Quadratic Sieve Among number theoretists, the so called multi polynomial quadratic sieve is considered to be the best method for factoring integers in the range between (roughly) 30 and 120 digits.
	citeseer.lcs.mit.edu /context/134541/0 (2028 words)

	Factoring
		Probably, most quadratic sieve programs are written so that one computer does all the work.
		In contrast to the quadratic sieve, implementations of the Number Field Sieve are (probably) almost always written so that sieving can take place on several computers, with the results being combined later.
		This is done by sieving, which is the time consuming part of factoring N. After enough factorizations have been collected, we perform some "magic" and come up with a set of (a,b)'s such that the product over all values of (a + bM) and over all values of norm(a + b alpha) are squares.
	home.netcom.com /~jrhowell/math/factor.htm (3357 words)

	Modern Cryptography and Distributed Computing (Site not responding. Last check: 2007-09-17)
		Because the numbers used in cryptographic applications are "hard" numbers, a general factoring algorithm is required.
		This paper will address the possibility of using the quadratic sieve as an alternative to the general number field sieve for numbers with between 130 and 160 decimal digits.
		Specifically, it covers the author's implementation of the quadratic sieve, designed for massively distributed use on the Internet.
	brandt.kurowski.net /projects/mpqs (188 words)

	Citations: The multiple polynomial quadratic sieve - Silverman (ResearchIndex) (Site not responding. Last check: 2007-09-17)
		The number field sieve is the newest and best performing member of the family.
		Several impressive factorisations of RSA moduli were performed using MPQS before the number field sieve came into being.
		By way of background to the number field sieve we now explain the factorisation strategy of algorithms in the family, concentrating on MPQS and the number field sieve.
	citeseer.ist.psu.edu /context/270989/0 (483 words)

	Factorization of RSA-129 (Site not responding. Last check: 2007-09-17)
		Quadratic Sieve algorithm, although not the fastest known, suffices for factorization of large numbers with about 120 digits.
		Choose a multiplier m such that mN is a quadratic resudue modulo many small primes.
		Then compute the factor base P consisting of primes for which mN is a quadratic residue.
	www.cs.berkeley.edu /~yozo/cs267.fa01/hw0.html (577 words)

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