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Topic: General number field sieve

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	General number field sieve (Site not responding. Last check: 2007-11-07)
		In mathematics, the general number field sieve is the most efficient algorithm known for factoring integers.
		It is derived from the special number field sieve.
		It is better than the general number field sieve when factors are of small size, as it works by finding smooth values of order of the smallest prime divisor of n, and its running time depends on the size of this divisor.
	www.serebella.com /encyclopedia/article-General_number_field_sieve.html (852 words)

	General number field sieve - Wikipedia, the free encyclopedia
		In mathematics, the general number field sieve is the most efficient algorithm known for factoring integers larger than 100 digits.
		It is a generalization of the special number field sieve: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number (apart from prime powers, but this is a minor issue).
		The principle of the number field sieve (both special and general) can be understood as an extension of the simpler rational sieve.
	en.wikipedia.org /wiki/General_number_field_sieve (767 words)

	Sieve theory - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-07)
		Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers.
		In one of the major strands of number theory in the twentieth century, ways were found of avoiding some of the difficulties of a frontal attack with a naive idea of what sieving should be.
		Sieve theory is somewhat related to sieve algorithms such as the general number field sieve, used for factoring large numbers, although sieve theory and sieve algorithms serve different purposes.
	encyclopedia.worldsearch.com /sieve_theory.htm (676 words)

	Sieve - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-07)
		A sieve is a device for segregating aggregate based on particle size.
		Sieves commonly consist of a wire mesh on which the aggregate is placed.
		A sieve used in cooking is a colander.
	www.encyclopedia-online.info /Sieve (139 words)

	General number field sieve (Site not responding. Last check: 2007-11-07)
		In mathematics, the general number field sieve is themost efficient algorithm known for factoring integers.
		When the term "number fieldsieve" is used without qualification, it refers to the general number field sieve.
		It is better than the general numberfield sieve when factors are of small size, as it works by finding smooth values of order of the smallest prime divisor of n, and its running time depends on the size of this divisor.
	www.therfcc.org /general-number-field-sieve-137872.html (622 words)

	General number field sieve - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-07)
		Now, we consider number field rings Z and Z where r1 and r2 are roots of polynomials f and g, and look for values a and b such that r=b
		We need a slightly stronger condition - that they are norms of squares in our number fields, but we can get that condition by this method too.
		Since m is root of both f and g mod n, there are homomorphisms from the rings Z and Z to the ring Z/nZ, which map r1 and r2 to m, and these homomorphisms will map each "square root" (typically not represented as a rational number) into its integer representative.
	encyclopedia.worldsearch.com /general_number_field_sieve.htm (653 words)

	General number field sieve - Term Explanation on IndexSuche.Com
		It is an improvement of the quadratic_sieve, which factors ''n'' by finding numbers ''ki'' such that ''ri=ki2-n'' factor completely over a fixed set (called ''basis'') of small primes.
		Now, we consider number field rings Z[r1] and Z[r2] where r1 and r2 are roots of polynomials ''f'' and ''g'', and look for values ''a'' and ''b'' such that ''r=bd
		Thus, we get two numbers ''x'' and ''y'', with ''x2-y2'' divisible by ''n'' and again with probability at least one half we get a factor of ''n'' by finding greatest_common_divisor of ''n'' and ''x-y'' The second-best-known algorithm for integer factorization is the Lenstra_elliptic_curve_factorization method.
	www.indexsuche.com /General_number_field_sieve.html (684 words)

	NFSNET
		The number field sieve is particularly efficient for numbers of the form: N=r^e+s where r and s are small.
		For numbers of this special form, the number field sieve is particularly efficient and when applied to them, the technique is called the special number field sieve (SNFS).
		Even a basic description of the number field sieve is beyond the scope of this FAQ however there are several books that provide excellent descriptions CFRAC and the QS and go further to explain at least the basics of the number field sieve.
	www.nfsnet.org /faq-nfs.html (510 words)

	Project LIMA
		"The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic.
		The efficiency of the former depends on the unknown factors, whereas the efficiency of the latter depends on the number to be factored.
		Special-purpose algorithms are best for factoring numbers with small factors, while general-purpose factoring algorithms cope better with larger factors and are more important in the context of cryptographic systems and their security.
	www.srcf.ucam.org /lima/introduction.htm (676 words)

	Zomd :: Science :: Math :: Number Theory :: Factoring (Site not responding. Last check: 2007-11-07)
		It also computes the number and sum of divisors, the Euler's totient and moebius functions, and the decomposition of the number in a sum of up to four squares.
		A listing of all the known pairs of numbers, each of which is the sum of the aliquot divisors of the other.
		Numbers representative of those used in the RSA cryptosystem are offered for factor attempts with prizes.
	www.zomd.org /category_26949.html (415 words)

	sieve (Site not responding. Last check: 2007-11-07)
		See sieve of Eratosthenes, sieve theory and general number field sieve.
		sieve is a mail filtering language proposed as Internet standard that can be found at this address.
		sieve is a popular taunt among crowds at American college ice hockey games, used as a reference to a goaltender.
	en.mcfly.org /sieve (109 words)

	[No title]
		This factorization was found using the Number Field Sieve (NFS) factoring algorithm, and beats the 140-digit record RSA-140 that was set on February 2, 1999, also with the help of NFS [RSA140].
		For the lattice sieve, a factor base bound of 16 777 216 (2^24) was chosen, both for the rational and for the algebraic side.
		For both sieves the large prime bound 1 000 000 000 was used both for the rational and for the algebraic primes.
	www.crypto-world.com /announcements/RSA155.txt (1050 words)

	NFSNET
		The goal of the NFSNET project is to use the Number Field Sieve to find the factors of increasingly large numbers.
		A small number of machines are still sieving for 2^761-1 because initial attempts at post processing showed that the resulting matrix was too large.
		Sieving is proceeding as normal but the stats reports are not reflecting all work done.
	www.nfsnet.org (274 words)

	RSA Security - 2.3.4 What are the best factoring methods in use today?
		Factoring algorithms come in two flavors, special purpose and general purpose; the efficiency of the former depends on the unknown factors, whereas the efficiency of the latter depends on the number to be factored.
		The ECM is often used in practice to find factors of randomly generated numbers; it is not fast enough to factor a large modulus of the kind used in the RSA cryptosystem.
		A "general number" is one with no special form that might make it easier to factor; moduli used in the RSA cryptosystem are created to be general numbers.
	www.rsasecurity.com /rsalabs/node.asp?id=2190 (582 words)

	Factorization of RSA-130 (12 Apr 1996)
		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.
		Sieving started in September 1995, initially on a very limited number of workstations.
	primes.utm.edu /notes/rsa130.html (944 words)

	Citations: Modifications to the number field sieve - Coppersmith (ResearchIndex)
		Coppersmith, Modifications to the Number Field Sieve, J. Cryptology 6, pp.
		....number field sieve can factor integers q of special forms (such as q = 2 n 1) in time of the form (2.8) with c = 1:5262 : The general number field sieve factors arbitrary integers with running time bound of the form (2.
		Coppersmith, Modifications to the number field sieve, J. Cryptology 6 (1993), 169--180.
	citeseer.ist.psu.edu /context/470098/0 (803 words)

	ScienceDaily: General number field sieve (Site not responding. Last check: 2007-11-07)
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	www.sciencedaily.com /encyclopedia/general_number_field_sieve (1471 words)

	Citations: A general number field sieve implementation - Bernstein, Lenstra (ResearchIndex)
		There is a whole family of number field sieves, of which the GNFS is unique in that it makes no special assumptions about the form of the number N to be factored.
		In contrast, the basic number field sieve is designed to work on numbers of the form r e Sigma s, where r and s are both small....
		On Quadratic Polynomials for the Number Field Sieve - Murphy, Brent (1997)
	citeseer.ist.psu.edu /context/21614/0 (1360 words)

	Integer factorization (Site not responding. Last check: 2007-11-07)
		If a fast method were found for solving the integer factorization problem, then several important cryptographic systems would be broken, including the RSA public-key algorithm and the Blum Blum Shub random number generator.
		In addition, there are a number of probabilistic algorithm s that can test primality very quickly if one is willing to accept the small possibility of error.
		This is the type of algorithm used to factor RSA number s.
	www.serebella.com /encyclopedia/article-Integer_factorization.html (966 words)

	Sieve of Eratosthenes (Site not responding. Last check: 2007-11-07)
		In mathematics, the Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to a specified integer.
		It is the predecessor to the modern Sieve of Atkin, which is faster but more complex.
		It is the square root of the last number you care about, rounded down.
	www.tocatch.info /en/Sieve_of_Eratosthenes.htm (358 words)

	Record Number Field Sieve Factorizations
		The Number Field Sieve algorithm has four main phases: polynomial selection, sieving, linear algebra, and square root.
		The sieving took eight weeks during April-June, 1993, using 30 SPARC workstations in the OSU Mathematics Department.
		Sieving took 700 workstation-hours in May, 1994 using the polynomials X^6 - 10 * X^4 + 24 * X^2 - 8 and 2^36 * X - (2^73 + 1).
	krum.rz.uni-mannheim.de /cabench/sieve-record.html (626 words)

	Sieve - Wikipedia, the free encyclopedia
		sieve theory, technique for counting or filtering sets of numbers
		sieve (category theory), a way of writing down how objects in a category glue to give other ones
		Sieve (mail filtering language), a proposed standard for specifying mail filters.
	en.wikipedia.org /wiki/Sieve (137 words)

	Special number field sieve Information (Site not responding. Last check: 2007-11-07)
		The special number field sieve (SNFS) is a special-purpose integer factorization algorithm.
		The general number field sieve (GNFS) was derived from it.
		When the term "number field sieve" is used without qualification, it refers to GNFS.
	www.articleshead.com /show_article/special-number-field-sieve (129 words)

	Factsheet
		In the 1980s it was generally held that prime numbers of a fifty odd digits would suffice.
		Thus, having found such numbers x and y, we also have found factors of N. The numbers are found by a sieving process, in which possible values of x and y are excluded.
		Factoring numbers on this list measures how secure the RSA method actually is. For special numbers, for example of the form ab ±1, factoring has proceeded to well over two hundred digits.
	www.cwi.nl /research/2001/TeRiele_Eng (890 words)

	GGNFS
		GGNFS is a rudimentary implementation of the General Number Field Sieve (GNFS) for factoring integers.
		I have done special numbers between 110 and 150 digits with it, with a special c150 taking about 2 days on an Athlon XP 3200+.
		In short, you probably won't be factoring any 512-bit general numbers with this code any time soon.
	www.math.ttu.edu /~cmonico/software/ggnfs (637 words)

	Sieve (Site not responding. Last check: 2007-11-07)
		A year ago Seattle was a sieve of a defense, particularly against the run.
		Drain in sieve lined with moistened paper towels, reserving soaking liquid.
		It should also make clear what too many still fail to see: Our border appears to be an open sieve.
	www.wikiverse.org /sieve (196 words)

	quadratic factoring algorithm (Site not responding. Last check: 2007-11-07)
		The quadratic sieve is a method of factoring large numbers, specifically large numbers that are a product of two large...
		Quadratic Sieve -- from MathWorld Quadratic Sieve -- from MathWorld A sieving procedure that can be used in conjunction with Dixon's factorization method to factor large numbers n.
		on the quadratic sieve factoring algorithm, the algorithm that was considered to be the most...
	www.finance-best.com /articles/72/quadratic-factoring-algorithm.html (424 words)

	general number field sieve (Site not responding. Last check: 2007-11-07)
		The general number field sieve''' works as follows:
		Each r is a norm of a- r1b and hence we get that product of corresponding factors a- r1b is a square in Z[r1], with a "square root" which can be determined (as a product of known factors in Z[r1]) - it will typically be represented as a nonrational algebraic number.
		Similary we get that product of factors a- r2*b is a square in Z[r2], with a "square root" which we can also compute.
	www.yourencyclopedia.net /general_number_field_sieve.html (698 words)

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