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Topic: Special number field sieve

Related Topics

General number field sieve

Integer factorization

Quadratic sieve

Lenstra elliptic curve factorization

Homomorphism

Autistic savant

Idiot Savants (game show)

Integer factorization

	General number field sieve - Wikipedia, the free encyclopedia
		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.
		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.
	en.wikipedia.org /wiki/General_number_field_sieve (745 words)

	Special number field sieve - Wikipedia, the free encyclopedia
		The special number field sieve (SNFS) is a special-purpose integer factorization algorithm.
		The general number field sieve (GNFS) was derived from it.
		First, find a large number of multiplicative relations among a factor base of elements of Z/nZ, such that the number of multiplicative relations is larger than the number of elements in the factor base.
	www.wikipedia.org /wiki/NFSNET (780 words)

	Encyclopedia: General number field sieve (Site not responding. Last check: 2007-10-20)
		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.
		In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form anxn + an−1xn−1 + ··· + a1x + a0 = 0 where n is a positive integer called the degree...
	www.nationmaster.com /encyclopedia/General-number-field-sieve (939 words)

	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.
		The quadratic sieve is a modification of Dixon's factorization method.
		Until the discovery of the number field sieve (NFS), QS was the asymptotically-fastest known general-purpose factoring algorithm.
	en.wikipedia.org /wiki/Quadratic_sieve (2116 words)

	NFSNET (Site not responding. Last check: 2007-10-20)
		The number field sieve (NFS) is the asymptotically fastest known algorithm for factoring large composite integers which have no known special form.
		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).
	www.nfsnet.org /faq-nfs.html (510 words)

	General Number Field Sieve - Monico (Site not responding. Last check: 2007-10-20)
		The general number field sieve is currently the best known asymptotic algorithm for factoring integers.
		Indeed, the Special Number Field Sieve is precisely the case where we want to factor a special number like 2^{523}-1 for which we can easily cook up a polynomial with the right degree and very small coefficients.
		The choice of polynomial has such a strong impact on the rest of the runtime of the algorithm that we can factor such special numbers with far less work than their randomly chosen counterparts of the same size.
	www.nd.edu /~cam/ima/GNFS_monico.htm (578 words)

	RSA Labs FAQ - 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.
		A "general number" is one with no special form that might make it easier to factor; RSA moduli are created to be general numbers.
		Numbers with up to 155 digits or more that have a special form are easier to factor than general numbers [LLM93].
	crypto.nknu.edu.tw /infosec/faq/html/2-3-4.html (556 words)

	Elementary Number Theory - Course Information
		This establishes a new record for the Special Number Field Sieve.
		The sieving was done on about 150 SGI workstations and Sun workstations and servers running at 180-450 MHz, and on about 100 PCs running at 266-600 MHz.
		The previous SNFS record was the 211-digit repunit number 10,211- = (10^211 - 1)/9, factored on April 8, 1999, also by the Cabal.
	www.math.usf.edu /~eclark/numtheory_links.html (308 words)

	Factoring Record starts Attack on RSA Code (Site not responding. Last check: 2007-10-20)
		This number, compactly written as 3263341 - 1, was factored with the ‘Special Number Field Sieve’, a method particularly suited to numbers of this form.
		At NEC Research one is interested in fast and reliable algorithms generating random number sets which are applied, eg, in physics (experiment simulation) and the financial world (assessing the value of investment portfolios).
		Testing the quality of certain random number generators requires the knowledge of the prime factors of numbers having the form of the number now factored at CWI.
	www.ercim.org /publication/Ercim_News/enw35/nieland.html (277 words)

	Binary GCD algorithm -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-20)
		We know the division by 2 is possible because the difference of two odd numbers is even.
		Following is a straightforward implementation of the algorithm in the (Click link for more info and facts about C programming language) C programming language, taking two positive arguments u and v and using only bit operations and subtraction.
		Efficiency can be enhanced on platforms with special instructions to find the least significant 1 bit in a word, allowing trailing zero bits to be removed in much larger groups.
	www.absoluteastronomy.com /encyclopedia/B/Bi/Binary_GCD_algorithm2.htm (881 words)

	General number field sieve (Site not responding. Last check: 2007-10-20)
		It is an improvement of the quadratic sieve, which factors n by finding numbers k
		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=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.
	www.sciencedaily.com /encyclopedia/general_number_field_sieve (709 words)

	Mersenne: M751 factored
		The sieving was primarily done at CWI, at Lehigh University, the Institute for Pure Math, Bonn University and the Institute for Applied Math, Bonn University; with additional contributions from Arjen Lenstra, Citicorp and Paul Leyland, Microsoft Research.
		Lattice sieving with algebraic special-q was done by Dodson, Lenstra and Leyland, with factor base bounds 16.7M.
		Franke's initial relations were found with a small sieving region and factor bases; but most were found with a sieving region of 16385x8192 (similar to the region used on the algebraic side), and factor base bounds 32M.
	www.mail-archive.com /mersenne@base.com/msg06931.html (788 words)

	Advanced Factorization Techniques: The Number Field Sieve (Site not responding. Last check: 2007-10-20)
		In this stage of the algorithm, "cycles"[LD95] are detected in the partial relations from the sieving stage, and quadratic characters are calculated for the relations.
		Returns the number of cycles in the partial relations of the NFS process P. This function is mainly intended for factoring with multiple processors.
		In this stage, number field square roots are extracted and we attempt to factor the dependencies found in the linear algebra stage.
	magma.maths.usyd.edu.au /magma/htmlhelp/text533.htm (3083 words)

	Record Number Field Sieve Factorizations
		The sieving took eight weeks during April-June, 1993, using 30 SPARC workstations in the OSU Mathematics Department.
		The sieving took four weeks during March-April, 1993, using 40 workstations in the Forest Science, Mathematics, and Statistics Departments at OSU.
		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)

	[No title]
		The previous record was the 181-digit number 12^167 + 1, factored by the CWI factoring group in September 1997.
		The Special Number Field Sieve requires the input number to be of the form b^n +- 1 for small b (or other nice polynomial form), so the known factors do not make the number any easier for SNFS.
		Of course, an alternative was to apply the quadratic sieve method or GNFS to that 144-digit cofactor but then the sieving time would be much larger than 5.1 CPU years (we estimate at least 100 CPU years).
	www.math.temple.edu /~wds/homepage/teriele.factor (837 words)

	RSA Laboratories Bulletin #13: A Cost-Based Security Analysis of Symmetric and Asymmetric Key Lengths (Site not responding. Last check: 2007-10-20)
		The amount of time it takes to factor a number of x bits is asymptotically the same as the time it takes to solve a discrete log over a field of size x bits.
		The sieving phase of the number field sieve depends for its speed upon the ability to rapidly retrieve values from memory, add to them, and put them back.
		The number of points on the randomly chosen elliptic curve taken modulo p is a random integer near p.
	www.nullify.org /docs/bulletin13/bulletin13.html (9701 words)

	APPLIED CRYPTOGRAPHY, SECOND EDITION: Protocols, Algorithms, and Source Code in C:Key Length
		Table 7.4 gives the number of mips-years required to factor numbers of different sizes, given current implementations of the general number field sieve [1190].
		A related algorithm, the special number field sieve, can already factor numbers of a certain specialized form—numbers not generally used for cryptography—much faster than the general number field sieve can factor general numbers of the same size.
		It is not unreasonable to assume that the general number field sieve can be optimized to run this fast [1190]; it is possible that the NSA already knows how to do this.
	friedo.szm.sk /krypto/AC/ch07/07-05.html (836 words)

	[No title]
		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.
		It is not unreasonable to assume that the general number field sieve can be optimized to run this fast; it is possible that the NSA already knows how to do this.
		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)

	Computational number theory
		The six largest numbers factored so far with NFS have 174 (factored on December 3, 2003), 160 (factored on April 1, 2003), 158 (factored on January 19, 2002), 155, 140, and 130 decimal digits.
		The previous three SNFS records (established on November 14, 2000, on April 8, 1999 and in September 1998) are numbers of 233, 211, and 186 decimal digits, respectively.
		The number and the size of known amicable number pairs has grown explosively in recent years: from 1108 amicable pairs in 1972 (the largest pair consisting of two 25-digit numbers), to more than five million amicable pairs in May 2003 (the largest pair consisting of two 5577-digit numbers).
	db.cwi.nl /projecten/project.php4?prjnr=84 (1003 words)

	Advanced Factorization Techniques: The Number Field Sieve
		For the sieve, the rounded off natural logarithms of primes are used to mark the sieve interval.
		During the sieve stage, relations that fully factor into primes within the factor bases ("full relations") will be output to the data file specified by the user in the tuple.
		Performs the factorization of n using the number field sieve with algebraic polynomial F, where m1 and m2 are such that F(m1, m2) = 0 mod n.
	www.umich.edu /~gpcc/scs/magma/text544.htm (2191 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		The previous SNFS-record was the 186-digit number 32633^41 - 1 factored in September 1998 by a CWI group (ftp: ftp.cwi.nl/pub/herman/SNFSrecords/SNFS-186).
		The sieving was done on about 125 SGI and Sun workstations running at 175 MHz on average, and on about 60 PCs running at 300 MHz on average.
		For the lattice sieve the special q's were chosen from selected intervals in the range 2^24 -- 10^8.
	www.loria.fr /~zimmerma/records/10,211- (435 words)

	[No title]
		Polynomial selection When using the Special Number Field Sieve, the algebraic form of the number will dictate which polynomials to use.
		This is approximately six factors in [10^4, 10^6] per non-free relation (3 per norm, compared to the log(6/4) ~ 0.4 for a ``typical'' number.) Another filter run with -filtmin 300000 -maxrels 5.5 -merge3 -maxdiscard 45000 had 46560 heavy discards by pass 3.
		For the RSA Challenge numbers, a checksum is included with the number.
	ftp.cwi.nl /factory/programs.doc (2708 words)

	Computing A Square Root For The Number Field Sieve - Couveignes (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		The number field sieve is a method proposed by Lenstra, Lenstra, Manasse and Pollard for integer factorization (this volume,pp.
		3: The Lattice Sieve (context) - Pollard - 1993
		35 Factoring integers with the number field sieve (context) - Buhler, Lenstra et al.
	citeseer.ist.psu.edu /256874.html (495 words)

	A Survey on the Number Field Sieve (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		The Number Field Sieve is expected to be the fastest among the recent major factoring methods.
		In this note, a survey on the Number Field Sieve, or NFS, will be given as one of the most important of these methods.
		7 Modications to the number eld sieve (context) - Coppersmith - 1993
	citeseer.ist.psu.edu /312752.html (546 words)

	Factsheet
		A number of 512 bits (155 digits) was factored at CWI on August 22, 1999.
		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)

	Factoring Cunningham Numbers
		This is a project that factors a 169-digit number, using the Special Number Field Sieve (SNFS).
		This smallish number most likely will not be distributed, but the input file is available here r647.
		An account of the earlier factorizations of 5^283+1 and the Mersenne number M727 posted by Peter Montgomery (to the nmbr_theory list and the Mersenne list) is available here.
	www.lehigh.edu /~bad0/cu-p647.html (715 words)

	[No title]
		The other two factored numbers (of the five) are M727, with smallest prime factor having 98-digits; and M809, with smallest prime factor having 61-digits.
		For each of these numbers the matrix step was run by CWI under a grant on the SARA computing center's SGI Origin 3800 using Montgomery's parallel Lanczos method (the matrix for M727 having 3.8 million rows/columns; for M809 there were 8.9 million rows/columns).
		No number has actually been the target of such a search, much less a large sieving candidate for which the runtime required would represent an efficient balance of effort between ecm pretest and sieving.
	www.lehigh.edu /~bad0/p57post.txt (1176 words)

	CWI factors Giant Number in Record-Time (Site not responding. Last check: 2007-10-20)
		The 180-digit number is the largest factored so far with the method employed here the Special Number Field Sieve (SNFS).
		The previous record (see: http://www.loria.fr/~zimmerma/records/), the factoring of a 167-digit number, was established last February by an international group of researchers who joined efforts through Internet.
		With a related method, the General Number Field Sieve, the reliability of widely used cryptographic codes is tested.
	www.ercim.org /publication/Ercim_News/enw31/nieland.html (245 words)

	Encyclopedia: Special number field sieve (Site not responding. Last check: 2007-10-20)
		People who viewed "Special number field sieve" also viewed:
		When the term "number field sieve" is used without qualification, it refers to GNFS.
		As such, it is ideal for factoring Fermat numbers.
	www.nationmaster.com /encyclopedia/Special-number-field-sieve (151 words)

	New Prime Factorisation Record obtained using the General Number Field Sieve
		Using a new implementation of the general number field sieve (GNFS), we have factored a 158-digit divisor of 2
		The previous record was a 155-digit RSA challenge number factored by a team of mathematicians led by CWI in 1999.
		This is followed by the sieving step, which is also massively parallel and takes most of the CPU time.
	www.ercim.org /publication/Ercim_News/enw49/franke.html (457 words)

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