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# Topic: Integer factorization

 Integer factorization - Encyclopedia, History, Geography and Biography In number theory, the integer factorization problem is the problem of finding a non-trivial divisor of a composite number; for example, given a number like 91, the challenge is to find a number such as 7 which divides it. A fast integer factorization algorithm would mean that the RSA public-key algorithm was insecure. Integer factorization, Prime decomposition, Practical applications, Current state of the art, Difficulty and complexity, Factoring algorithms, Special-purpose, General-purpose, Other notable algorithms, External links, References and Integer factorization algorithms. www.arikah.net /encyclopedia/Integer_factorization   (1060 words)

 Integer factorization : Information and resources about Integer factorization : School Work Guru   (Site not responding. Last check: 2007-10-07) 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. It has been proved to be exactly as hard as integer factorization. There is no way to break it without also solving integer factorization quickly. www.schoolworkguru.org /encyclopedia/i/in/integer_factorization.html   (778 words)

 Encyclopedia: Integer factorization   (Site not responding. Last check: 2007-10-07) In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. Wikinews has news related to this article: Two hundred digit number factored In mathematics, RSA-200 is one of the RSA numbers, large semiprimes that are part of the RSA Factoring Challenge. It has been proved to be exactly as hard as integer factorization: if you can break the generator in polynomial time then you can factorize integers in polynomial time. www.nationmaster.com /encyclopedia/Integer_factorization   (593 words)

 Factorization After that, if a composite factor of at least 25 digits remains, MPQS is used; it is the best method available for factoring integers of more than about 40 decimal digits especially for products of two primes of roughly equal size. The integer n not=0 is subjected to trial division by primes up to a certain bound B (the sign of n is ignored). Given a set or sequence S of integers, return a coprime basis of S in the form of a factorization sequence Q whose integer value is the same as the product of the elements of S but Q has coprime bases (i.e., the first components of tuples from Q are coprime). magma.maths.usyd.edu.au /magma/htmlhelp/text529.htm   (3030 words)

 Entries: 5.2.4 Integer Factorization and Prime Numbers Thus any factor less than 4012009 is known to be a prime, for greater factors a primality test is used before calling the actual Pollard Rho. Since the factorization can potentially take a very long time, an execution time test is used to abort factoring very long integers (limit is 60s for each composite). Modular * + ^ mod for Brent-Pollard factorization. staff.science.uva.nl /~dominik/hpcalc/entries/hp49g/entries_178.html   (341 words)

 Integer factorization - Wikipedia, the free encyclopedia Given the state of the art as of 2006, the hardest instances of these problems are those where the factors are two randomly-chosen prime numbers of about the same size. If a large, b-bit number is the product of two primes that are roughly the same size, then no algorithm is known that can factor in polynomial time. This is because both YES and NO answers can be checked if given the prime factors along with their primality certificates. en.wikipedia.org /wiki/Integer_factorization   (1033 words)

 Integral Domains, Gaussian Integer, Unique Factorization This is the set of complex numbers with integer coefficients. belongs to Z, or is a root of a quadratic monic polynomial with integer coefficients. In 1847, G.Lamé (1795-1870) gave a talk at a meeting of the French Academy where he announced a solution to Fermat's Theorem and suggested that the glory of solving the famous theorem should be shared with J.Liouville (1809-1882) to whom he ascribed the method of solution. www.cut-the-knot.org /arithmetic/int_domain4.shtml   (981 words)

 World War 1 and 2 - Talk:Integer factorization   (Site not responding. Last check: 2007-10-07) Discrete logarithm for integers modulo a prime appears to be related somehow it factorisation, but a factorisation breakthrough is not guaranteed to solve discrete logarithm. I would interpret this as meaning that a deeper understanding of integer factorization would lead to a deeper understanding of complexity theory (or quantum computers). It's rather difficult to say...this article was named "integer factorization" because there are other types of factorization (you can factorize polynomials, for example). www.worldwardiary.com /history/Talk:Integer_factorization   (528 words)

 Factoring Papers Factoring Integers with Large Prime Variations of the Quadratic Sieve by H. Boender and H.J.J. te Riele (compressed postscript, 109K) from CWI (http://www.cwi.nl/ftp/CWIreports/NW/NM-R9513.ps.Z). Factoring estimates for a 1024-bit RSA modulus by Arjen K. Lenstra, Eran Tromer, Adi Shamir, Wil Kortsmit, Bruce Dodson, James Hughes, Paul Leyland. Factorization of the Eighth Fermat Number by Richard Brent and John Pollard, from Richard Brent's homepage (http://web.comlab.ox.ac.uk/oucl/work/richard.brent/pub/pub061.html). www.crypto-world.com /FactorPapers.html   (629 words)

 Thirty Years of Integer Factorization Even if the security of RSA is not equivalent to integer factorization, factoring the RSA key is the simplest way to decode everything, so a lot of people tried to factor. Factoring is of great interest since it allows to use the properties of prime number in arithmetic. The linear algebra is often the limiting factor, and unless there is a new idea on the subject, RSA can still be used for some times if used with a key big enough. algo.inria.fr /seminars/sem00-01/morain.html   (1633 words)

 11Y05: Factorization and primality testing Naturally this is related to number theory, and when we replace the ring of integers with more general rings, (in particular rings of polynomials) we are in the realm of commutative rings. An older summary of factorization techniques (with citations to the literature). Factor n, n-1, n+1 where n is the order of the Monster finite simple group. www.math.niu.edu /~rusin/known-math/index/11Y05.html   (611 words)

 Cryptosystems Based on Integer Factorization 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. There have also been developed over the years much improved factoring algorithms [13], but despite this progress, factoring a general composite number with as few as say 200 digits is still out of reach of the fastest computers using the best algorithms known today. It is not our purpose here to delve into the theory of primality testing and integer factorization (for which we refer the reader to [13, 20] for recent developments). www.math.clemson.edu /faculty/Gao/crypto_mod/node3.html   (1781 words)

 Integer factorization Article, Integerfactorization Information   (Site not responding. Last check: 2007-10-07) In mathematics, the integer prime-factorization (alsoknown as prime decomposition) problem is this: given a positive integer, write it as a product of prime numbers. If a fast method were found for solving the integer factorization problem, then severalimportant cryptographic systems would be broken, including the RSA public-key algorithm and the Blum Blum Shub pseudo-random number generator. The decision-problem form of it ("does N have a factor less thanM?") is known to be in both NP and co-NP. www.anoca.org /algorithm/prime/integer_factorization.html   (797 words)

 [No title]   (Site not responding. Last check: 2007-10-07) This project is concerned with the implementation and testing of integer factorization algorithms on the AP1000 and clarification of the important question of how hard it is to factor numbers of a given size, or numbers with prime factors of a given size. The implementation of scalable parallel algorithms for integer factorisation and related problems such as discrete logarithm. Prime factors of up to 43 decimal digits have been found on the AP1000, and up to 49 decimal digits on other parallel machines using a slight modification of the program. cap.anu.edu.au /cap/reports/report97/integer.html   (494 words)

 Demetrius at The Australian National University: Item 1885/40810 Factor is a program which accesses a large database of factors of integers of the form an_1. The program factor implements a simple version of the Elliptic Curve algorithm if it is unable to complete a factorization using trial division and the factor database. Factor is written in Turbo Pascal and runs on IBM PC or compatible computers. hdl.handle.net /1885/40810   (161 words)

 Amazon.com: Primality Testing and Integer Factorization in Public-Key Cryptography (Advances in Information Security): ...   (Site not responding. Last check: 2007-10-07) Primality testing and integer factorization, as identified by Gauss in his "Disquisitiones Arithmeticae", Article 329, in 1801, are the two most fundamental problems (as well as the two most important research fields) in computational number theory. The final chapter presents the applications of the problems/techniques of primality testing, integer factorization, square roots, discrete logarithms and quadratic residuosity in public-key cryptography. Text introduces various algorithms for primality testing and integer factorization, with their applications in public-key cryptography and information security. www.code4u.com /buy/1402076495   (685 words)

 digital certificate mumbai Suppose an integer n is to be factored. QRP is as difficult as the problem of factoring integers, although no proof of this is known. The prime factorization of n is 250 = 2 ยท 53. services.eliteral.com /digital-certificate-mumbai/chap3.php   (15117 words)

 The circuits for integer factorization from [1] 1] that the cost of factorization is ``the product of the time and the cost of the machine.'' We refer to this cost function as throughput cost, since it can be interpreted as measuring the equipment cost per unit problem-solving throughput. In circuit-NFS (i.e., the mesh) a parallelization factor How does the comparison between circuit-NFS and standard-NFS with respect to their throughput costs turn out if standard-NFS is first properly tuned (Remark 3.4) to the throughput cost function, given the state of the art in, say, 1990 (cf. www.wisdom.weizmann.ac.il /~tromer/papers/meshc/node3.html   (1323 words)

 Citations: An implementation of the elliptic curve integer factorization method - Bosma, Lenstra (ResearchIndex)   (Site not responding. Last check: 2007-10-07) Bosma and A. Lenstra, An implementation of the elliptic curve integer factorization method, Computational Algebra and Number Theory (edited by W. Bosma and A. van der Poorten), Kluwer Academic Publishers, Dordrecht, 1995, 119--136. This k is gradually increased for each new attempt, until a factor is found or until the factoring attempt is aborted. Bosma, A.K. Lenstra, An implementation of the elliptic curve integer factorization method, chapter 9 in Computational algebra and number theory (W. Bosma, A. van der Poorten, eds.), Kluwer Academic Press (1995). citeseer.lcs.mit.edu /context/170492/0   (982 words)

 Citations: The future of integer factorization - Odlyzko (ResearchIndex)   (Site not responding. Last check: 2007-10-07) The Factorization of RSA-140 - Laboratories Division Of ....time required to factor small(k) number of integers of size k2 should be moderate, whereas the time required to factor large(k) number of integers of size k2 should be infeasible. 5 Since the result of this is just to transfer the factorization of M i to T i one might ask why we do it this way, as opposed to just sending the factors of M i to T i. citeseer.lcs.mit.edu /context/244734/0   (3496 words)

 [No title]   (Site not responding. Last check: 2007-10-07) General-purpose Algorithms: the largest integer factored with a general-purpose algorithm is RSA200 (200 digits), which was factored on May 9, 2005 by Bahr, Boehm, Franke and Kleinjung. The previous record was RSA-576 (174 digits), which was factored on December 3rd, 2003 into two 87-digit factors using GNFS by Franke, Kleinjung, Montgomery, te Riele, Bahr, Leclair, Leyland, Wackerbarth. A few days later, on December 19th, a 164-digit number was factored by Aoki, Kida, Shimoyama, Sonoda and Ueda. www.loria.fr /~zimmerma/records/factor.html   (216 words)

 integer factorization - OneLook Dictionary Search   (Site not responding. Last check: 2007-10-07) Tip: Click on the first link on a line below to go directly to a page where "integer factorization" is defined. Integer Factorization : Eric Weisstein's World of Mathematics [home, info] Phrases that include integer factorization: integer factorization algorithms, integer factorization problem www.onelook.com /?w=integer+factorization   (92 words)

 C Board - Integer Factorization I want the function to take an integer, return a pointer to an array of ints on the heap. The const is fine, and no, it couldn't have been avoided with a linked list, although a linked list could have a place in an application like this if you needed the factors for something other than printing to the screen. The other trick to note, is that if n has no factors < sqrt(n) then ether sqrt(n) is a factor (n is the square of a prime), or n is a prime. cboard.cprogramming.com /archive/index.php/t-42323.html   (387 words)

 WIFC (World Integer Factorization Center) These are factorization results of various kind of numbers. After finished to factor under 100 digits, several members reported further results over 100 digits, so I extended the search range up to 150 or certain digits, and added the new series of numbers, Wolstenholme numbers 3, 4 (numerator of sigma(1/n^i)), fourier coefficients of j(tau) (elliptic modular function), partition numbers. Factors for 2^n+1 and 2^n-1 for large n (Arjen Bot) www.asahi-net.or.jp /~KC2H-MSM/mathland/matha1   (1762 words)

 RSA Security - RSA-155 is factored! On August 22, 1999, a group of researchers completed the factorization of the 155 digit (512 bit) RSA Challenge Number. This is within the rough range of estimates based on the factorization of RSA-140, though the CPU time was somewhat less than predicted due perhaps to statistical variations, as well as to the improved polynomial selection. The total calendar time for factoring RSA-155 was 5.2 months (March 17 - August 22) excluding polynomial selection time. www.rsasecurity.com /rsalabs/node.asp?id=2098   (403 words)

 Integer Factorization   (Site not responding. Last check: 2007-10-07) Factoring Estimates for a 1024-bit RSA-Modulus, AK Lenstra, E Tromer, A Shamir, W Kortsmit, B Dodson, J Hughes and PC Leyland, Asiacrypt 2003, LNCS 2894 (2003) This one is the second-largest SNFS factorization yet attempted anywhere. Paul Zimmermann is running ECMNET which is another distributed factoring project. www.leyland.vispa.com /numth/factorization/main.htm   (325 words)

 ifp   (Site not responding. Last check: 2007-10-07) The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated. Deals with the potential for distributing implementations of various factoring schemes. Description of the algorithm, implementation notes, and some factorizations obtained thus far. www.upl.cs.wisc.edu /~hamblin/ifp.html   (190 words)

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