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Integer factorization is the process of determining which prime numbers divide a given positive integer. Doing this quickly has applications in cryptography. The difficulty depends on both the size and form of the number and its prime factors; it is currently very difficult to factorize large semiprimes (and, indeed, most numbers which have no small factors). == Numbers of a general form == The first very large distributed factorisation was RSA129, a challenge number described in the Scientific American article of 1977 which first popularised the RSA cryptosystem. It was factorised between September 1993 and April 1994, using MPQS, with relations contributed by about 600 people from all over the Internet, and the final stages of the calculation performed on a MasPar supercomputer at Bell Labs. Between January and August 1999, RSA-155, a challenge number prepared by the RSA company, was factorised using GNFS with relations again contributed by a large group, and the final stages of the calculation performed in just over nine days on the Cray C916 supercomputer at the SARA Amsterdam Academic Computer Center. In January 2002, Franke et al. announced the factorisation of a 158-digit cofactor of 2953+1, using a couple of months on about 25 PCs at the University of Bonn, with the final stages done using a cluster of six Pentium-III PCs. In April 2003, the same team factored RSA-160 using about a hundred CPUs at BSI, with the final stages of the calculation done using 25 processors of an SGI Origin supercomputer. The 174-digit RSA-576 was factored by Franke, Kleinjung and members of the NFSNET collaboration in December 2003, using resources at BSI and the University of Bonn; soon afterwards, Aoki, Kida, Shimoyama, Sonoda and Ueda announced that they had factored a 164-digit cofactor of 21826+1. A 176-digit cofactor of 11281+1 was factored by Aoki, Kida, Shimoyama and Ueda between February and May 2005 using machines at NTT and Rikkyo University in Japan.〔(【引用サイトリンク】accessdate=2007-05-23 )〕 The RSA-200 challenge number was factored by Franke, Kleinjung et al. between December 2003 and May 2005, using a cluster of 80 Opteron processors at BSI in Germany; the announcement was made on 9 May 2005.〔(【引用サイトリンク】accessdate=2007-05-23 )〕 They later (November 2005) factored the slightly smaller RSA-640 challenge number. On December 12, 2009, a team including researchers from the CWI, the EPFL, INRIA and NTT in addition to the authors of the previous record factored RSA-768, a 232-digit semiprime.〔(【引用サイトリンク】accessdate=2013-04-11 )〕 They used the equivalent of almost 2000 years of computing on a single core 2.2 GHz AMD Opteron. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Integer factorization records」の詳細全文を読む スポンサード リンク
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