We are happy to announce that RSA-129 = 1143816257578888676692357799761466120102182967212423625625618429\ 35706935245733897830597123563958705058989075147599290026879543541 = 3490529510847650949147849619903898133417764638493387843990820577 * 32769132993266709549961988190834461413177642967992942539798288533 The encoded message published was 968696137546220614771409222543558829057599911245743198746951209308162\ 98225145708356931476622883989628013391990551829945157815154 This number came from an RSA encryption of the `secret' message using the public exponent 9007. When decrypted with he `secret' exponent 106698614368578024442868771328920154780709906633937862801226224496631\ 063125911774470873340168597462306553968544513277109053606095 this becomes 200805001301070903002315180419000118050019172105011309190800151919090\ 618010705 Using the decoding scheme 01=A, 02=B, ..., 26=Z, and 00 a space between words, the decoded message reads THE MAGIC WORDS ARE SQUEAMISH OSSIFRAGE To find the factorization of RSA-129, we used the double large prime variation of the multiple polynomial quadratic sieve factoring method. The sieving step took approximately 5000 mips years, and was carried out in 8 months by about 600 volunteers from more than 20 countries, on all continents except Antarctica. Combining the partial relations produced a sparse matrix of 569466 rows and 524338 columns. This matrix was reduced to a dense matrix of 188614 rows and 188160 columns using structured Gaussian elimination. Ordinary Gaussian elimination on this matrix, consisting of 35489610240 bits (4.13 gigabyte), took 45 hours on a 16K MasPar MP-1 massively parallel computer. The first three dependencies all turned out to be `unlucky' and produced the trivial factor RSA-129. The fourth dependency produced the above factorization. We would like to thank everyone who contributed their time and effort to this project. Without your help this would not have been possible. Derek Atkins Michael Graff Arjen Lenstra Paul Leyland
Derek Atkins reports to us:
We are happy to announce that
RSA-129 = 1143816257578888676692357799761466120102182967212423625625618429\ 35706935245733897830597123563958705058989075147599290026879543541 = 3490529510847650949147849619903898133417764638493387843990820577 * 32769132993266709549961988190834461413177642967992942539798288533
Of course. What else could it be? First, to check your result, firing up Mathematica 2.2 gives: Timing[3490529510847650949147849619903898133417764638493387843990820577 32769132993266709549961988190834461413177642967992942539798288533] {0.0666667 Second, 11438162575788886766923577997614661201021\ 829672124236256256184293570693524573389783059712356395870\ 5058989075147599290026879543541} That is, it took MMA only 0.066 second, mostly overhead, to multiply your two factors to the product you gave. But much more interesting is seeing how long MMA's "FactorInteger" function takes to find the factors: Timing[FactorInteger [11438162575788886766923577997614661201021\ 829672124236256256184293570693524573389783059712356395870\ 5058989075147599290026879543541]] {4194 Second, {{3490529510847650949147849619903898133417764638493387843990820577, 1}, {32769132993266709549961988190834461413177642967992942539798288533, 1}}} So, this took slightly longer, 4194 seconds, or a bit over an hour, but MMA had no problem factoring this number. Why such a big deal? MMA was even able to extract the magic words: ExtractMagicWords [%] { NOTE THAT THE TIMING ABOVE HAS A CERTAIN DATE VALUE } You people at the universities sure do know how to waste taxpayer money! --Tim May P.S. My congratulations. No practical use to factor just one such number, given 10^72 particles in the Universe, but the methods used to harness so many machines may be useful in all kinds of problems. -- .......................................................................... Timothy C. May | Crypto Anarchy: encryption, digital money, tcmay@netcom.com | anonymous networks, digital pseudonyms, zero 408-688-5409 | knowledge, reputations, information markets, W.A.S.T.E.: Aptos, CA | black markets, collapse of governments. Higher Power: 2^859433 | Public Key: PGP and MailSafe available. "National borders are just speed bumps on the information superhighway."
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That is, it took MMA only 0.066 second, mostly overhead, to multiply your two factors to the product you gave. [more stuff deleted] So, this took slightly longer, 4194 seconds, or a bit over an hour, but MMA had no problem factoring this number. Why such a big deal?
Cute, Tim! (Uhh, you're about 3 weeks too late for this ;-) Actually, the *first* thing I did when I received these factors was fire up a trusty mathematics package and verify the product: bc. :-) Although I admit that RSA-129 dprobably does not have any cosmic significance with regards to protecting any vital data, it is a data point: it is the largest number of its type to ever have been factored. As a result, it tells us that 425-bit keys are not secure, and keys not much bigger are not secure, either, today! But you are right, we are learning alot about factoring and distributed problems as a result of this exercise (at least I feel that I have learned alot). -derek Derek Atkins, SB '93 MIT EE, G MIT Media Laboratory Member, MIT Student Information Processing Board (SIPB) Home page: http://www.mit.edu:8001/people/warlord/home_page.html warlord@MIT.EDU PP-ASEL N1NWH PGP key available
Derek Atkins wrote:
We are happy to announce that
RSA-129 = 1143816257578888676692357799761466120102182967212423625625618429\ 35706935245733897830597123563958705058989075147599290026879543541 = 3490529510847650949147849619903898133417764638493387843990820577 * 32769132993266709549961988190834461413177642967992942539798288533
To find the factorization of RSA-129, we used the double large prime variation of the multiple polynomial quadratic sieve factoring method. The sieving step took approximately 5000 mips years, and was carried out in 8 months by about 600 volunteers from more than 20 countries, on all continents except Antarctica. Combining the partial relations
Now let's see, where's my slide rule, let's see 5,000 mips years at $30,000 /mips = damn, where is that calculator. :-)
We would like to thank everyone who contributed their time and effort to this project. Without your help this would not have been possible.
Derek Atkins
Nahh, couldn't be,
Michael Graff Arjen Lenstra Paul Leyland
participants (3)
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Derek Atkins -
Istvan Oszaraz von Keszi -
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