# RSA-576 Factored

By Eric W. Weisstein

December 5, 2003 -- On December 3, the day after the announcement of the discovery of the largest known prime by the Great Internet Mersenne Prime Search on December 2 (MathWorld headline news: December 2, 2003), a team at the German Bundesamt für Sicherheit in der Informationstechnik (Federal Bureau for Security in Information Technology; BIS) announced the factorization of the 174-digit number 1881 9881292060 7963838697 2394616504 3980716356 3379417382 7007633564 2298885971 5234665485 3190606065 0474304531 7388011303 3967161996 9232120573 4031879550 6569962213 0516875930 7650257059 known as RSA-576.

RSA numbers are composite numbers having exactly two prime factors (i.e., so-called semiprimes) that have been listed in the Factoring Challenge of RSA Security(R).

While composite numbers are defined as numbers that can be written as a product of smaller numbers known as factors (for example, 6 = 2 x 3 is composite with factors 2 and 3), prime numbers have no such decomposition (for example, 7 does not have any factors other than 1 and itself). Prime factors therefore represent a fundamental (and unique) decomposition of a given positive integer. RSA numbers are special types of composite numbers particularly chosen to be difficult to factor, and are identified by the number of digits they contain.

While RSA-576 is a much smaller number than the 6,320,430-digit monster Mersenne prime announced earlier this week, its factorization is significant because of the curious property of numbers that proving or disproving a number to be prime ("primality testing") seems to be much easier than actually identifying the factors of a number ("prime factorization"). Thus, while it is trivial to multiply two large numbers p and q together, it can be extremely difficult to determine the factors if only their product pq is given. With some ingenuity, this property can be used to create practical and efficient encryption systems for electronic data.

RSA Laboratories sponsors the RSA Factoring Challenge to encourage research into computational number theory and the practical difficulty of factoring large integers and because it can be helpful for users of the RSA encryption public-key cryptography algorithm for choosing suitable key lengths for an appropriate level of security. A cash prize is awarded to the first person to factor each Challenge number.

RSA numbers were originally spaced at intervals of 10 decimal digits between 100 and 500 digits, and prizes were awarded according to a complicated formula. These original numbers were named according to the number of decimal digits, so RSA-100 was a hundred-digit number. As computers and algorithms became faster, the unfactored challenge numbers were removed from the prize list and replaced with a set of numbers with fixed cash prizes. At this point, the naming convention was also changed so that the trailing number would indicate the number of digits in the binary representation of the number. Hence, RSA-576 has 576 binary digits, which translates to 174 digits in decimal.

RSA numbers received widespread attention when a 129-digit number known as RSA-129 was used by R. Rivest, A. Shamir, and L. Adleman to publish one of the first public-key messages together with a \$100 reward for the message's decryption (Gardner 1977). Despite widespread belief at the time that the message encoded by RSA-129 would take millions of years to break, it was factored in 1994 using a distributed computation which harnessed networked computers spread around the globe performing a multiple polynomial quadratic sieve (Leutwyler 1994). The result of all the concentrated number crunching was decryption of the encoded message to yield the profound plaintext message "The magic words are squeamish ossifrage." (For the benefit of non-ornithologists, an ossifrage is a rare predatory vulture found in the mountains of Europe.)

Factorization of RSA-129 followed earlier factorizations of RSA-100, RSA-110, and RSA-120. The Challenge numbers RSA-130, RSA-140, RSA-155, and RSA-160 were also subsequently factored between 1996 and April of this year. (Amusingly, RSA-150 apparently remains unfactored following its withdrawal from the RSA Challenge list.) On December 2, Jens Franke circulated an email announcing factorization of the smallest prize number RSA-576. The factorization was accomplished using a prime factorization algorithm known as the general number field sieve (GNFS). The two 87-digit factors found using this sieve are

3980750 8642406493 7397125500 5503864911 9906436234 2526708406 3851895759 4638895726 1768583317
x 4727721 4610743530 2536223071 9730482246 3291469530 2097116459 8521711305 2071125636 3590397527

which can easily be multiplied to verify that they do indeed give the original number.

Franke's note detailed the factorization process in which "lattice" sieving was done by J. Franke and T. Kleinjung using hardware at the Scientific Computing Institute and the Pure Mathematics Institute at Bonn University, Max Planck Institute for Mathematics in Bonn, and Experimental Mathematics Institute in Essen; and "line" sieving was done by P. Montgomery and H. te Riele at the CWI, F. Bahr and his family, and NFSNET (which at that time consisted of D. Leclair, Paul Leyland, and R. Wackerbarth). Post-processing of this data to construct the actual factors was then done with the support of the BSI.

For their efforts, the team will receive a cash prize of \$10,000 from RSA Security. However, award seekers need not be deterred. As the following table shows, RSA-640 to RSA-2048 remain open, carrying awards from \$20,000 to \$200,000 to whoever is clever and persistent enough to track them down. A list of the open Challenge numbers may be downloaded from RSA or in the form of a Mathematica package from the MathWorld package archive.

 number digits prize factored RSA-100 100 Apr. 1991 RSA-110 110 Apr. 1992 RSA-120 120 Jun. 1993 RSA-129 129 \$100 Apr. 1994 RSA-130 130 Apr. 10, 1996 RSA-140 140 Feb. 2, 1999 RSA-150 150 withdrawn? open RSA-155 155 Aug. 22, 1999 RSA-160 160 Apr. 1, 2003 RSA-576 174 \$10,000 Dec. 3, 2003 RSA-640 193 \$20,000 open RSA-704 212 \$30,000 open RSA-768 232 \$50,000 open RSA-896 270 \$75,000 open RSA-1024 309 \$100,000 open RSA-1536 463 \$150,000 open RSA-2048 617 \$200,000 open

References:

Franke, J. "RSA576." Privately circulated email reposted to primenumbers Yahoo! Group.

Gardner, M. "Mathematical Games: A New Kind of Cipher that Would Take Millions of Years to Break." Sci. Amer. 237, 120-124, Aug. 1977.

Leutwyler, K. "Superhack: Forty Quadrillion Years Early, a 129-Digit Code is Broken." Sci. Amer. 271, 17-20, 1994.

NFSNet: Large-Scale Distributed Factoring. http://www.nfsnet.org

RSA Security(R). "RSA Factoring Challenge." http://www.rsasecurity.com/rsalabs/challenges/factoring

RSA Security(R). "The RSA Challenge Numbers." http://www.rsasecurity.com/rsalabs/challenges/factoring/numbers.html

Weisstein, E. W. RSANumbers.m Mathematica package.