# RSA Factoring Challenge

The RSA Factoring Challenge was a challenge put forward by RSA Laboratories on March 18, 1991[1] to encourage research into computational number theory and the practical difficulty of factoring large integers and cracking RSA keys used in cryptography. They published a list of semiprimes (numbers with exactly two prime factors) known as the RSA numbers, with a cash prize for the successful factorization of some of them. The smallest of them, a 100-decimal digit number called RSA-100 was factored by April 1, 1991. Many of the bigger numbers have still not been factored and are expected to remain unfactored for quite some time, however advances in quantum computers make this prediction uncertain due to Shor's algorithm.

In 2001, RSA Laboratories expanded the factoring challenge and offered prizes ranging from \$10,000 to \$200,000 for factoring numbers from 576 bits up to 2048 bits.[2][3][4]

The RSA Factoring Challenges ended in 2007.[5] RSA Laboratories stated: "Now that the industry has a considerably more advanced understanding of the cryptanalytic strength of common symmetric-key and public-key algorithms, these challenges are no longer active."[6] When the challenge ended in 2007, only RSA-576 and RSA-640 had been factored from the 2001 challenge numbers.[7]

The factoring challenge was intended to track the cutting edge in integer factorization. A primary application is for choosing the key length of the RSA public-key encryption scheme. Progress in this challenge should give an insight into which key sizes are still safe and for how long. As RSA Laboratories is a provider of RSA-based products, the challenge was used by them as an incentive for the academic community to attack the core of their solutions — in order to prove its strength.

The RSA numbers were generated on a computer with no network connection of any kind. The computer's hard drive was subsequently destroyed so that no record would exist, anywhere, of the solution to the factoring challenge.[6]

The first RSA numbers generated, RSA-100 to RSA-500 and RSA-617, were labeled according to their number of decimal digits; the other RSA numbers (beginning with RSA-576) were generated later and labelled according to their number of binary digits. The numbers in the table below are listed in increasing order despite this shift from decimal to binary.

## The mathematics

RSA Laboratories states that: for each RSA number n, there exists prime numbers p and q such that

n = p × q.

The problem is to find these two primes, given only n.

## The prizes and records

The following table gives an overview over all RSA numbers. Note that the RSA Factoring Challenge ended in 2007[5] and no further prizes will be awarded for factoring the higher numbers.

The challenge numbers in white lines are numbers expressed in base 10, while the challenge numbers in yellow lines are numbers expressed in base 2
RSA number Decimal digits Binary digits Cash prize offered Factored on Factored by
RSA-100 100 330 US\$1,000[8] April 1, 1991[9] Arjen K. Lenstra
RSA-110 110 364 US\$4,429[8] April 14, 1992[9] Arjen K. Lenstra and M.S. Manasse
RSA-120 120 397 US\$5,898[8] July 9, 1993[10] T. Denny et al.
RSA-129 [a] 129 426 US\$100 April 26, 1994[9] Arjen K. Lenstra et al.
RSA-130 130 430 US\$14,527[8] April 10, 1996 Arjen K. Lenstra et al.
RSA-140 140 463 US\$17,226 February 2, 1999 Herman te Riele et al.
RSA-150 150 496   April 16, 2004 Kazumaro Aoki et al.
RSA-155 155 512 US\$9,383[8] August 22, 1999 Herman te Riele et al.
RSA-160 160 530   April 1, 2003 Jens Franke et al., University of Bonn
RSA-170 [b] 170 563   December 29, 2009 D. Bonenberger and M. Krone [c]
RSA-576 174 576 US\$10,000 December 3, 2003 Jens Franke et al., University of Bonn
RSA-180 [b] 180 596   May 8, 2010 S. A. Danilov and I. A. Popovyan, Moscow State University[11]
RSA-190 [b] 190 629   November 8, 2010 A. Timofeev and I. A. Popovyan
RSA-640 193 640 US\$20,000 November 2, 2005 Jens Franke et al., University of Bonn
RSA-200 [b] ? 200 663   May 9, 2005 Jens Franke et al., University of Bonn
RSA-210 [b] 210 696 September 26, 2013[12] Ryan Propper
RSA-704 [b] 212 704 US\$30,000 July 2, 2012 Shi Bai, Emmanuel Thomé and Paul Zimmermann
RSA-220 [b] 220 729   May 13, 2016 S. Bai, P. Gaudry, A. Kruppa, E. Thomé and P. Zimmermann
RSA-230 [b] 230 762   August 15, 2018 Samuel S. Gross, Noblis, Inc.
RSA-232 [b] 232 768   February 17, 2020[13] N. L. Zamarashkin, D. A. Zheltkov and S. A. Matveev.
RSA-768 [b] 232 768 US\$50,000 December 12, 2009 Thorsten Kleinjung et al.[14]
RSA-240 [b] 240 795   Dec 2, 2019[15] F. Boudot, P. Gaudry, A. Guillevic, N. Heninger, E. Thomé and P. Zimmermann
RSA-250 [b] 250 829   Feb 28, 2020[16] F. Boudot, P. Gaudry, A. Guillevic, N. Heninger, E. Thomé and P. Zimmermann
RSA-260 260 862
RSA-270 270 895
RSA-896 270 896 US\$75,000[d]
RSA-280 280 928
RSA-290 290 962
RSA-300 300 995
RSA-309 309 1024
RSA-1024 309 1024 US\$100,000[d]
RSA-310 310 1028
RSA-320 320 1061
RSA-330 330 1094
RSA-340 340 1128
RSA-350 350 1161
RSA-360 360 1194
RSA-370 370 1227
RSA-380 380 1261
RSA-390 390 1294
RSA-400 400 1327
RSA-410 410 1360
RSA-420 420 1393
RSA-430 430 1427
RSA-440 440 1460
RSA-450 450 1493
RSA-460 460 1526
RSA-1536 463 1536 US\$150,000[d]
RSA-470 470 1559
RSA-480 480 1593
RSA-490 490 1626
RSA-500 500 1659
RSA-617 617 2048
RSA-2048 617 2048 US\$200,000[d]
1. ^ RSA-129 was not part of the RSA Factoring Challenge, but was related to a column by Martin Gardner in Scientific American.
2. The number was factored after the challenge ended.
3. ^ RSA-170 was also independently factored by S. A. Danilov and I. A. Popovyan two days later.[11]
4. ^ a b c d The challenge ended before this prize was awarded.

## Notes

1. ^ Kaliski, Burt (18 Mar 1991). "Announcement of "RSA Factoring Challenge"". Retrieved 8 March 2021.
2. ^ Leyden, John (25 Jul 2001). "RSA poses \$200,000 crypto challenge". The Register. Retrieved 8 March 2021.
3. ^ RSA Laboratories. "The New RSA Factoring Challenge". Archived from the original on 2001-07-14.
4. ^ RSA Laboratories. "The RSA Challenge Numbers". Archived from the original on 2001-08-05.
5. ^ a b RSA Laboratories. "RSA Factoring Challenge". Archived from the original on 2013-09-21. Retrieved 2008-08-05.
6. ^ a b RSA Laboratories. "The RSA Factoring Challenge FAQ". Archived from the original on 2013-09-21. Retrieved 2008-08-05.
7. ^ RSA Laboratories. "The RSA Challenge Numbers". Archived from the original on 2013-09-21. Retrieved 2008-08-05.
8. ^ a b c RSA Honor Roll
9. ^ Denny, T.; Dodson, B.; Lenstra, A. K.; Manasse, M. S. (1994). On the factorization of RSA-120. Advances in Cryptology — CRYPTO' 93. pp. 166–174. doi:10.1007/3-540-48329-2_15.
10. ^ a b Danilov, S. A.; Popovyan, I. A. (9 May 2010). "Factorization of RSA-180" (PDF). Cryptology ePrint Archive.
11. ^ RSA-210 factored, mersenneforum.org
12. ^ INM RAS news
13. ^ Kleinjung, Thomas (18 Feb 2010). "Factorization of a 768-bit RSA modulus" (PDF). Cite journal requires `|journal=` (help)
14. ^ Thomé, Emmanuel (December 2, 2019). "795-bit factoring and discrete logarithms". cado-nfs-discuss (Mailing list).
15. ^ Zimmermann, Paul (February 28, 2020). "Factorization of RSA-250". cado-nfs-discuss (Mailing list).