In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the RSA numbers are a set of large
semiprimes
In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers.
Because there are infinitely many prime ...
(numbers with exactly two
prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s) that were part of the
RSA Factoring Challenge
The RSA Factoring Challenge was a challenge put forward by RSA Laboratories on March 18, 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers and cracking RSA keys used in cryptograp ...
. The challenge was to find the prime factors of each number. It was created by
RSA Laboratories
RSA Security LLC, formerly RSA Security, Inc. and doing business as RSA, is an American computer and network security company with a focus on encryption and encryption standards. RSA was named after the initials of its co-founders, Ron Rivest, ...
in March 1991 to encourage research into
computational number theory
In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of
computational methods for investigating and solving problems in number theory and arithmetic geometry, including algorithm ...
and the practical difficulty of
factoring large
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s. The challenge was ended in 2007.
RSA Laboratories
RSA Security LLC, formerly RSA Security, Inc. and doing business as RSA, is an American computer and network security company with a focus on encryption and encryption standards. RSA was named after the initials of its co-founders, Ron Rivest, ...
(which is an
acronym
An acronym is a word or name formed from the initial components of a longer name or phrase. Acronyms are usually formed from the initial letters of words, as in ''NATO'' (''North Atlantic Treaty Organization''), but sometimes use syllables, as ...
of the creators of the technique; Rivest, Shamir and Adleman) published a number of semiprimes with 100 to 617
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
digits. Cash prizes of varying size, up to
US$
The United States dollar (symbol: $; code: USD; also abbreviated US$ or U.S. Dollar, to distinguish it from other dollar-denominated currencies; referred to as the dollar, U.S. dollar, American dollar, or colloquially buck) is the official ...
200,000 (and prizes up to $20,000 awarded), were offered for factorization of some of them. The smallest RSA number was factored in a few days. Most of the numbers have still not been factored and many of them are expected to remain unfactored for many years to come. , the smallest 23 of the 54 listed numbers have been factored.
While the RSA challenge officially ended in 2007, people are still attempting to find the factorizations. According to RSA Laboratories, "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."
Some of the smaller prizes had been awarded at the time. The remaining prizes were retracted.
The first RSA numbers generated, from RSA-100 to RSA-500, were labeled according to their number of decimal digits. Later, beginning with RSA-576,
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...
digits are counted instead. An exception to this is RSA-617, which was created before the change in the numbering scheme. The numbers are listed in increasing order below.
RSA-100
RSA-100 has 100 decimal digits (330 bits). Its factorization was announced on April 1, 1991, by
Arjen K. Lenstra. Reportedly, the factorization took a few days using
the multiple-polynomial quadratic sieve algorithm on a
MasPar
MasPar Computer Corporation was a minisupercomputer vendor that was founded in 1987 by Jeff Kalb. The company was based in Sunnyvale, California.
History
While Kalb was the vice-president of the division of Digital Equipment Corporation (DEC) t ...
parallel computer.
The value and factorization of RSA-100 are as follows:
RSA-100 = 1522605027922533360535618378132637429718068114961380688657908494580122963258952897654000350692006139
RSA-100 = 37975227936943673922808872755445627854565536638199
× 40094690950920881030683735292761468389214899724061
It takes four hours to repeat this factorization using the progra
Msieveon a 2200 MHz
Athlon 64
The Athlon 64 is a ninth-generation, AMD64-architecture microprocessor produced by Advanced Micro Devices (AMD), released on September 23, 2003. It is the third processor to bear the name ''Athlon'', and the immediate successor to the Athlon XP. T ...
processor.
The number can be factorized in 72 minutes on overclocked to 3.5 GHz Intel Core2 Quad q9300, usin
GGNFSan
Msievebinaries running b
distributed version of the factmsieve Perl script
RSA-110
RSA-110 has 110 decimal digits (364 bits), and was factored in April 1992 by
Arjen K. Lenstra and Mark S. Manasse in approximately one month.
The number can be factorized in less than four hours on overclocked to 3.5 GHz Intel Core2 Quad q9300, usin
GGNFSan
Msievebinaries running b
distributed version of the factmsieve Perl script
The value and factorization are as follows:
RSA-110 = 35794234179725868774991807832568455403003778024228226193532908190484670252364677411513516111204504060317568667
RSA-110 = 6122421090493547576937037317561418841225758554253106999
× 5846418214406154678836553182979162384198610505601062333
RSA-120
RSA-120 has 120 decimal digits (397 bits), and was factored in June 1993 by Thomas Denny, Bruce Dodson, Arjen K. Lenstra, and Mark S. Manasse. The computation took under three months of actual computer time.
The value and factorization are as follows:
RSA-120 = 227010481295437363334259960947493668895875336466084780038173258247009162675779735389791151574049166747880487470296548479
RSA-120 = 327414555693498015751146303749141488063642403240171463406883
× 693342667110830181197325401899700641361965863127336680673013
RSA-129
RSA-129, having 129 decimal digits (426 bits), was not part of the 1991 RSA Factoring Challenge, but rather related to
Martin Gardner
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis ...
's
Mathematical Games column
Over a period of 24 years (January 1957 – December 1980), Martin Gardner wrote 288 consecutive monthly "Mathematical Games" columns for ''Scientific American'' magazine. During the next years, through June 1986, Gardner wrote 9 more columns, ...
in the August 1977 issue of ''
Scientific American
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...
''.
RSA-129 was factored in April 1994 by a team led by
Derek Atkins
Derek A Atkins is a computer scientist specializing in computer security. He studied electrical engineering and computer science at the Massachusetts Institute of Technology. In June 2014, he became the chief technology officer of SecureRF.
At ...
,
Michael Graff
Michael Graff is a co-author and one of several architects of BIND 9.
In April 1994, he co-authored ''The Magic Words are Squeamish Ossifrage''.
Graff graduated from Iowa State University
Iowa State University of Science and Technology ...
,
Arjen K. Lenstra and
Paul Leyland
Paul Leyland is a British number theorist who has studied integer factorization and primality testing.
He has contributed to the factorization of RSA-129, RSA-140, and RSA-155, as well as potential factorial primes as large as 400! + 1. He has ...
, using approximately 1600 computers from around 600 volunteers connected over the
Internet
The Internet (or internet) is the global system of interconnected computer networks that uses the Internet protocol suite (TCP/IP) to communicate between networks and devices. It is a '' network of networks'' that consists of private, pub ...
. A
US$
The United States dollar (symbol: $; code: USD; also abbreviated US$ or U.S. Dollar, to distinguish it from other dollar-denominated currencies; referred to as the dollar, U.S. dollar, American dollar, or colloquially buck) is the official ...
100 token prize was awarded by RSA Security for the factorization, which was donated to the
Free Software Foundation
The Free Software Foundation (FSF) is a 501(c)#501(c)(3), 501(c)(3) non-profit organization founded by Richard Stallman on October 4, 1985, to support the free software movement, with the organization's preference for software being distributed ...
.
The value and factorization are as follows:
RSA-129 = 114381625757888867669235779976146612010218296721242362562561842935706935245733897830597123563958705058989075147599290026879543541
RSA-129 = 3490529510847650949147849619903898133417764638493387843990820577
× 32769132993266709549961988190834461413177642967992942539798288533
The factorization was found using the
Multiple Polynomial Quadratic Sieve algorithm.
The factoring challenge included a message encrypted with RSA-129. When decrypted using the factorization the message was revealed to be "
The Magic Words are Squeamish Ossifrage "The Magic Words are Squeamish Ossifrage" was the solution to a challenge ciphertext posed by the inventors of the RSA cipher in 1977. The problem appeared in Martin Gardner's ''Mathematical Games column'' in the August 1977 issue of ''Scientific A ...
".
RSA-130
RSA-130 has 130 decimal digits (430 bits), and was factored on April 10, 1996, by a team led by
Arjen K. Lenstra and composed of
Jim Cowie,
Marije Elkenbracht-Huizing Marije is a given name. Notable people with this name include:
* Marije van Hunenstijn (born 1995), Dutch sprinter
* Marije Joling (born 1987), Dutch female allround speed skater
* Marije Tolman
Marije Tolman (born 1976) is a Dutch illustrator ...
,
Wojtek Furmanski,
Peter L. Montgomery,
Damian Weber Damian ( la, links=no, Damianus) may refer to:
*Damian (given name)
*Damian (surname)
*Damian Subdistrict, in Longquanyi District, Chengdu, Sichuan, China
See also
*Damiani, an Italian surname
*Damiano (disambiguation)
*Damien (disambiguation)
*Dam ...
and
Joerg Zayer.
The value and factorization are as follows:
RSA-130 = 1807082088687404805951656164405905566278102516769401349170127021450056662540244048387341127590812303371781887966563182013214880557
RSA-130 = 39685999459597454290161126162883786067576449112810064832555157243
× 45534498646735972188403686897274408864356301263205069600999044599
The factorization was found using the
Number Field Sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form
:\exp\left( ...
algorithm and the
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
5748302248738405200 x
5 + 9882261917482286102 x
4
- 13392499389128176685 x
3 + 16875252458877684989 x
2
+ 3759900174855208738 x
1 - 46769930553931905995
which has a root of 12574411168418005980468 modulo RSA-130.
RSA-140
RSA-140 has 140 decimal digits (463 bits), and was factored on February 2, 1999, by a team led by
Herman te Riele
Hermanus Johannes Joseph te Riele (born 5 January 1947) is a Dutch mathematician at CWI in Amsterdam with a specialization in computational number theory. He is known for proving the correctness of the Riemann hypothesis for the first 1.5 billio ...
and composed of
Stefania Cavallar, Bruce Dodson,
Arjen K. Lenstra, Paul Leyland,
Walter Lioen, Peter L. Montgomery,
Brian Murphy Brian Murphy may refer to:
Sportspeople
* Brian Murphy (Jamaican cricketer) (born 1973), Jamaican cricketer
* Brian Murphy (Zimbabwean cricketer) (born 1976), Zimbabwean cricketer
* Brian Murphy (baseball) (born 1980), American head baseball coach ...
and
Paul Zimmermann.
The value and factorization are as follows:
RSA-140 = 21290246318258757547497882016271517497806703963277216278233383215381949984056495911366573853021918316783107387995317230889569230873441936471
RSA-140 = 3398717423028438554530123627613875835633986495969597423490929302771479
× 6264200187401285096151654948264442219302037178623509019111660653946049
The factorization was found using the
Number Field Sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form
:\exp\left( ...
algorithm and an estimated 2000
MIPS-years of computing time.
RSA-150
RSA-150 has 150 decimal digits (496 bits), and was withdrawn from the challenge by RSA Security. RSA-150 was eventually factored into two 75-digit primes by Aoki et al. in 2004 using the
general number field sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form
:\exp\left( ...
(GNFS), years after bigger RSA numbers that were still part of the challenge had been solved.
The value and factorization are as follows:
RSA-150 = 155089812478348440509606754370011861770654545830995430655466945774312632703463465954363335027577729025391453996787414027003501631772186840890795964683
RSA-150 = 348009867102283695483970451047593424831012817350385456889559637548278410717
× 445647744903640741533241125787086176005442536297766153493419724532460296199
RSA-155
RSA-155 has 155 decimal digits (512 bits), and was factored on August 22, 1999, in a span of six months, by a team led by Herman te Riele and composed of Stefania Cavallar, Bruce Dodson,
Arjen K. Lenstra, Walter Lioen, Peter L. Montgomery, Brian Murphy,
Karen Aardal,
Jeff Gilchrist
Jeff is a masculine name, often a short form (hypocorism) of the English given name Jefferson or Jeffrey, which comes from a medieval variant of Geoffrey.
Music
* DJ Jazzy Jeff, American DJ/turntablist record producer Jeffrey Allen Townes
* ...
,
Gerard Guillerm, Paul Leyland,
Joel Marchand Joel or Yoel is a name meaning "Yahweh Is God" and may refer to:
* Joel (given name), origin of the name including a list of people with the first name.
* Joel (surname), a surname
* Joel (footballer, born 1904), Joel de Oliveira Monteiro, Brazili ...
,
François Morain
François () is a French masculine given name and surname, equivalent to the English name Francis.
People with the given name
* Francis I of France, King of France (), known as "the Father and Restorer of Letters"
* Francis II of France, King ...
,
Alec Muffett
Alec David Edward Muffett (born April 22, 1968) is an Anglo-American internet security evangelist, architect, and software engineer. His work includes Crack, the original Unix password cracker, and for the CrackLib password-integrity testing li ...
, Craig Putnam,
Chris Putnam and Paul Zimmermann.
The value and factorization are as follows:
RSA-155 = 10941738641570527421809707322040357612003732945449205990913842131476349984288934784717997257891267332497625752899781833797076537244027146743531593354333897
RSA-155 = 1026395928297411057720541965739916759007165678080380668033419335217907113077
79
× 1066034883801684548209272203600128786792079585759892915222706082371930628086
43
The factorization was found using the
general number field sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form
:\exp\left( ...
algorithm and an estimated 8000
MIPS-years of computing time.
RSA-160
RSA-160 has 160 decimal digits (530 bits), and was factored on April 1, 2003, by a team from the
University of Bonn
The Rhenish Friedrich Wilhelm University of Bonn (german: Rheinische Friedrich-Wilhelms-Universität Bonn) is a public research university located in Bonn, North Rhine-Westphalia, Germany. It was founded in its present form as the ( en, Rhine U ...
and the
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ger ...
Federal Office for Information Security
The Federal Office for Information Security (german: Bundesamt für Sicherheit in der Informationstechnik, abbreviated as BSI) is the German upper-level federal agency in charge of managing computer and communication security for the German go ...
(BSI). The team contained
J. Franke, F. Bahr,
T. Kleinjung, M. Lochter, and M. Böhm.
The value and factorization are as follows:
RSA-160 = 2152741102718889701896015201312825429257773588845675980170497676778133145218859135673011059773491059602497907111585214302079314665202840140619946994927570407753
RSA-160 = 4542789285848139407168619064973883165613714577846979325095998470925000415733
5359
× 4738809060383201619663383230378895197326892292104095794474135464881202849390
9367
The factorization was found using the
general number field sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form
:\exp\left( ...
algorithm.
RSA-170
RSA-170 has 170 decimal digits (563 bits) and was first factored on December 29, 2009, by D. Bonenberger and M. Krone from
Fachhochschule Braunschweig/Wolfenbüttel. An independent factorization was completed by S. A. Danilov and I. A. Popovyan two days later.
The value and factorization are as follows:
RSA-170 = 26062623684139844921529879266674432197085925380486406416164785191859999628542069361450283931914514618683512198164805919882053057222974116478065095809832377336510711545759
RSA-170 = 3586420730428501486799804587268520423291459681059978161140231860633948450858
040593963
× 7267029064107019078863797763923946264136137803856996670313708936002281582249
587494493
The factorization was found using the
general number field sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form
:\exp\left( ...
algorithm.
RSA-576
RSA-576 has 174 decimal digits (576 bits), and was factored on December 3, 2003, by J. Franke and T. Kleinjung from the University of Bonn. A cash prize of $10,000 was offered by RSA Security for a successful factorization.
The value and factorization are as follows:
RSA-576 = 188198812920607963838697239461650439807163563379417382700763356422988859715234665485319060606504743045317388011303396716199692321205734031879550656996221305168759307650257059
RSA-576 = 3980750864240649373971255005503864911990643623425267084063851895759463889572
61768583317
× 4727721461074353025362230719730482246329146953020971164598521711305207112563
63590397527
The factorization was found using the
general number field sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form
:\exp\left( ...
algorithm.
RSA-180
RSA-180 has 180 decimal digits (596 bits), and was factored on May 8, 2010, by S. A. Danilov and I. A. Popovyan from
Moscow State University
M. V. Lomonosov Moscow State University (MSU; russian: Московский государственный университет имени М. В. Ломоносова) is a public research university in Moscow, Russia and the most prestigious ...
, Russia.
RSA-180 = 1911479277189866096892294666314546498129862462766673548641885036388072607034
3679905877620136513516127813425829612810920004670291298456875280033022177775
2773957404540495707851421041
RSA-180 = 4007800823297508779525813391041005725268293178158071765648821789984975727719
50624613470377
× 4769396887386118369955354773570708579399020760277882320319897758246062255957
73435668861833
The factorization was found using the
general number field sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form
:\exp\left( ...
algorithm implementation running on three Intel Core i7 PCs.
RSA-190
RSA-190 has 190 decimal digits (629 bits), and was factored on November 8, 2010, by I. A. Popovyan from Moscow State University, Russia, and A. Timofeev from
CWI, Netherlands.
RSA-190 = 1907556405060696491061450432646028861081179759533184460647975622318915025587
1841757540549761551215932934922604641526300932385092466032074171247261215808
58185985938946945490481721756401423481
RSA-190 = 3171195257690152709485171289740475929805147316029450327784761927832793642798
1256542415724309619
× 6015260020444561641587641685526676183243543359471811072599763828083615704046
0481625355619404899
RSA-640
RSA-640 has 193 decimal digits (640 bits). A cash prize of US$20,000 was offered by RSA Security for a successful factorization. On November 2, 2005, F. Bahr, M. Boehm, J. Franke and T. Kleinjung of the German Federal Office for Information Security announced that they had factorized the number using GNFS as follows:
RSA-640 = 3107418240490043721350750035888567930037346022842727545720161948823206440518
0815045563468296717232867824379162728380334154710731085019195485290073377248
22783525742386454014691736602477652346609
RSA-640 = 1634733645809253848443133883865090859841783670033092312181110852389333100104
508151212118167511579
× 1900871281664822113126851573935413975471896789968515493666638539088027103802
104498957191261465571
The computation took five months on 80 2.2 GHz
AMD
Advanced Micro Devices, Inc. (AMD) is an American multinational semiconductor company based in Santa Clara, California, that develops computer processors and related technologies for business and consumer markets. While it initially manufactur ...
Opteron
Opteron is AMD's x86 former server and workstation processor line, and was the first processor which supported the AMD64 instruction set architecture (known generically as x86-64 or AMD64). It was released on April 22, 2003, with the ''SledgeHa ...
CPUs
A central processing unit (CPU), also called a central processor, main processor or just processor, is the electronic circuitry that executes instructions comprising a computer program. The CPU performs basic arithmetic, logic, controlling, and ...
.
The slightly larger RSA-200 was factored in May 2005 by the same team.
RSA-200
RSA-200 has 200 decimal digits (663 bits), and factors into the two 100-digit primes given below.
On May 9, 2005, F. Bahr, M. Boehm, J. Franke, and T. Kleinjung announced
Thorsten Kleinjung
Thorsten (Thorstein, Torstein, Torsten) is a Scandinavian given name. The Old Norse name was ''Þórsteinn''. It is a compound of the theonym ''Þór'' (''Thor'') and ''steinn'' "stone", which became ''Thor'' and ''sten'' in Old Danish and Old Swe ...
(2005-05-09)
We have factored RSA200 by GNFS
. Retrieved on 2008-03-10. that they had factorized the number using GNFS as follows:
RSA-200 = 2799783391122132787082946763872260162107044678695542853756000992932612840010
7609345671052955360856061822351910951365788637105954482006576775098580557613
579098734950144178863178946295187237869221823983
RSA-200 = 3532461934402770121272604978198464368671197400197625023649303468776121253679
423200058547956528088349
× 7925869954478333033347085841480059687737975857364219960734330341455767872818
152135381409304740185467
The CPU time spent on finding these factors by a collection of parallel computers amounted – very approximately – to the equivalent of 75 years work for a single 2.2
GHz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that one he ...
Opteron
Opteron is AMD's x86 former server and workstation processor line, and was the first processor which supported the AMD64 instruction set architecture (known generically as x86-64 or AMD64). It was released on April 22, 2003, with the ''SledgeHa ...
-based computer.
[ Note that while this approximation serves to suggest the scale of the effort, it leaves out many complicating factors; the announcement states it more precisely.
]
RSA-210
RSA-210 has 210 decimal digits (696 bits) and was factored in September 2013 by Ryan Propper:
RSA-210 = 2452466449002782119765176635730880184670267876783327597434144517150616008300
3858721695220839933207154910362682719167986407977672324300560059203563124656
1218465817904100131859299619933817012149335034875870551067
RSA-210 = 4359585683259407917999519653872144063854709102652201963187054821445240853452
75999740244625255428455944579
× 5625457617268841037562770073044474817438769440075105451049468510945483965774
79473472146228550799322939273
RSA-704
RSA-704 has 212 decimal digits (704 bits), and was factored by Shi Bai, Emmanuel Thomé and Paul Zimmermann. The factorization was announced July 2, 2012. A cash prize of US$30,000 was previously offered for a successful factorization.
RSA-704 = 7403756347956171282804679609742957314259318888923128908493623263897276503402
8266276891996419625117843995894330502127585370118968098286733173273108930900
552505116877063299072396380786710086096962537934650563796359
RSA-704 = 9091213529597818878440658302600437485892608310328358720428512168960411528640
933367824950788367956756806141
× 8143859259110045265727809126284429335877899002167627883200914172429324360133
004116702003240828777970252499
RSA-220
RSA-220 has 220 decimal digits (729 bits), and was factored by S. Bai, P. Gaudry, A. Kruppa, E. Thomé and P. Zimmermann. The factorization was announced on May 13, 2016.
RSA-220 = 2260138526203405784941654048610197513508038915719776718321197768109445641817
9666766085931213065825772506315628866769704480700018111497118630021124879281
99487482066070131066586646083327982803560379205391980139946496955261
RSA-220 = 6863656412267566274382371499288437800130842239979164844621244993321541061441
4642667938213644208420192054999687
× 3292907439486349812049301549212935291916455196536233952462686051169290349309
4652463337824866390738191765712603
RSA-230
RSA-230 has 230 decimal digits (762 bits), and was factored by Samuel S. Gross on August 15, 2018.
RSA-230 = 1796949159794106673291612844957324615636756180801260007088891883553172646034
1490933493372247868650755230855864199929221814436684722874052065257937495694
3483892631711525225256544109808191706117425097024407180103648316382885188526
89
RSA-230 = 4528450358010492026612439739120166758911246047493700040073956759261590397250
033699357694507193523000343088601688589
× 3968132623150957588532394439049887341769533966621957829426966084093049516953
598120833228447171744337427374763106901
RSA-232
RSA-232 has 232 decimal digits (768 bits), and was factored on February 17, 2020, by N. L. Zamarashkin, D. A. Zheltkov and S. A. Matveev.[INM RAS news](_blank)
/ref>
RSA-232 = 1009881397871923546909564894309468582818233821955573955141120516205831021338
5285453743661097571543636649133800849170651699217015247332943892702802343809
6090980497644054071120196541074755382494867277137407501157718230539834060616
2079
RSA-232 = 2966909333208360660361779924242630634742946262521852394401857157419437019472
3262390744910112571804274494074452751891
× 3403816175197563438006609498491521420547121760734723172735163413276050706174
8526506443144325148088881115083863017669
RSA-768
RSA-768 has 232 decimal digits (768 bits), and was factored on December 12, 2009, over the span of two years, by Thorsten Kleinjung, Kazumaro Aoki, Jens Franke, Arjen K. Lenstra, Emmanuel Thomé, Pierrick Gaudry, Alexander Kruppa, Peter Montgomery, Joppe W. Bos, Dag Arne Osvik, Herman te Riele, Andrey Timofeev, and Paul Zimmermann.[Cryptology ePrint Archive: Report 2010/006](_blank)
/ref>
RSA-768 = 1230186684530117755130494958384962720772853569595334792197322452151726400507
2636575187452021997864693899564749427740638459251925573263034537315482685079
1702612214291346167042921431160222124047927473779408066535141959745985690214
3413
RSA-768 = 3347807169895689878604416984821269081770479498371376856891243138898288379387
8002287614711652531743087737814467999489
× 3674604366679959042824463379962795263227915816434308764267603228381573966651
1279233373417143396810270092798736308917
The CPU time spent on finding these factors by a collection of parallel computers amounted approximately to the equivalent of almost 2000 years of computing on a single-core 2.2 GHz AMD Opteron-based computer.
RSA-240
RSA-240 has 240 decimal digits (795 bits), and was factored in November 2019 by Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann.
RSA-240 = 1246203667817187840658350446081065904348203746516788057548187888832896668011
8821085503603957027250874750986476843845862105486553797025393057189121768431
8286362846948405301614416430468066875699415246993185704183030512549594371372
159029236099
RSA-240 = 5094359522858399145550510235808437141326483820241114731866602965218212064697
46700620316443478873837606252372049619334517
× 2446242088383181505678131390240028966538020925789314014520412213365584770951
78155258218897735030590669041302045908071447
The CPU time spent on finding these factors amounted to approximately 900 core-years on a 2.1 GHz Intel Xeon Gold 6130 CPU. Compared to the factorization of RSA-768, the authors estimate that better algorithms sped their calculations by a factor of 3–4 and faster computers sped their calculation by a factor of 1.25–1.67.
RSA-250
RSA-250 has 250 decimal digits (829 bits), and was factored in February 2020 by Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé, and Paul Zimmermann. The announcement of the factorization occurred on February 28.
RSA-250 = 2140324650240744961264423072839333563008614715144755017797754920881418023447
1401366433455190958046796109928518724709145876873962619215573630474547705208
0511905649310668769159001975940569345745223058932597669747168173806936489469
9871578494975937497937
RSA-250 = 6413528947707158027879019017057738908482501474294344720811685963202453234463
0238623598752668347708737661925585694639798853367
× 3337202759497815655622601060535511422794076034476755466678452098702384172921
0037080257448673296881877565718986258036932062711
The factorisation of RSA-250 utilised approximately 2700 CPU core-years, using a 2.1 GHz Intel Xeon Gold 6130 CPU as a reference. The computation was performed with the Number Field Sieve algorithm, using the open sourc
CADO-NFS software
The team dedicated the computation to Peter Montgomery, an American mathematician known for his contributions to computational number theory
In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of
computational methods for investigating and solving problems in number theory and arithmetic geometry, including algorithm ...
and cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
who died on February 18, 2020.
RSA-260
RSA-260 has 260 decimal digits (862 bits), and has not been factored so far.
RSA-260 = 2211282552952966643528108525502623092761208950247001539441374831912882294140
2001986512729726569746599085900330031400051170742204560859276357953757185954
2988389587092292384910067030341246205457845664136645406842143612930176940208
46391065875914794251435144458199
RSA-270
RSA-270 has 270 decimal digits (895 bits), and has not been factored so far.
RSA-270 = 2331085303444075445276376569106805241456198124803054490429486119684959182451
3578286788836931857711641821391926857265831491306067262691135402760979316634
1626693946596196427744273886601876896313468704059066746903123910748277606548
649151920812699309766587514735456594993207
RSA-896
RSA-896 has 270 decimal digits (896 bits), and has not been factored so far. A cash prize of $75,000 was previously offered for a successful factorization.
RSA-896 = 4120234369866595438555313653325759481798116998443279828454556264338764455652
4842619809887042316184187926142024718886949256093177637503342113098239748515
0944909106910269861031862704114880866970564902903653658867433731720813104105
190864254793282601391257624033946373269391
RSA-280
RSA-280 has 280 decimal digits (928 bits), and has not been factored so far.
RSA-280 = 1790707753365795418841729699379193276395981524363782327873718589639655966058
5783742549640396449103593468573113599487089842785784500698716853446786525536
5503525160280656363736307175332772875499505341538927978510751699922197178159
7724733184279534477239566789173532366357270583106789
RSA-290
RSA-290 has 290 decimal digits (962 bits), and has not been factored so far.
RSA-290 = 3050235186294003157769199519894966400298217959748768348671526618673316087694
3419156362946151249328917515864630224371171221716993844781534383325603218163
2549201100649908073932858897185243836002511996505765970769029474322210394327
60575157628357292075495937664206199565578681309135044121854119
RSA-300
RSA-300 has 300 decimal digits (995 bits), and has not been factored so far.
RSA-300 = 2769315567803442139028689061647233092237608363983953254005036722809375824714
9473946190060218756255124317186573105075074546238828817121274630072161346956
4396741836389979086904304472476001839015983033451909174663464663867829125664
459895575157178816900228792711267471958357574416714366499722090015674047
RSA-309
RSA-309 has 309 decimal digits (1,024 bits), and has not been factored so far.
RSA-309 = 1332943998825757583801437794588036586217112243226684602854588261917276276670
5425540467426933349195015527349334314071822840746357352800368666521274057591
1870128339157499072351179666739658503429931021985160714113146720277365006623
6927218079163559142755190653347914002967258537889160429597714204365647842739
10949
RSA-1024
RSA-1024 has 309 decimal digits (1,024 bits), and has not been factored so far. $100,000 was previously offered for factorization.
RSA-1024 = 135066410865995223349603216278805969938881475605667027524485143851526510604
859533833940287150571909441798207282164471551373680419703964191743046496589
274256239341020864383202110372958725762358509643110564073501508187510676594
629205563685529475213500852879416377328533906109750544334999811150056977236
890927563
RSA-310
RSA-310 has 310 decimal digits (1,028 bits), and has not been factored so far.
RSA-310 = 1848210397825850670380148517702559371400899745254512521925707445580334710601
4125276757082979328578439013881047668984294331264191394626965245834649837246
5163148188847336415136873623631778358751846501708714541673402642461569061162
0116380982484120857688483676576094865930188367141388795454378671343386258291
687641
RSA-320
RSA-320 has 320 decimal digits (1,061 bits), and has not been factored so far.
RSA-320 = 2136810696410071796012087414500377295863767938372793352315068620363196552357
8837094085435000951700943373838321997220564166302488321590128061531285010636
8571638978998117122840139210685346167726847173232244364004850978371121744321
8270343654835754061017503137136489303437996367224915212044704472299799616089
2591129924218437
RSA-330
RSA-330 has 330 decimal digits (1,094 bits), and has not been factored so far.
RSA-330 = 1218708633106058693138173980143325249157710686226055220408666600017481383238
1352456802425903555880722805261111079089882303717632638856140900933377863089
0634828167900405006112727432172179976427017137792606951424995281839383708354
6364684839261149319768449396541020909665209789862312609604983709923779304217
01862444655244698696759267
RSA-340
RSA-340 has 340 decimal digits (1,128 bits), and has not been factored so far.
RSA-340 = 2690987062294695111996484658008361875931308730357496490239672429933215694995
2758588771223263308836649715112756731997946779608413232406934433532048898585
9176676580752231563884394807622076177586625973975236127522811136600110415063
0004691128152106812042872285697735145105026966830649540003659922618399694276
990464815739966698956947129133275233
RSA-350
RSA-350 has 350 decimal digits (1,161 bits), and has not been factored so far.
RSA-350 = 2650719995173539473449812097373681101529786464211583162467454548229344585504
3495841191504413349124560193160478146528433707807716865391982823061751419151
6068496555750496764686447379170711424873128631468168019548127029171231892127
2886825928263239383444398948209649800021987837742009498347263667908976501360
3382322972552204068806061829535529820731640151
RSA-360
RSA-360 has 360 decimal digits (1,194 bits), and has not been factored so far.
RSA-360 = 2186820202343172631466406372285792654649158564828384065217121866374227745448
7764963889680817334211643637752157994969516984539482486678141304751672197524
0052350576247238785129338002757406892629970748212734663781952170745916609168
9358372359962787832802257421757011302526265184263565623426823456522539874717
61591019113926725623095606566457918240614767013806590649
RSA-370
RSA-370 has 370 decimal digits (1,227 bits), and has not been factored so far.
RSA-370 = 1888287707234383972842703127997127272470910519387718062380985523004987076701
7212819937261952549039800018961122586712624661442288502745681454363170484690
7379449525034797494321694352146271320296579623726631094822493455672541491544
2700993152879235272779266578292207161032746297546080025793864030543617862620
878802244305286292772467355603044265985905970622730682658082529621
RSA-380
RSA-380 has 380 decimal digits (1,261 bits), and has not been factored so far.
RSA-380 = 3013500443120211600356586024101276992492167997795839203528363236610578565791
8270750937407901898070219843622821090980641477056850056514799336625349678549
2187941807116344787358312651772858878058620717489800725333606564197363165358
2237779263423501952646847579678711825720733732734169866406145425286581665755
6977260763553328252421574633011335112031733393397168350585519524478541747311
RSA-390
RSA-390 has 390 decimal digits (1,294 bits), and has not been factored so far.
RSA-390 = 2680401941182388454501037079346656065366941749082852678729822424397709178250
4623002472848967604282562331676313645413672467684996118812899734451228212989
1630084759485063423604911639099585186833094019957687550377834977803400653628
6955344904367437281870253414058414063152368812498486005056223028285341898040
0795447435865033046248751475297412398697088084321037176392288312785544402209
1083492089
RSA-400
RSA-400 has 400 decimal digits (1,327 bits), and has not been factored so far.
RSA-400 = 2014096878945207511726700485783442547915321782072704356103039129009966793396
1419850865094551022604032086955587930913903404388675137661234189428453016032
6191193056768564862615321256630010268346471747836597131398943140685464051631
7519403149294308737302321684840956395183222117468443578509847947119995373645
3607109795994713287610750434646825511120586422993705980787028106033008907158
74500584758146849481
RSA-410
RSA-410 has 410 decimal digits (1,360 bits), and has not been factored so far.
RSA-410 = 1965360147993876141423945274178745707926269294439880746827971120992517421770
1079138139324539033381077755540830342989643633394137538983355218902490897764
4412968474332754608531823550599154905901691559098706892516477785203855688127
0635069372091564594333528156501293924133186705141485137856845741766150159437
6063244163040088180887087028771717321932252992567756075264441680858665410918
431223215368025334985424358839
RSA-420
RSA-420 has 420 decimal digits (1,393 bits), and has not been factored so far.
RSA-420 = 2091366302476510731652556423163330737009653626605245054798522959941292730258
1898373570076188752609749648953525484925466394800509169219344906273145413634
2427186266197097846022969248579454916155633686388106962365337549155747268356
4666583846809964354191550136023170105917441056517493690125545320242581503730
3405952887826925813912683942756431114820292313193705352716165790132673270514
3817744164107601735413785886836578207979
RSA-430
RSA-430 has 430 decimal digits (1,427 bits), and has not been factored so far.
RSA-430 = 3534635645620271361541209209607897224734887106182307093292005188843884213420
6950355315163258889704268733101305820000124678051064321160104990089741386777
2424190744453885127173046498565488221441242210687945185565975582458031351338
2070785777831859308900851761495284515874808406228585310317964648830289141496
3289966226854692560410075067278840383808716608668377947047236323168904650235
70092246473915442026549955865931709542468648109541
RSA-440
RSA-440 has 440 decimal digits (1,460 bits), and has not been factored so far.
RSA-440 = 2601428211955602590070788487371320550539810804595235289423508589663391270837
4310252674800592426746319007978890065337573160541942868114065643853327229484
5029942332226171123926606357523257736893667452341192247905168387893684524818
0307729497304959710847337973805145673263119916483529703607405432752966630781
2234597766390750441445314408171802070904072739275930410299359006059619305590
701939627725296116299946059898442103959412221518213407370491
RSA-450
RSA-450 has 450 decimal digits (1,493 bits), and has not been factored so far.
RSA-450 = 1984634237142836623497230721861131427789462869258862089878538009871598692569
0078791591684242367262529704652673686711493985446003494265587358393155378115
8032447061155145160770580926824366573211993981662614635734812647448360573856
3132247491715526997278115514905618953253443957435881503593414842367096046182
7643434794849824315251510662855699269624207451365738384255497823390996283918
3287667419172988072221996532403300258906083211160744508191024837057033
RSA-460
RSA-460 has 460 decimal digits (1,526 bits), and has not been factored so far.
RSA-460 = 1786856020404004433262103789212844585886400086993882955081051578507634807524
1464078819812169681394445771476334608488687746254318292828603396149562623036
3564554675355258128655971003201417831521222464468666642766044146641933788836
8932452217321354860484353296131403821175862890998598653858373835628654351880
4806362231643082386848731052350115776715521149453708868428108303016983133390
0416365515466857004900847501644808076825638918266848964153626486460448430073
4909
RSA-1536
RSA-1536 has 463 decimal digits (1,536 bits), and has not been factored so far. $150,000 was previously offered for successful factorization.
RSA-1536 = 184769970321174147430683562020016440301854933866341017147178577491065169671
116124985933768430543574458561606154457179405222971773252466096064694607124
962372044202226975675668737842756238950876467844093328515749657884341508847
552829818672645133986336493190808467199043187438128336350279547028265329780
293491615581188104984490831954500984839377522725705257859194499387007369575
568843693381277961308923039256969525326162082367649031603655137144791393234
7169566988069
RSA-470
RSA-470 has 470 decimal digits (1,559 bits), and has not been factored so far.
RSA-470 = 1705147378468118520908159923888702802518325585214915968358891836980967539803
6897711442383602526314519192366612270595815510311970886116763177669964411814
0957486602388713064698304619191359016382379244440741228665455229545368837485
5874455212895044521809620818878887632439504936237680657994105330538621759598
4047709603954312447692725276887594590658792939924609261264788572032212334726
8553025718835659126454325220771380103576695555550710440908570895393205649635
76770285413369
RSA-480
RSA-480 has 480 decimal digits (1,593 bits), and has not been factored so far.
RSA-480 = 3026570752950908697397302503155918035891122835769398583955296326343059761445
7144169659817040125185215913853345598217234371231338324773210726853524776378
4105186549246199888070331088462855743520880671299302895546822695492968577380
7067958428022008294111984222973260208233693152589211629901686973933487362360
8129660418514569063995282978176790149760521395548532814196534676974259747930
6858645849268328985687423881853632604706175564461719396117318298679820785491
875674946700413680932103
RSA-490
RSA-490 has 490 decimal digits (1,626 bits), and has not been factored so far.
RSA-490 = 1860239127076846517198369354026076875269515930592839150201028353837031025971
3738522164743327949206433999068225531855072554606782138800841162866037393324
6578171804201717222449954030315293547871401362961501065002486552688663415745
9758925793594165651020789220067311416926076949777767604906107061937873540601
5942747316176193775374190713071154900658503269465516496828568654377183190586
9537640698044932638893492457914750855858980849190488385315076922453755527481
1376719096144119390052199027715691
RSA-500
RSA-500 has 500 decimal digits (1,659 bits) and has not been factored so far.
RSA-500 = 1897194133748626656330534743317202527237183591953428303184581123062450458870
7687605943212347625766427494554764419515427586743205659317254669946604982419
7301601038125215285400688031516401611623963128370629793265939405081077581694
4786041721411024641038040278701109808664214800025560454687625137745393418221
5494821277335671735153472656328448001134940926442438440198910908603252678814
7850601132077287172819942445113232019492229554237898606631074891074722425617
39680319169243814676235712934292299974411361
RSA-617
RSA-617 has 617 decimal digits (2,048 bits) and has not been factored so far.
RSA-617 = 2270180129378501419358040512020458674106123596276658390709402187921517148311
9139894870133091111044901683400949483846818299518041763507948922590774925466
0881718792594659210265970467004498198990968620394600177430944738110569912941
2854289188085536270740767072259373777266697344097736124333639730805176309150
6836310795312607239520365290032105848839507981452307299417185715796297454995
0235053160409198591937180233074148804462179228008317660409386563445710347785
5345712108053073639453592393265186603051504106096643731332367283153932350006
7937107541955437362433248361242525945868802353916766181532375855504886901432
221349733
RSA-2048
RSA-2048 has 617 decimal digits (2,048 bits). It is the largest of the RSA numbers and carried the largest cash prize for its factorization, $200,000. The RSA-2048 may not be factorizable for many years to come, unless considerable advances are made in integer factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization.
When the numbers are suf ...
or computational power in the near future.
RSA-2048 = 2519590847565789349402718324004839857142928212620403202777713783604366202070
7595556264018525880784406918290641249515082189298559149176184502808489120072
8449926873928072877767359714183472702618963750149718246911650776133798590957
0009733045974880842840179742910064245869181719511874612151517265463228221686
9987549182422433637259085141865462043576798423387184774447920739934236584823
8242811981638150106748104516603773060562016196762561338441436038339044149526
3443219011465754445417842402092461651572335077870774981712577246796292638635
6373289912154831438167899885040445364023527381951378636564391212010397122822
120720357
See also
* Integer factorization records
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 ...
* RSA Factoring Challenge
The RSA Factoring Challenge was a challenge put forward by RSA Laboratories on March 18, 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers and cracking RSA keys used in cryptograp ...
(includes table with size and status of all numbers)
* RSA Secret-Key Challenge
The RSA Secret-Key Challenge was a series of cryptographic contests organised by RSA Laboratories with the intent of helping to demonstrate the relative security of different encryption algorithms. The challenge ran from 28 January 1997 until May ...
Notes
References
* RSA Factoring Challenge Administrator (1997-10-12)
RSA Challenge List
* RSA Laboratories
The RSA Challenge Numbers
(archived by the Internet Archive
The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
in 2006 before the RSA challenge ended).
* RSA Laboratories,
* Kazumaro Aoki, Yuji Kida, Takeshi Shimoyama, Hiroki Ueda
GNFS Factoring Statistics of RSA-100, 110, ..., 150
Cryptology ePrint Archive, Report 2004/095, 2004.
External links
* Steven Levy
Steven Levy (born 1951) is an American journalist and Editor at Large for ''Wired'' who has written extensively for publications on computers, technology, cryptography, the internet, cybersecurity, and privacy. He is the author of the 1984 book ...
(March 1996)
Wisecrackers
in Wired News
''Wired'' (stylized as ''WIRED'') is a monthly American magazine, published in print and online editions, that focuses on how emerging technologies affect culture, the economy, and politics. Owned by Condé Nast, it is headquartered in San Fran ...
. Has coverage on RSA-129.
{{DEFAULTSORT:Rsa Numbers
RSA Factoring Challenge
Integer factorization algorithms
Large integers