RSA numbers

Key takeaways

RSA numbers belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect RSA numbers to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of RSA numbers from memory before moving on to harder problems.

Reference excerpt

In mathematics, the RSA numbers are a set of large semiprimes (numbers with exactly two prime factors) that were part of the RSA Factoring Challenge. The challenge was to find the prime factors of each number. It was created by RSA Laboratories in March 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers. The challenge was ended in 2007. RSA Laboratories (which is an initialism of the creators of the technique, Rivest, Shamir and Adleman) published a number of semiprimes with 100 to 617 decimal digits. Cash prizes of varying size, up to US$ 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. As of February 2020, 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understanding of the cryptanalytic strength of common symmetric-key and public-key algorithms, these challenges are no longer active."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 digits are counted instead. An exception to this is RSA-617, which was created before the change in the numbering scheme. The numbers (sequence A391940 in the OEIS) 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 parallel computer. The value and factorization of RSA-100 are as follows:

RSA-100 = 1522605027922533360535618378132637429718068114961380688657908494580122963258952897654000350692006139

RSA-100 = 37975227936943673922808872755445627854565536638199 × 40094690950920881030683735292761468389214899724061

RSA-100 is often used to benchmark new factorization software or new hardware. As of December 2009, it took four hours to repeat this factorization using the program Msieve on a 2200 MHz Athlon 64 processor. As of June 2015, the number could be factorized in 72 minutes on overclocked to 3.5 GHz Intel Core2 Quad q9300, using GGNFS and Msieve binaries running by distributed version of the factmsieve Perl script. As of June 2025, the number was factored in 108 seconds on 32 Epyc 9174 server cores using YAFU's implementation of the self initializing quadratic sieve. As of January 2026, the number was reported to have been factored in 4 minutes and 57 seconds on an NVIDIA RTX 5070 Ti, the first such factorization on a complete quadratic sieve factorization pipeline on GPU.

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, using GGNFS and Msieve binaries running by 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's Mathematical Games column in the August 1977 issue of Scientific American. RSA-129 was factored in April 1994 by a team led by Derek Atkins, Michael Graff, Arjen K. Lenstra and Paul Leyland, using approximately 1600 computers from around 600 volunteers connected over the Internet. A US$100 token prize was awarded by RSA Security for the factorization, which was donated to the Free Software Foundation. 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". In 2015, RSA-129 was factored in about one day, with the CADO-NFS open source implementation of number field sieve, using a commercial cloud computing service for about $30.

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, Wojtek Furmanski, Peter L. Montgomery, Damian Weber and Joerg Zayer. The factorization was found in the third trial. The value and factorization are as follows:

RSA-130 = 1807082088687404805951656164405905566278102516769401349170127021450056662540244048387341127590812303371781887966563182013214880557

RSA-130 = 39685999459597454290161126162883786067576449112810064832555157243 × 45534498646735972188403686897274408864356301263205069600999044599

The factorization was found using the Number Field Sieve algorithm and the polynomial

5748302248738405200 x5 + 9882261917482286102 x4 - 13392499389128176685 x3 + 16875252458877684989 x2 + 3759900174855208738 x1 - 46769930553931905995

which has a root of 12574411168418005980468 modulo RSA-130.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with RSA numbers

Start with the simplest possible case. Write down what RSA numbers claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to RSA numbers before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about RSA numbers ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of RSA numbers

In research: RSA numbers appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses RSA numbers in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: RSA numbers is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integer factorization algorithms, Large integers, RSA Factoring Challenge, so understanding it makes those chapters shorter.
In everyday life: Look for RSA numbers outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study RSA numbers in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what RSA numbers means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain RSA numbers out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is RSA numbers in simple terms?

In mathematics, the RSA numbers are a set of large semiprimes (numbers with exactly two prime factors) that were part of the RSA Factoring Challenge. The challenge was to find the prime factors of each number.

Why does RSA numbers matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study RSA numbers?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on RSA numbers.