# RSA Factoring Challenge

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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 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, but 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.

## Contents

The RSA challenges ended in 2007. 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understanding of the cryptanalytic strength of common symmetric-key and public-key algorithms, these challenges are no longer active."

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.

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 exist 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.

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. The prizes for RSA-576 and RSA-640 have been awarded. The remaining prizes have been retracted since the challenge became inactive in 2007.

^ * The number was factored after the challenge became inactive.

^ ** RSA-129 was not part of the RSA Factoring Challenge, but was related to a column by Martin Gardner in Scientific American.

^ *** RSA-170 was also independently factored by S. A. Danilov and I. A. Popovyan two days later.

## References

RSA Factoring Challenge Wikipedia

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