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# RSA Laboratories

## The RSA Factoring Challenge FAQ

This challenge is no longer active

Why is the RSA Factoring Challenge no longer active?

What was the RSA Factoring Challenge?

How do I get the challenge numbers?

What does it mean when a challenge number is factored?

How were the challenge numbers generated?

How do I know which challenge numbers have been factored?

What are the best factoring methods?

### Why is the RSA Factoring Challenge no longer active?

Various cryptographic challenges — including the RSA Factoring Challenge — served in the early days of commercial cryptography to measure the state of progress in practical cryptanalysis and reward researchers for the new knowledge they have brought to the community. 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understanding of the cryptanalytic strength of common symmetric-key and public-key algorithms, these challenges are no longer active.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 records, however, are presented here for reference by interested cryptographers.

### What was the RSA Factoring Challenge?

The RSA Factoring challenge was an effort, sponsored by RSA Laboratories, to learn about the actual difficulty of factoring large numbers of the type used in RSA keys. Posted here for historical interest is the set of eight challenge numbers, ranging in size from 576 bits (174 decimal digits) to 2048 bits (617 decimal digits) that made up the challenge. Each number is the product of two large primes, similar to the modulus of an RSA key pair.

Factoring a number means representing it as the product of prime numbers. Prime numbers, such as 2, 3, 5, 7, 11, and 13, are those numbers that are not evenly divisible by any smaller number, except 1. A non-prime, or composite number, can be written as the product of smaller primes, known as its prime factors. 665, for example is the product of the primes 5, 7, and 19. A number is said to be factored when all of its prime factors are identified. As the size of the number increases, the difficulty of factoring increases rapidly.

Factoring 100-digit numbers is easy with today's hardware and algorithms. No public effort has yet resulted in successful factoring of numbers of more than 200 digits. Advances in both computer hardware and number theory, though, are expected to advance the state of the art. One purpose of this contest was to "track" the state of the art in factoring.

The first person to submit a correct factorization for any of the challenge numbers was eligible for a cash prize. Given the amount of computation required for such a factorization, the prizes were mainly symbolic. They served as a small incentive for public demonstrations of factoring on a large scale.

To date, the largest number of this type to be factored, in 2005, was 640 bits. The 704-bit or 768-bit value is likely to be factored soon. On the other hand, barring fundamental algorithmic or computing advances, RSA-2048 should stand for decades.

### How do I get the RSA Challenge numbers?

There are eight RSA challenge numbers, ranging in size from 576 bits to 2048 bits. They are available here. To obtain a single challenge number, select its entry and a page will display containing the decimal value of the challenge number, its current status (whether or not it has yet been factored), and the prize awarded for the first factorization. The value may be copied directly from this page, or downloaded as ASCII text.

If you prefer to obtain all the challenge numbers at once, the URL listed above allows you to download a single file, in text format, that contains all eight challenge numbers.

### What does it mean when a Challenge Number is factored?

Users of the RSA public-key cryptosystem may wonder what the factoring of a challenge number implies about the security of their keys. Should they immediately replace their keys with larger ones? Should they stop using RSA altogether?

Clearly, the factoring of a challenge-number of specific length does not mean that the RSA cryptosystem is "broken." It does not even mean, necessarily, that keys of the same length as the factored challenge number must be discarded. It simply gives us an idea of the amount of work required to factor a modulus of a given size. This can be translated into an estimate of the cost of breaking a particular RSA key pair.

Suppose, for example, that in the year 2010 a factorization of RSA-768 is announced that requires 6 months of effort on 100,000 workstations. In this hypothetical situation, would all 768-bit RSA keys need to be replaced? The answer is no. If the data being protected needs security for significantly less than six months, and its value is considerably less than the cost of running 100,000 workstations for that period, then 768-bit keys may continue to be used.

Applications that require longer-term security or have data with a high financial value should migrate to longer keys before the factoring of the corresponding challenge number is announced. In either case, the results of the Factoring Challenge provide real data to help the cryptosystem user choose the appropriate key size.

RSA Laboratories' Frequently Asked Questions About Today's Cryptography provides more information on choosing RSA key lengths for various applications. RSA Laboratories Bulletin #13 discusses key length requirements for various cryptosystems.

### How were the challenge numbers generated?

The RSA challenge numbers were generated using a secure process that guarantees that the factors of each number cannot be obtained by any method other than factoring the published value. No one, not even RSA Laboratories, knows the factors of any of the challenge numbers.

The generation took place on a Compaq laptop PC with no network connection of any kind. The process proceeded as follows:

1. First, 30,000 random bytes were generated using a ComScire QNG hardware random number generator, attached to the laptop's parallel port.
2. The random bytes were used as the seed values for the B_GenerateKeyPair function, in version 4.0 of the RSA BSAFE library. The private portion of the generated keypair was discarded. The public portion was exported, in DER format to a disk file.
3. The moduli were extracted from the DER files and converted to decimal for posting on the Web page.
4. The laptop's hard drive was destroyed.

### How do I know which challenge numbers have been factored?

The status of each of the challenge numbers is available here. The status will be shown as "Not Factored" for values for which no correct factorization has been submitted. If the number has been factored, the status will identify the submitter and the date of submission.

For challenge numbers that have been factored, the individual page will provide a brief description of the factoring effort. A pointer to a web site, if available, that has the details of the effort will also be provided.

### What are the best factoring methods?

The best known algorithm for factoring large numbers is the General Number Field Sieve (GNFS).

GNFS consists of a sieving phase that searches a fixed set of prime numbers for candidates that have a particular algebraic relationship, modulo the number to be factored. This is followed by a matrix solving phase that creates a large matrix from the candidate values, then solves it to determine the factors.

The sieving phase may be done in distributed fashion, on a large number of processors simultaneously. The matrix solving phase requires massive amounts of storage and is typically performed on a large supercomputer.

More information on factoring algorithms is available on the RSA Laboratories FAQ.

The TWIRL design by Adi Shamir and Eran Tromer represents the state of the art in hardware circuits for integer factoring, which is likely the most efficient approach for very large numbers. More information about TWIRL and its impact on RSA key size can be found in an RSA Laboratories technical note.

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