November 8, 2005--A team at the German Federal Agency for Information Technology Security (BSI) recently announced the factorization of the 193-digit number
310 7418240490 0437213507 5003588856 7930037346 0228427275 4572016194 8823206440 5180815045 5634682967 1723286782 4379162728 3803341547 1073108501 9195485290 0733772482 2783525742 3864540146 9173660247 7652346609
known as RSA-640 (Franke 2005). The team responsible for this factorization is the same one that previously factored the 174-digit number known as RSA-576 (MathWorld headline news, December 5, 2003) and the 200-digit number known as RSA-200 (MathWorld headline news, May 10, 2005).
RSA numbers are composite numbers having exactly two prime factors (i.e., so-called semiprimes) that have been listed in the Factoring Challenge of RSA Security®.
While composite numbers are defined as numbers that can be written as a product of smaller numbers known as factors (for example, 6 = 2 x 3 is composite with factors 2 and 3), prime numbers have no such decomposition (for example, 7 does not have any factors other than 1 and itself). Prime factors therefore represent a fundamental (and unique) decomposition of a given positive integer. RSA numbers are special types of composite numbers particularly chosen to be difficult to factor, and they are identified by the number of digits they contain.
While RSA-640 is a much smaller number than the 7,816,230-digit monster Mersenne prime known as M42 (which is the largest prime number known), its factorization is significant because of the curious property that proving or disproving a number to be prime ("primality testing") seems to be much easier than actually identifying the factors of a number ("prime factorization"). Thus, while it is trivial to multiply two large numbers p and q together, it can be extremely difficult to determine the factors if only their product pq is given. With some ingenuity, this property can be used to create practical and efficient encryption systems for electronic data.
Gotta go update my "Unsolved Codes" webpage . . .