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May 23, 2007
Loooooooooong Division
A team of mathematicians has set a
new record for factoring a large number into primes,
breaking a massive 307-digit number into its three indivisible
factors and besting the previous mark by 30 digits.
Written as a binary string of zeros and ones,
the number is 1017 places or "bits" long--nearly as
long as the 1024-bit numbers currently used to encode
electronic messages--and the researchers' method of using a network
of computers raises the prospect of hijacking PC and
video-game systems to try to crack codes. However, security
experts say they're confident they can stay ahead of
would-be hackers.
To keep financial transactions or military intelligence under
wraps, these secret messages are transformed into strings of
numbers, which are then multiplied or otherwise mixed up
with other large numbers that serve as "keys" for
scrambling and decoding the message. For example, the widely
used RSA encryption scheme uses two keys. A public
key is essentially a very large number that is
the product of two prime numbers and is available
to everyone. A private key consists of the two
prime numbers. The public key allows anyone to send
an encrypted message, but only the holder of the
private key can read it, using the two prime
numbers to unlock the original message. The only known
way to crack RSA is to factor the public
key, which requires long, brute-force calculation. Nevertheless, in 1999
researchers mustered enough computing power to show that 512-bit
public keys then commonly used in Europe could be
factored into the primes (Science, 3 September 1999, p.
1472). That result led to an increased adoption of
1024-bit keys.
To factor the 1017-bit behemoth, Thorsten Kleinjung of
Bonn University and colleagues distributed the number crunching over
hundreds of computers that among them tallied a total
of 95 years of processing time. Previously, one important
step in the computation, called the matrix step, had
to be performed by a large cluster of computers
in a single location. "What is a really new
aspect of this work is that we have been
able to distribute this matrix step over [computers at]
different locations," says Arjen Lenstra of the Swiss Federal
Institute of Technology in Lausanne. The team reported the
result in an open memo to their fellow mathematicians.
The
result raises the possibility that criminals might someday commandeer
computers on the Internet to crack codes, Lenstra says.
In fact, Play Station 3 video-game systems, which are
optimized for number crunching and typically connected to the
Internet, could provide a useful resource for such chicanery.
Kleinjung and his colleagues are now trying to get
their hands on a substantial number of Play Stations.
"We want to have thousands of them, or ten
thousands, and see what analytic potential they may have,"
says Lenstra.
Ronald Rivest, a cryptographer at the Massachusetts Institute
of Technology in Cambridge and one of the developers
of the RSA code, says that a real breakthrough
in the basic mathematics of factoring large numbers might
cause a problem, but meanwhile he is confident that
RSA encryption will remain secure. "There are already recommendations
to switch to keys of 2048 bits; ... the
software is flexible," he says.
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