A mighty number falls

[R.A. Hettinga]

<http://www.physorg.com/printnews.php?newsid=98962171>

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A mighty number falls

Mathematicians and number buffs have their records. And today, an
international team has broken a long-standing one in an impressive feat of
calculation.

On March 6, computer clusters from three institutions - the EPFL, the
University of Bonn and NTT in Japan -- reached the end of eleven months of
strenuous calculation, churning out the prime factors of a well-known,
hard-to-factor number that is a whopping 307 digits long.

"This is the largest 'special' hard-to-factor number factored to date,"
explains EPFL cryptology professor Arjen Lenstra. (The number is 'special'
because it has a special mathematical form -- it is close to a power of
two.) The news of this feat will grab the attention of information security
experts and may eventually lead to changes in encryption techniques.

Although it is relatively easy to identify huge prime numbers, factoring,
or breaking a number down into its prime components, is extremely
difficult. RSA encryption, named for the three individuals who devised the
technique (Ronald Rivest, Adi Shamir and Leonard Adleman), takes advantage
of this. Using the RSA method, information is encrypted using a large
composite number, usually 1024 bits in size, created by multiplying
together two 150-or-so digit prime numbers. Only someone who knows those
two numbers, the "keys", can read the message. Because there is a vast
supply of large prime numbers, it's easy to come up with unique keys.
Information encrypted this way is secure, because no one has ever been able
to factor these huge numbers. At least not yet.

The most recent factoring record is RSA200, a 200-digit 'non-special'
number whose two prime factors were identified in 2005 after 18 months of
calculations that took over a half century of computer time.

The international team factored the current 307-digit behemoth using the
"special number field sieve," a method devised in the late 1980s by Lenstra
(then at Bellcore), his brother Hendrik, then a professor at UC Berkeley,
English mathematician John Pollard and Mark Manasse from DEC. The 11-month
job took a century of computer time.

A feat like this would have been unthinkable back in 1990 when Lenstra
started applying number theory and distributed computing to the task of
breaking factoring records. Increased computer power and refined
computational techniques have raised the bar, and will continue to do so.
"We have more powerful computers, we have come up with better ways to map
the algorithm onto the architecture, and we take better advantage of cache
behavior," Lenstra explains.

Is the writing on the wall for 1024-bit encryption" "The answer to that
question is an unqualified yes," says Lenstra. For the moment the standard
is still secure, because it is much more difficult to factor a number made
up of two huge prime numbers, such as an RSA number, than it is to factor a
number like this one that has a special mathematical form. But the clock is
definitely ticking. "Last time, it took nine years for us to generalize
from a special to a non-special hard-to factor number (155 digits). I won't
make predictions, but let's just say it might be a good idea to stay tuned."

Source: Ecole Polytechnique Fidirale de Lausanne




-- 
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R. A. Hettinga <mailto: rah at ibuc.com>
The Internet Bearer Underwriting Corporation <http://www.ibuc.com/>
44 Farquhar Street, Boston, MA 02131 USA
"... however it may deserve respect for its usefulness and antiquity,
[predicting the end of the world] has not been found agreeable to
experience." -- Edward Gibbon, 'Decline and Fall of the Roman Empire'