Cryptology - Unofficial St. Mary's College of California Web Site

12.10. HOW TO BREAK RSA 247
had to find it. This task is almost infinitely easier than the factoring N using a
brute force factoring method!
OK, perhaps you say that we know that none of the small prime numbers
divide N, so let’s not waste our time with those. In fact, let’s only check the
primes that have between 299 and 301 digits. This can’t be so many, right
Well, the Prime Number Theorem, again, tells us there are
10 301 / ln(10 301 ) − 10 299 / ln(10 299 ) > 10 298
primes in this region. We haven’t eliminated too many! 16
OK, perhaps you will argue that somebody will someday figure out how to
factor such big numbers. Really, it is a very simple idea: just factor the darned
thing. There are indeed many many people working on exactly this question,
developing fancy methods with exotic names like “Pollard’s ρ-method” and the
“Number Field Sieve”, and these method have been shown to have the ability
to factor numbers of up to 155 digits. 17 So it is possible that in the future 18
people will be able to quickly factor 600 digit numbers. I guess our hope is that
by then whatever messages we send today will be so outdated that no one will
care to go back and break them. And by then we will have switched so that P
and Q are about 300 digits each.
The final “OK”. OK, perhaps you will say but isn’t there this thing called
the “National Security Agency” and isn’t the US government spending billions
of dollars to fund them every year to do cryptographic work for the FBI and
CIA and weren’t they smart enough to break the Russians one-time pads when
the Russians didn’t use them properly and don’t they hire many many really
smart mathematicians that they swear to secrecy Might not they have figured
out a way to break RSA ciphers and not be telling us Huh Hey smart guy,
what’s your answer to this And I’d have to answer, “dunno”. Maybe they
have. No one seems to have any real evidence that they did, but we just don’t
know since they aren’t telling.
The upshot is that, outside of possibly the NSA (or its equivalents in other
countries), as far as I know no one is currently able to break a carefully constructed
and properly used RSA cipher in any reasonable amount of time. This
is an area of much research in mathematics, so I’m making no promises about
the future. But for the time being a well-constructed RSA system appears to
be quite secure.
16 In fact, intuitively this makes sense. $10, 000 is a lot of money, but removing that much
from $1, 000, 000 still leaves a whole lot left over. Those two extra 0’s are a big deal!
17 See the rsasecurity.com homepage.
18 Two big shots in the field, Arjen Lenstra and Eric Verheul, have guessed that in another
5 or so years (2009) it will be possible to build a computer that can break a 1024-bit RSA
key in about a day for $250 million. The National Institute of Standards and Technology
recommends that to protect information until 2015 one should use primes of roughly 300
digits.