Re: 2047 bit keys in PGP

[Rick Busdiecker <rfb@lehman.com>]

    From: Cedric Tefft <CedricT@datastorm.com>
    Date: Thu, 04 Jan 96 14:12:00 PST

    >And they strongly imply that brute-force attacks against 256-bit
    >keys will be infeasible until computers are built from something
    >other than matter and occupy something other than space."

    Hmmm... Well, the 384-bit Blacknet PGP key was cracked in just a
    few months.  How?

Factoring a 384-bit number is not equivalent to searching a 384-bit
keyspace.  Consider that there are 78498 primes less than 1000000.
This means that you can do a brute force search of a keyspace of under
17-bits to find a prime factor of any composite number less than
1000000000000 -- a bit under 40 bits.  I've done this to verify the
results of an implementation of the Rabin-Miller primality test on
relatively small numbers.  I'm not sure how many primes there are with
192 or fewer bits, but it's far fewer than 2^384.

There are better techniques around for factoring large numbers than
this sort of brute force testing.  While I didn't follow the thread
very closely, they probably used the quadratic sieve or number field
sieve algorithm.  See Schneier's _Applied_Cryptography_ for more on
factoring, including references to more detailed works.

--
Rick Busdiecker                        Please do not send electronic junk mail!
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A `hacker' is one who writes code.  Breaking into systems is `cracking'.