There are presently three families of public key cryptographic algorithms.
Only the IF (Integer Factorization) family is based on the concept of using
the product of two primes where these primes must be kept secret. The other
two families DL (Discrete Logarithm) and EC (Elliptic Curve) are based on
entirely different principles. The DL family uses a prime as the basis for
arithmetic over the Galois field GF (p). This prime is public information.
The other family, EC, uses a prime number as the basis for either GF (p) or
GF(2**p). Again the prime p is public information. The significance of
Mersenne primes in DL and EC cryptographic systems is that arithmetic can
be sped up if p is a Mersenne prime. I believe that this is part of the
trick in Crandall's scheme, although I have not looked at his patents in
detail.
What does this mean for GIMPS? Not a whole lot, since all of the Mersenne
primes that are of practical use in cryptography are already known.
IMHO the entire RC5 effort had the primary goal of making a political
statement that served to line the pockets of a small number of people. This
primary goal did not prove anything that was not already known. It did have
the secondary product of providing and example of an effective distributed
computation on PCs. That was a worthwhile goal in its own right.
At 03:15 PM 10/1/1999 , Aaron Blosser wrote in flowing prose:
> > >The types of primes found by GIMPS are so incredibly huge
> > that they would
> > >have no practical purpose for encryption.
> >
> > Nope, there is a crypto system using the Mersennes. Look for
> > a link to it
> > at www.mersenne.org. (It is also mentioned in one of the Mxx
> > press releases
> > from George... or at least in an article ;-) )
>
>Hmm...no kidding. Now, correct me if I'm wrong (I probably am) but aren't
>those types of encryption schemes based on multiplying large primes together
>to generate the "key", and the fact that it would take a VERY long time to
>factor the product means it's relatively secure?
>
>So...how would it be helpful to use certain types of primes during the key
>generation? That takes away a huge chunk of the security, especially since
>with Mersenne primes being one of the number multiplied, you only need try
>38 (not 37...doh!) numbers to try and factor with in order to find the pair.
>And of those 38, alot are small enough to be worthless for encryption
>anyway.
>
>I know that certain of the algorithms we have available, such as ECM, do
>make it faster to hunt for factors, meaning that factoring an unknown pair
>can be done faster, but I didn't think that Mersenne primes in particular
>are useful for cryptography at all.
>
>Aaron
>
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