What is the use of Mersenne Primes in cryptography There is an international search for Mersenne Primes. The project is huge. But what is the use of Mersenne Primes in cryptography? Do they have any other properties other than the $2^n-1$ form?
I can think of two places where we use a Mersenne Prime within cryptogaphy:


As a modulus within a prime elliptic curve.  $2^{521}-1$ is a prime, and so we can define an elliptic curve using $GF(2^{521}-1)$, which is in moderately common use.  One reason we use such a modulus (rather than another prime of approximately the same size) is that the special form of the prime makes computing $x \bmod 2^{521}-1$ easy.
In the (rather obscure) Carter Wegmen Counter mode, we use the fact that $2^{127}-1$ is prime; we use that prime, rather than another value, because it is approximately the correct size, and (as above) computing $x \bmod 2^{127}-1$ is easy.


In both cases, the special property that we take advantage of is that they make computing the modulo operation cheap (by using shifts and adds).