# Estimate Security level of the Rabin Signature

I'm trying to figure out how long the modulus $$n$$ has to be in the Rabin Signature scheme, to provide 128 bit security.

We assume that the used hash function is "secure enough". Then the naive approach would be the following:

Since forging a signature is prooven to be as hard as factorization of $$n$$, the security level of the signature scheme is approx. equal to the bit length of $$n$$. Therefore for 128 bit security, $$n$$ should be at least a number,such that the fastest factoring algorithms take 2^180 tries. But how long is such a number?

Of course I know, that its said that RSA keys should be around 3072bit to achieve 128bit sec. On the other hand RSA has no tight reduction to number factorization in contrast to Rabin.

Can someone therefore explain how to estimate Rabin modulus size?