### Question 46. What is the Significance
of Factoring in Cryptography?

Factoring is the underlying, presumably hard problem upon which several
public-key cryptosystems are based, including RSA. Factoring an RSA modulus
(see Question 8) would allow an attacker to figure
out the private key; thus, anyone who can factor the modulus can decrypt
messages and forge signatures. The security of RSA depends on the factoring
problem being difficult and the presence of no other types of attack. Unfortunately,
it has not been proven that factoring must be difficult, and there remains
a possibility that a quick and easy factoring method might be discovered
(see Question 49), although factoring researchers
consider this possibility remote.

Factoring large numbers takes more time than factoring smaller numbers.
This is why the size of the modulus in RSA determines how secure an actual
use of RSA is; the larger the modulus, the longer it would take an attacker
to factor, and thus the more resistant to attack the RSA modulus is.