Cryptography: Integral Domain Zn

•Earlier, when we were looking at how to characterize Zn, we said that, although it possessed a multiplicative identity element, it could not be an integral domain because Zn allowed for the equality a × b = 0 even for non-zero a and b.

•Now we have decided that Zp is a finite field for prime p because every element in Zp has a unique multiplicative inverse, are we sure that we can now also guarantee that if a × b = 0 then either a or b must be 0?

–Yes, we have that guarantee because a × b = 0 for general Zn occurs only when non-zero a and b are factors of the modulus n. When n is a prime, its only factors are 1 and n. So with the elements of Zn being in the range 0 through n − 1, the only time we will see a×b = 0 is when either a is 0 or b is 0.