# Does the prime modulus have to be bigger that the generator?

In the DH algorithm $a^x \bmod b$ where $a$ is the generator and $b$ is the prime modulus does $b$ have to be bigger than $a$ or could $a$ be larger for example $19^x \bmod 17$ . Because I don't see a problem with the generator being larger because the answer can still only be $b \ge \text{answer} > 0$ . Obviously the prime modulus would still have to be large enough to provide good security but I was just curious if the generator could be larger.

I suppose there is really no requirement to have $a <b$. But then again, if you are using an $a>b$ why not reduce it modulo $b$ and save space?