## Showing the Gaussian Intergers are a euclidean domain under a certain mapping

I am using the Euclidean mapping:

$e(a+bi) = \lfloor log_2(a^2+b^2)\rfloor$

I was trying to use the fact that $e(a+bi) = a^2+b^2$ is a valid mapping while considering the Gaussian integers as a Euclidean Domain, but have essentially ended up nowhere.

I am sorry if this seems like a sophomoric question, I am new to number theory (abstract algebra as well), and I just can't seem to make any leeway with this problem. A hint to an approach to this problem would be greatly appreciated.