It is known that nimbers (Grundy numbers) below $2^{2^n}$ form a field with the nim addition $\oplus$ and the nim product $\cdot$.
Generally, one can develop an algorithm to compute the product of two nimbers based on the fact that for $a \neq b$ it holds that $2^{2^a} \cdot 2^{2^b} = 2^{2^a+2^b}$ and $2^{2^a} \cdot 2^{2^a} =\frac{3}{2}2^{2^a}$. But typical assumption here is that $2^{2^n}$ fits in the machine word.
Are there any known fast algorithms to compute nim products when it is not the case?
I'd be interested in any algorithms that are practical for, say, $n=16$.
