Computing modular exponent given order

Question

I want to compute $g^{mn}$ mod $n^2$ where $n=pq$ and I know that $g$ has order $kn$ mod $n^2$ where $m<k$. Is there any clever way of doing it utilizing the order? I have tried other methods of computing $g^{mn}$ directly.

My question stems from an implementation of the Paillier cryptosystem that I'm working on. Right now, I'm following the original paper "Public-key cryptosystems based on composite degree residuosity classes", scheme 3 with the recommendation of generating $g$ using DSA. I have looked at different methods of computing modular exponentiation directly such as square and multiply, k-ary, window sliding, etc. I want an algorithm for computing $g^{mn} \bmod n^2$ that does not rely upon knowledge of the factorization of $n$.