FFT of k*k matrix from FFT of a j*j matrix

Question

FFT of matrix a j by j matrix, A $\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ = $\begin{bmatrix}10 & -2\\-4 & 0\end{bmatrix}$
From that result, is there a way to quickly compute FFT of k by k matrix B$\begin{bmatrix}1 & 2 & 0 & 0\\3 & 4 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\end{bmatrix}$ ? .
Two helpful conditions.

1. j and k are powers of 2 and j < k.
2. B is basically zero padded A.

Of lesser importance, I am providing background for this, if that helps. I am trying to compute correlation of two 2d matrices by FFT. The size of one of the matrices is significantly higher than other. For e.g. I have a 1024 *1024 matrix and a 16*16 matrix. Currently I am zero padding the 16*16 matrix to 1024*1024 and then do the FFT on it. This increases the computational costs as I am computing FFT of 16*16 matirx as a zero padded 1024*1024 matrix.