Find the Hill cipher key matrix that can realize this permutation 
Find the Hill cipher key matrix $K$ that can realize the permutation $$f: (1,2,3,4,5) \to  (3,5,1,4,2).$$

I am not sure how to find a $5\times 5$ matrix that satisfies this.
My guess is
$$K=\begin{bmatrix} 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0\\0 & 1 & 0 & 0 & 0\\ \end{bmatrix}.$$
I am not sure if this is correct.
 A: I hope that I understand you question right. The block size corresponds to the size of the matrix. So the block size in this case is five.
When your key is $K$ and your plaintext is for example $HELLO$ corresponding to the vector
$$
\pmatrix{8 \\ 5 \\ 12\\ 12\\ 15}
$$
then you get
$$
\pmatrix{ 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0\\0 & 1 & 0 & 0 & 0}
\pmatrix{8 \\ 5 \\ 12\\ 12\\ 15} = \pmatrix{12\\15\\8\\12\\5}.$$
So the ciphertext is $LOHLE$. This corresponds exactly to the permutation that you gave. 
A: I know this is a little late. But I think this can be generalised as follows.
Say the Key Matrix (for the Hill Cipher ) we need is K (which is an mxm matrix, with 'm' being the plain text's and cipher text's length) . 
Let 'f' be the permutation function we have in hand .
(For example , in your case f(1)=3 , f(2)=5 , f(3)=1 and so on)
Then the Key matrix can be formalised as a matrix full of zeros except that
              k(i,j) =  1   , if j=f(i)   ( where k(i,j) is the element in the intersection of i'th row and j'th column )  
