What is the best CPA distinguisher for the function $F_k:\{0,1\}^{8n}\to\{0,1\}^{16n}$ described below?
Let $E_k$ be a $2n\times2n$ matrix with elements in $GF(2^8)$, selected by generating bit strings of length $32n^2$ using a predefined function $PRF_k$, until one is found that is invertible e.g. using Gaussian elimination.
To encode an input bit string $M$, generate random bit string $Pad$ of length $8n$. Let the $2n$ element vector $T$ correspond to the concatenation $M|Pad$. Calculate the vector $C$ using the formula $E_kT^T=C^T$.
The decoding function is given by the $n\times2n$ matrix corresponding to the first $n$ rows of $E_k^{-1}$.
Obviously, you need no more than $4n$ chosen cipher texts and access to a decryption oracle, to be able to derive the decoding matrix. But is there a better CPA distinguisher than statistical analysis of the skew caused by a constant $M$ for a relatively large number of $Pad$ values?