Quantitatively measure the private information revealed by a matrix?

Suppose I have a private matrix $A$ that I want to conceal. I do the matrix multiplication $A^\prime =AK$ and $A^\prime$ is public.

How can I quantitatively measure "how well" the matrix $A$ is concealed? To be specific, I may select different matrices $K$ and want to analyze the different protection levels provided by multiplying $K$.

I way we can think about is the probability to recover $A$ given $A^\prime$. However, even if $A$ cannot be recovered, some information (such as ratio between the entries) will be leaked. How can I quantify such information?

• Are the $K$ matrices also made public? I'll point out that multiplying by many different $K$s is analogous to multiplying by a single $[ K_1 \| K_2 \| \cdots \| K_n]$. Presumably the rank of this combined matrix has some bearing about how much uncertainty remains about $A$. – Mikero Sep 21 '15 at 5:20
• The matrix $K$ is secret. I am not meant to multiply many different $K$s at one time. Every time, I just multiply one $K$. The types of $K$ may be different. I want to analyze the security of different types of $K$ in a quantitative way. – Paradox Sep 21 '15 at 15:40
• Is the matrix $A$ from a subgroup of all matriices ( e.g. is it invertible)? – user27950 Oct 4 '15 at 7:02