To prove $r \cdot f_1 +f_2 \cdot (s+1)$ is secure We define the polynomials $r, f_1,f_2,s \in R[x]$. Where $r$ is a random degree 1 polynomial and $s$ is a random polynomial such that: $degree(s)=degree(f_1)=degree(f_2)$, let $R$ be $\mathbb{Z}_p$ where $p$ is a large prime number. 

My question: " How to To prove: Given $\ r\cdot f_1 +f_2 \cdot (s+1)$ one who knows only $f_2$ cannot learn anything about $f_1$."



Or even simpler: Given $\ f_1 +f_2 \cdot s$, one who knows $f_2$ can learn anything about $f_1$?

Thanks
First, a fact.

For some polynomial $f_x$ and some random polynomial of the same degree, say $t$, an adversary given only $f_x+t$, knows no additional information about $f_x$. Basically (due to the finite ring), this operation is the same thing as the one-time-pad.

On to the problem at hand.

Let $f_3=f_2\cdot(s+1)$. Since $f_2$ and $s$ have the same degree, we know that $f_3$ has double that degree. Say the degree of $s$ (and $f_2$ and $f_1$) is $d$, so the degree of $f_3$ is $2d$. Since $2d&gt;d$ (when $d&gt;0$), if you could prove that $f_3$ was a random polynomial, then clearly an attacker given the values you specified above would learn no additional information about $f_1$. Problem is, $f_3$ is not a random polynomial since the coefficients would have interdependencies. For example (using $\mathbb{Z}_{11}$), let $f_2=3x+5$ and $s=7x+2$, then $f_3=(7*3)x^7+(2*3+5*7)x+(5*2)=10x^2+8x+10$. The terms (before I reduced the multiplication) have interdependencies.

Since $f_3$ is not random, Adding $f_3$ to another non-random polynomial ($f_1$ or $r\cdot f_1$ in your case) would not be indistinguishable from a random polynomial. So, I don't think you can make an information theoretic argument.