Looking for a formula for "Given $x_1$, $\ldots$, $x_{n-1}$, the output $F(x_1,\ldots,x_n)$ is dependent of $x_n$ and is of $l$-bits security"

Question

Suppose I have a function $F(x_1,\ldots,x_n)$, where each $x_i$ is of $l$-bits security (suppose each $x_i$ is a binary string of length $l$). At the same time, suppose there are $n$ persons, each one having a value $x_i = b_i$.

I want to write down in a mathematical formula the following: "given $x_1=b_1$, $\ldots$, $x_{n-1}=b_{n-1}$, the output of $F$ is dependent of $x_n$ and is of $l$-bits security". I.e., if any $n-1$ persons plug in their values, there are $2^l$ possibilities for $F(b_1,\ldots,b_{n-1},x_n)$.

For example, consider the function $F(x_1,\ldots,x_n) = x_1$. The output of $F$ is only dependent on $x_1$. This function offers no security: if player 1 plugs $x_1=b_1$, there is one unique possibily for $F(x_1,\ldots,x_n)$ ($l=0$).

Second example (bit-wise sum). Consider $F(x_1,\ldots,x_n) = x_1 + \cdots + x_n \pmod{2}$. Given $x_1,\ldots,x_{n-1}$, the output depends on $x_n$ and is of $l$-bits security.

Now, returning to the question, I have an intuition that the formula that I want is

$$H(F(x_1,\ldots,x_n) | x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n) = H(h_i) \quad \forall i=1,\ldots,n$$

where $H$ is the Shannon Entropy.

Can someone point if I am in the right path and formally describe the formula that I am looking for and explain it?

Thanks in advance.

Answer 1

To be well defined as a function, you need to specify lengths of the $x_i$, which is not the same thing as their entropy. Also, the $H(h_i)$ in your equation should probably be $H(x_i).$

Let's assume each $x_i$ are binary vectors of length $n$ for concreteness. Then what you want in a more general version of this problem would be $$ \mathbb{H}(F(x_1,\ldots,x_n) | x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n) \geq \mathbb{H}(x_i) \quad \forall i=1,\ldots,n $$ which also takes care of the unequal entropy case for the different $x_i.$