# 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"

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?

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 $$x_i.$$