Deterministic outputs based on non-connected inputs I am looking for a one way function that can generate outputs deterministically. However, revealing any of the inputs dont allow the person to generate other inputs. 
E.g: 
lets say we have a list of inputs: $i_{1},i_{2},i_{3},...$ that look completely random and a list of corresponding outputs: $o_{1},o_{2},o_{3},...$ that also look completely random which are each mapped by one way functions: $f_{1},f_{2},f_{3},...$ 
These outputs can be generated without knowing the inputs so anyone should be able to deterministically generate the outputs (perhaps by knowing a public shared data which is used in the function). But if one of these inputs is revealed, it should not reveal the other inputs.
If it is not mathematically possible, I need a proof for it.
"public shared data" $\:$ = $\:$ the ordered pair whose entries are

[a key for a full-domain statistically injective PRF] and [a signature verification key]

$\hspace{.04 in}f$ is an arbitrary one-way function.

For each positive integer $j$, [$o_j$ is the value of the PRF at $j\hspace{.03 in}$] and [$i_j$ is [[the signature for $o_j$] concatenated with [the string of zeros whose length equals the length of that signature]]]

and $\hspace{.04 in}f_j$ is given by

if $\hspace{.04 in}f_j$'s input has even length and its left half is a valid signature for $o_j$

and its right half is all-zero then output $o_j$ else output $\hspace{.04 in}f$ of $\hspace{.04 in}f_j$'s input.