Consider we have two vectors $v_1, v_2$ of size $n$, and each vector contains $n$ elements. We permute the vectors as: $\pi (k_1,v_1), \pi (k_2,v_2)$. Where $\pi (k_i,v)$ denotes a permutation of a vector, $v$, using key $k_i$.
I use the above technique (permutation), to hide the original positions of the elements in each vector. I want to give away the two permuted vectors to a malicious server, to do some computation. Therefore, I need to reveal to it the relationship between the two permuted vectors; for instance, position 2 in the permuted $v_1$ is related to position 6 in permuted $v_2$.
Question: Given two permuted vectors, $\pi (k_1,v_1), \pi (k_2,v_2)$, and mapping between them, can an adversary learn the original position of the elements in either vector.