I have distinct sets of pairs of values $(a_1, b_1), ..., (a_i, b_i)$. This set of pairs is ordered, and each $a_i$ and $b_i$ are related. I want to share the $a_i$ values publicly. I do not want to share the $b_i$ with anyone.
I want the $b_i$ values to correspond to a single Shamir secret $S$ (Perhaps concatenate them, and encrypt with AES).
Is there any way to use Shamir's secret to make $a_i$ be the shared points D($i$, f($x_i$)=$a_i$) that corresponds to the Shamir secret , such that the $a_i$ are the shared points?
In other words, have the $a_i$ be the solution to the (k-1) degree polynomials?
My end goal is to "transmit" all $a_i$, but be able to use compute the $b_i$ after I know ALL of the $a_i$. I thought polynomial interpretation might be useful for this scenario.
If not Shamir's, is there any other method that may work?