Pre-determined points in Shamir's secret

Question

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?

Answer 1

Based on your question, it is not entirely clear exactly how the $x_i$ and $y_i$ values are coupled.

As I understand it, you want to make a situation where when someone has all $x_i$, they can recover all $y_i$. A simple way to do this would be to encrypt all the $y_i$ with a strong symmetric algorithm, e.g. AES-256-GCM, using a key $k$ and make the encrypted blob public. Then use Shamir's with the $x_i$ to protect the key $k$. Then, when all $x_i$ are assembled, the key $k$ can be recovered and the encrypted blob decrypted, rendering the $y_i$ values.

This may or may not be what you actually want, depending on your use case.