In Shamir secret sharing scheme, we have a secret and then the shares are generated based on the secret. The question is can we generate the shares before knowing the 'secret' (which is uniformly random). Specifically, suppose $n$ users choose random numbers $r_1,r_2,\dots, r_n$, can we reconstruct a secret (random number) using $t$ of them with Shamir's technique?
I know of 2 types of threshold secret sharing. I believe you are looking for the Type II threshold secret sharing. These are the basics behind them:
- Type I threshold secret sharing
As proposed originally by Shamir in 1979, in this threshold secret sharing a single owner owns a secret which he splits up into shares and divides these among $n$ shareholders. To reconstruct the secret, $t$ shareholders (for threshold $t$) are required to work together to reconstruct the original secret.
- Type II threshold secret sharing
This secret sharing scheme was proposed by Wang et al in 2003. This secret sharing scheme does not have an owner of a secret. A set of $n$ users create a random secret and after all users sharing partial information with everyone else, a master secret is created of which only $t$ or more users are able to construct this secret.
The computational details of both these secret sharing schemes can be found in: https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=8076576
No, if $n$ users have independently chosen secret random numbers $r_i$, and for any given $t\in[1,n)$, we can't use Shamir secret sharing to construct a secret dependent on $t$ out of $n$ of the $r_i$, and independent of which $r_i$ are picked. The critical issue is that what Shamir secret sharing assigns to a user is dependent on what the others get.
More generally, I think there is no function taking $t\in[1,n)$ pairs $(i,r_i)$ with distinct $i$ and producing something independent of the picks but dependent on each $r_i$. I'm trying to think of a proof, but nothing has happened yet.