Noisy secret sharing

Question

I am looking for a secret sharing scheme that is robust against noise, the shares are going to be noisy. We do not want to reconstruct the secret perfectly and a noisy reconstruction with a bound on noise is good enough.

Assume we are using $(n,k)$ Shamir's scheme to distribute a secret among $n$ parties so that we need at least $k$ of them to recover. If we add noise to the shares, using $k$ points, the reconstructed polynomial would be very noisy. However, having access to an infinite amount of points would result in a perfect reconstruction. Are there any works that quantify this reconstruction noise?

If you know of any other idea or another sharing scheme, that would also be great!

Answer 1

Do you have a noise model in mind?

If the noise affects only a small portion of shares, regular shamir secret sharing is robust. You only need k complete shares.

For other scenarios, like having a few bit flips across all shares. You add any error correction code after creating the shares before distributing them. E.g Reed Solomon. so That each party still gets a single share but with extra error correction information.