# Noisy secret sharing

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!

• Did you have a look at Reed Solomon error correcting codes? Does that solve your problem in anyway? Oct 13, 2020 at 18:06
• also see this: mortendahl.github.io/2017/08/13/secret-sharing-part3 Oct 13, 2020 at 18:06
• – Mark
Oct 14, 2020 at 4:26
• Possibly a duplicate of crypto.stackexchange.com/q/1760/819. Oct 14, 2020 at 20:50
• The post you guys mention reconstructs the secret perfectly. However, here we can tolerate some level of noise. So the problem is a little bit different. Oct 15, 2020 at 23:07

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.

• We can assume that noise is normal. We add noise to all of the shares. Oct 15, 2020 at 23:08
• normal noise makes sense with analog systems. Not so much with digital systems. Where we discuss bit flips zeroing bits or setting bits noise can be independent or can affect runs of continious data. etc. anyway applying a suitable error correcting code to each share independently and sharing those will work. The error correcting code ensures each share reaches it's recepient. and we don't tell anyone anything about other shares. Oct 16, 2020 at 4:38