# When Shamir secret sharing used in one time pad scheme

In the following, all operations are defined over a finite field of prime order. Please note that what I'm explaining below is a part of a bigger protocol, however, to keep things simple I just mention the relevant part.

Let's assume we have a secret value: $$\beta$$ and we want to use a one time pad to mask it as $$c= \beta+r$$, where $$r$$ is a uniformly random value.

For some reason (as a part of the bigger protocol), I need to secret share $$r$$ into shares: $$r_1$$ and $$r_2$$, using Shamir secret sharing scheme.

Question: given $$c$$ and $$r_1$$ can an adversary learn $$r$$ and ultimately $$\beta$$?

• Is $(\beta + r_1 + r_2, r_1)$ independent of $\beta$, if $r_1$, $r_2$, and $\beta$ are all independent, and $r_1$ and $r_2$ are uniformly distributed? (This is not exactly SSSS but it's close.) – Squeamish Ossifrage May 15 at 15:34