I am trying to build a cryptographic system that has several components and ran into a problem with a secret sharing scheme.
Let $v$ be a value we are interested committing to. I generate a commitment $P = aG + vH$ where $a\in \mathbb{Z}_q$ , $G$ and $H$ are generators of that group (Pedersen commitment scheme, if given $a$ someone can derive the value $v$).
Now lets assume I want to break $a$ (lets call it the blinding key) to $n$ shares using shamir secret sharing scheme where $k$ shares can produce $a$ ($k,n$ sharing scheme). so if $P$ is publicly known, is there a way where players who received a share can verify that the share they got actually produce $a$ without uncovering $a$? meaning that they know the combined shares can generate the blinding key.
I know that there exist a scheme for verifying that all shares are derived from the same polynomial (Paul Feldman's scheme) but what can prevent the possibility of an adversary calculating and distributing shares derived from some randomly generated and unrelated polynomial (where the secret is $c\neq a$)?