Perfect verification of modular scheme

Secret sharing schemes are used to distribute a secret value among a group of users so that only authorized set of them can reconstruct the original secret correctly. The modular secret sharing scheme (MSSS) we are studying is based on the Chinese Remainder Theorem. In this scheme the secrets $s(x), S(x), s_{1}(x),\dots , s_{k}(x)$ are defined as follows $s(x)=S(x) ~mod\, m(x), s_{i}(x)=S(x) ~mod\, m_{i}(x), i=1,2,\dots , k$. All the secrets and moduli are chosen from polynomial ring $F_{p}[x]$, and the reconstruction of secret $s(x)$ is carried out by applying the above-mentioned Chinese Remainder Theorem. The verification of any secret sharing scheme is understood as the protocol of verification by the participants of their partial secrets and (or) the protocol for verifying the legitimacy of the actions of the dealer. In this paper, we introduce a perfect verification protocol of MSSS. It means that none information leaks under distribution and verification. Two verification protocols are introduced in this paper. The first one is simpler and it depends on assumption about dealer honesty. If there is no such assumption verification is more complex. Both protocols are based on one work by J. Benalo and generalize the protocol proposed earlier by M. Vaskovsky and G. Matveev in two ways. First, the general, not only the threshold access structure is verified, and secondly, the dealer is not necessarily honest. Earlier, N. Shenets found the perfection condition of MSSS. Thus, if these conditions аre met, both the MSSS and its verification protocol are perfect.