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CRT-based Verifiable Secret Sharing Scheme

In this paper, section 4 Verifiable Secret Sharing Scheme, the dealer phase and the combiner phase are as follows:

Dealer Phase

To share a secret $d \in Z_{m_0}$ among a group of $n$ users with verifiable shares, the dealer does the following:

1) The dealing procedure of the Asmuth-Bloom SSS to obtain the shares $y_i = y \mod m_i$ for each $1 \le i \le n$ where $y = d+ Am_0 < M$.

Note that the $m_i$'s are large primes where $p_i = 2m_i + 1$ is also a prime for $1 \le i \le n$.

2) Let $N$ be an integer whose prime factorization is not known by the users and the dealer.

Compute $E(y) = g^ y mod P N$ .Send $y_ i$ to the $i^{th}$ user secretly for all $1 \le i \le n$

3)The $i^{th}$ user checks $g^ { y_{i} }≡ E(y) \mod p_i$

to verify $y_i = y \mod m_i$ . Then he checks the validity of the range proof to verify $y < M$.

Combiner Phase

Let $S$ be a coalition gathered to construct the secret.

The share $y_i$ of user can be verified by the other users by the verification equation

$$g^ { y_{i}} ≡ E(y) \mod p_i$$

If all shares are valid then the coalition $S$ can obtain the secret $d$

In this scheme $y$ is private. So, after knowing all shares of $y_i$, users reconstruct the secret. But how to verify $y_i = y \mod m_i$ user checks the validity of the range proof to verify $y < M$ without knowing all shares of users.

Could you please explain how user can verify range

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    $\begingroup$ As there are several schemes described in the linked paper, it would have been nice to write which one you are asking about. I assume it's the one of Fig. 4. In section 4.1 the authors say that they use Cao's non-interactive range-proof scheme as a black box, and refer their readers to the papers [4] by Boudot and [6] by Cao and Liu. Did you try reading these papers? $\endgroup$ – j.p. Feb 15 at 7:08

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