# Explain the meaning of “$E_i = g^{F_i} g^{G_i}$ is uniformly distributed in $G_q$”?

In this paper[1] Qiong et al.’s CRT-based VSS} {Dealer Phase} To share a secret $$d \epsilon Z_{m_0}$$ among a group of $$n$$ users with verifiable shares, the dealer does the following:

  Use 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 $$y = d+ Am_0 < M$$ .

Let $$p, q$$ be primes such that $$q|(p − 1)$$.Construct the unique polynomial $$F (x) \epsilon Z_q [x]$$ where $$deg(F (x)) = n − 1$$ and $$F(m_ i ) = y_ i$$ .

Construct a random polynomial $$G(x) \epsilon q[x]$$ where $$deg(G(x)) = n− 1$$. Let $$z_i = G(m_i )$$ for all $$1 \le i \le n$$.

Let $$g \epsilon Z_p$$ with order $$q, h$$ be a random integer in the group generated by $$g$$ and $$E(a, b) = g^a h^ b ~mod~ p$$ for inputs $$a, b \epsilon Z_{q}^{*}$$.

Compute $$E _i = E(F_ i , G_i)= g^{F_ i} h^{{G_{i}} ~ mod ~ p$$

where $$F_i$$ and $$G_{i}$$ are the $$(i − 1)^{th}$$coefficients of $$F(x)$$ and $$G(x)$$, respectively,

for all $$1 \le i \le n$$. Broadcast $$E_i$$s to all users Send $$(y_i , z_i )$$ secretly to the $$i^{th}$$ user for all $$1 \le i \le n$$.

To verify the validity of his share, each user checks

$$E(y_ i , z_i) \equiv \prod_{i=1}^{n} (E_j)^{{m_i}^{j-1}}$$

$$E(y_ i , z_i) \equiv \prod_{i=1}^{n} (g^{F_j})^{{m_i}^{j-1}}\prod_{i=1}^{n} (h^{G_j})^{{m_i}^{j-1}}$$

$$E(y_ i , z_i) \equiv g^{y_i} h^{z_i}$$

The broadcasted verification information $$E_i = E (F_i, G_i)$$ does not reveal any information about $$F_i$$, , $$i = O,1,2,... ,n - 1$$

It is known that $$g, h \in G_q$$ , where $$G$$, is a cyclic group. According the closure attribute of a group: if $$A$$ and $$B$$ are two elements in G, then the product $$AB$$ is also in $$G$$ $$E_i, = g^{F_i} h^{G_i}$$, Let $$e$$ is the identity element of $$G$$. Since the degree of $$G$$ , is $$q$$, for any $$F_i \in q$$,

, and randomly selected $$G_i \in q$$ , we can randomly select integers $$a,b \in Z_p$$

such that $$g^{qa+F_i} F^{qb+G_i}$$

$${g^{q}}^{a} g^{F_i}{F^{q}}^{b} F^{G_i}$$

$$e^{a} g^{F_i} e^{b} g^{G_i}$$

$$g^{F_i} g^{G_i}$$

Therefore, $$E_i = g^{F_i} g^{G_i}$$ is uniformly distributed in $$G_q$$, , which means it does not reveal any information about $$F_i$$.

I understood theorem mathematical part but i didnot understood conclusion statement i.e

Therefore, $$E_i = g^{F_i} g^{G_i}$$ is uniformly distributed in $$G_q$$, , which means it does not reveal any information about $$F_i$$.