I'm reading this paper, which on page 3(Section IV.C) presents a Jointly Random Verifiable Secret Sharing Scheme for Elliptic curves. The algorithm makes sense to me save for this part:
"Each $P_{i\neq j}$ verifies that his share is consistent with other shares i.e. $a_0^{(i)}G = \sum_{j \in B}b_j f_i(j)G$"
Just to explain the terms for those that haven't read the whole paper:$E$ is an elliptic curve, the base point is $G \in E(Z_p)$ of order $r$.
Player $P_i$ shares secret $a_0^{(i)}$ via $f_i(x)$, a random polynomial of degree $t$. $P_i$ then sends other players their share, i.e. sends $f_i(j)$ to player $P_j \hspace{3pt}\forall j \in \{1,\cdots,n\}$. $P_i$ also broadcasts $f_i(j)G \hspace{3pt}\forall j \in \{1,\cdots,n\}$.
So, can someone please explain to me why this equation holds:$$a_0^{(i)}G = \sum_{j \in B}b_j f_i(j)G$$
The authors do not seem to define $B$ or $b_i$. However, later in the same paper,in a related context, they define $b_i = \prod\limits_{j \in B, j \neq i}\frac{j}{(j-i)}$. I'm not sure if that definition applies here too.