In the publication - "Communication complexity of secure computation", I have encountered a small problem - on page 5 of the literature under the heading "Theorem 1" they have described a $(c,d; k,n)$ multi-party secret sharing scheme.
Here, $d$ denotes the minimum number of parties that must come together to recover the secret, $k$ denotes the number of secrets, $n$ denotes the number of parties and $c$ denotes the maximum number of parties who together cannot gain any further information about the secret by collaborating.
I quote the text when it describes a $(t - k + 1, t + 1; k , n)$ multi-secret sharing scheme - (note $k \leq t< n$).
Theorem 1: There is a $(t - k + 1,t + 1; k,n)$- multi-secret sharing scheme to share $k$ elements of a finite field $\mathbb{F}$ among $n$ players, with $k \leq t < n,$ where each multi-share is a single element of $\mathbb{F}$.
Proof: Let $S_1, \ldots,S_k$ be the Dealer’s secrets. Assume that $a_1,...,a_k$ and $e_1,...,e_k$ are pre-selected elements of $\mathbb{F}$ that are known to the Dealer and all $n$ players.
A generalization of Shamir’s secret-sharing scheme suffices. In the Distribution Phase, each player $i$ receives the multi-share $p(a_i),$ where $p(x) \in \mathbb{F}[x]$ is an otherwise random degree $t$ polynomial such that $$p(e_i) = S_i, 1 \leq i \leq k.$$
More specifically, $$p(x) = q(x) \prod_{i = 1}^{k}(x - e_i) + \sum_{i = 1}^{k}S_i L_i(x),$$ where $q(x)$ is a completely random degree $t - k$ polynomial, and where $L_i(x)$ is the Lagrange polynomial $\frac{\prod_{j \neq i} (x - e_j)}{\prod_{j \neq i} (e_i - e_j)}$.
i. I seem to only see the $k$ multi-shares distributed to k out of n parties. Is this understanding of mine incorrect? What is the role of the other $(n-k)$ parties?
ii. Are there any further resources online that I could refer to for such multi-secret sharing schemes? (preferably not academic literature as I'd like to work through some examples.)
Thanks very much in advance.
