Multi-secret sharing (Packed secret sharing) - Some questions In the publication - &quot;Communication complexity of secure computation&quot;, I have encountered a small problem - on page 5 of the literature under the heading &quot;Theorem 1&quot; 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 $k \leq t&lt; n$).

Theorem 1: There is a $(t– k+ l,t + 1; k,n)$- multi-secret sharing
scheme to share $k$ elements of a finite field $\mathbb{F}$ among $n$
players, with $k &lt; n,$ where each multi-share is a single element of F.
Proof: Let $S_1, \ldots,S_k$ be the Dealer’s secrets. Assume that
$a_l,....,a_k$ and $e_l,....,e_k$ 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 $p(x) \in F[x]$
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.

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.
I believe there is a typo in the paper: instead of $k$ elements $a_1, \dots, a_k$ there should actually be $n$ fixed public elements, i.e. $a_1, \dots, a_n \in \mathbb F$. To secret-share, first determine $p(x)$ based on the dealer's secrets. Then, give $p(a_i)$ to player $i$ for each $1 \leq i \leq n$.