With this question I am referring to the BGW multiplication by Gennaro et al (PDF here).
The multiplication is described on the 4th page.
(Another source for me was "A pragmatic Introduction to Secure Multi-Party Computation" p. 43-44)
Summary of BGW Multiplication Procedure:
To do the multiplication of 2 secret values $\alpha$ and $\beta$ of every player $P_i$ has to have the share $f_{\alpha}(i)$ and $f_{\beta}(i)$ where $f_{\alpha}$ and $f_{\beta}$ are the random degree t polynomials from the Shamir secret sharing.
Now every player $P_i$ computes $f_{\alpha}(i) \cdot f_{\beta}$ and sends shares of this value $h_i(j)$ created with the random degree t polynomial $h_i$ (so that $h_i(0) = (f_{\alpha} \cdot f_{\beta})(i)$) to player $P_j$ for $1 \le j \le n$.
Next the paper from above describes how the players can obtain random degree t shares of the value $\alpha \cdot \beta$ (so that they can then reconstruct the result of the multiplication with these shares):
Every player $P_i$ computes the value $H(i)$ from the degree t polynomial $H$ which is defined as:
$$H(x) = \sum_{i=1}^{n} \lambda_i h_i(x)$$
($the \lambda_i$s are the appropiate Lagrange coefficients).
$H$ is a random polynomial with
$$H(0) = \sum_{i=1}^{n} \lambda_i h_i(0) = \sum_{i=1}^{n} \lambda_i (f_{\alpha} \cdot f_{\beta})(i) = (f_{\alpha} \cdot f_{\beta})(0) = \alpha \cdot \beta$$
My Question: Is H(x) really of degree t? Couldn't it also be bigger because $n$ points from different degree t polynomials $h_i$ for $1 \le i \le n$ are used for the interpolation? Usually it is argued that linear operations on $(t,n)$ shared shares result in new $(t,n)$ shares and because the $h_i$ functions are of degree $t$ linear combinations of values $h_i{j}$ for $1 \le i \le i$ should result in $(t,n)$ shares as well. Does this also hold in this scenario, since we always combine values from different degree $t$ polynomials at the same $x$ value?
Another question: It is also noted that $t$ hast to be such that $2t+1 \le n$. Is this really necessary? Wouldn't $t+1 \le n$ suffice because $H(x)$ is degree $t$ anyways or is the information from 2t+1 shares necessary to properly construct $H(x)$? (My hypothesis was that, without the $2t+1$ Lagrange coefficients $\lambda_i$, $H(0)$ would no be $\alpha \cdot \beta$)
The "Pragmatic Intro" p. 44 says that only with $2t+1 \le n$ the players have enough information to determine the value $(f_{\alpha} \cdot f_{\beta})(0)$. Why is this the case?
