Lagrange interpolation in the exponent

Question

I am going over the paper https://verifiable-timed-signatures.github.io/web/assets/paper.pdf and on page 5, it says the following:

Suppose I have a t-out-of-n threshold sharing scheme for the secret $\sigma=H(m)^\alpha$. The first $t-1$ shares are defined as $\sigma_i = H(m)^{\alpha_i}$, where $\alpha_i$ are sampled randomly in the field.

Then, for $i \in \{ t, t+1, \dots, n \}$, we defined the shares as follows: $\sigma_i = \left(\frac{\sigma}{\prod_{1\leq j \leq t-1} \sigma_j^{l_j(0)}}\right)^{l_i(0)^{-1}}$

where $l_i()$ is the i-th Lagrange polynomial basis.

My question is: what are the $l_i$ exactly? Because to calculate each $l_i$, wouldn't I need to know at least $t$ $\sigma_i$? It seems like I am using information I don't have it to calculate each $\sigma_i$.

Answer 1

Let $L$ be a Vector Space of Polynomials of $x$ of degree $\leq n-1$ with coefficients in some field $\mathbb{K}$.

Normally we define $$l_i(x) :=\prod _ {{j=1},{j\neq i}}^n \frac{x-a_j}{a_i-a_j}$$ as the Lagrange polynomial basis. Note that

$$l_i(0) :=\prod _ {{j=1},{j\neq i}}^n \frac{-a_j}{a_i-a_j}= (-1)^{n-1}\prod _ {{j=1},{j\neq i}}^n \frac{a_j}{a_i-a_j} $$ is a constant and it suffices to compute the expression in the question given only the first $t-1$ $\sigma_i$'s.