# The Multiopen Argument in Semacaulk

Oracles are: $q1, q2, q3, q4$
Roots are: $v, \alpha, w\alpha, w^{91}\alpha$

We define:
$$
s_1 = \{v\}, z_1 = (X - v) \\
s_2 = \{\alpha\}, z_2 = (X - \alpha) \\
s_3 = \{\alpha, w\alpha\}, z_3 = (X - \alpha)(X - w\alpha) \\
s_4 = \{\alpha, w\alpha, w^{91}\alpha\}, z_4 = (X - \alpha)(X - w\alpha)(X - w^{91}\alpha)
$$

We need to compute $r_1, r_2, r_3, r_4$ such that for $q_1, q_2, q_3, q_4$ we have that $z_i(X)$ divides $q_i(X) - r_i(X)$.

Then $Verifier$ has to evaluate all $r_i(X)$ and $z_i(X)$ in challenge point $\xi$ in order to evaluate $f_i(\xi) ={{q_i(\xi) - r_i(\xi)}\over{z_i(\xi)}}$.

### Lagrange poly form

$L_i(X) = {Z(X)\over{Z'(w^i) (X - w^i)}}$ where
$Z'(X) = \sum_{i \in |H|} Z(X)/(X - w^i)$

#### Each oracle unwrapped

$$
z_1(\xi) = (\xi - v) \\
z_2(\xi) = (\xi - \alpha) \\
z_3(\xi) = (\xi - \alpha)(\xi - w\alpha) \\
z_4(\xi) = (\xi - \alpha)(\xi - w\alpha)(\xi - w^{91}\alpha)
$$

---
$r_1$ and $r_2$ are just constants:
$r_1 = q_1(v)$ and $r_2 = q_2(\alpha)$

---
$r_3(X) = q_3(\alpha)L_{1,3}(X) + q_3(w\alpha)L_{2,3}(X)$

$z_3'(X) = (X - w\alpha) + (X - \alpha)$
$z_3'(\alpha) = (\alpha - w\alpha)$
$z_3'(w\alpha) = (w\alpha - \alpha)$

$L_{1,3}(X) = (X - w\alpha)/(\alpha - w\alpha)$
$L_{2,3}(X) = (X - \alpha)/(w\alpha - \alpha)$

$L_{1,3}(\xi) = (\xi - w\alpha)/(\alpha - w\alpha)$
$L_{2,3}(\xi) = (\xi - \alpha)/(w\alpha - \alpha)$

---
$r_4(X) = q_4(\alpha)L_{1,4}(X) + q_4(w\alpha)L_{2,4}(X) + q_4(w^{91}\alpha)L_{3,4}(X)$

$z_4'(X) = (X - w\alpha)(X - w^{91}\alpha) + (X - \alpha)(X - w^{91}\alpha) + (X - \alpha)(X - w\alpha)$
$z_4'(\alpha) = (\alpha - w\alpha)(\alpha - w^{91}\alpha)$
$z_4'(w\alpha) = (w\alpha - \alpha)(w\alpha - w^{91}\alpha)$
$z_4'(w^{91}\alpha) = (w^{91}\alpha - \alpha)(w^{91}\alpha - w\alpha)$

$L_{1,4}(X) = (X - w\alpha)(X - w^{91}\alpha)/(\alpha - w\alpha)(\alpha - w^{91}\alpha)$
$L_{2,4}(X) = (X - \alpha)(X - w^{91}\alpha)/(w\alpha - \alpha)(w\alpha - w^{91}\alpha)$
$L_{3,4}(X) = (X - \alpha)(X - w\alpha)/(w^{91}\alpha - \alpha)(w^{91}\alpha - w\alpha)$

$L_{1,4}(\xi) = (\xi - w\alpha)(\xi - w^{91}\alpha)/(\alpha - w\alpha)(\alpha - w^{91}\alpha)$
$L_{2,4}(\xi) = (\xi - \alpha)(\xi - w^{91}\alpha)/(w\alpha - \alpha)(w\alpha - w^{91}\alpha)$
$L_{3,4}(\xi) = (\xi - \alpha)(\xi - w\alpha)/(w^{91}\alpha - \alpha)(w^{91}\alpha - w\alpha)$

#### Required computation

We can derive everything from:
$$
(\xi - v) \\
(\xi - \alpha) \\
(\xi - w\alpha) \\
(\xi - w^{91}\alpha) \\
(\alpha - w\alpha) \\
(\alpha - w^{91}alpha) \\
(w\alpha - w^{91}alpha)
$$