Equation function
Turn a function in binary field to its multilinear extension.
let \(\mathcal{G}\) be a function in binary field:
$$\mathcal{G}(x), x \in \{0,1\}^m$$
its multi-linear extension can be presented as:
$$\mathcal{G}(r_1,…,r_m)=\sum\limits_{x\in\{0,1\}^m}\mathcal{G}(x)\prod\limits^{m}_{i=1}\underbrace{(r_i\cdot x_i + (1-r_i)(1-x_i))}_{x_i=0 \rightarrow 1-r_i, x=1 \rightarrow r_i }$$
Or another equivalent expression:
$$\mathcal{G}(r_1,…,r_m)=\sum\limits_{x\in\{0,1\}^m}\mathcal{G}(x)eq(x,r)$$
emphasize:
\(r_i \in \mathbb{F}\)
\(x \in \{0,1\}^m\)
\(x_i \) is 0 or 1