Multivariate Sum-Check Protocol

Given a $v$-variate polynomial $g$ defined over field $\mathbb{F}$, the protocol convinces the verifier that the she computed the following some correctly:

$$H:=\sum _{b_1\in \{0,1\}}\sum _{b_2\in \{0,1\}}...\sum _{b_v\in \{0,1\}}g(b_1,b_2,...,b_v)$$

The advantage of using the protocol is that the verifier runtime is only $O(v)$ plus the time to evaluate $g$ at a single input in ${\mathbb{F}}^v$.