Sumcheck

\gdef\vec#1{\mathbf{#1}} \gdef\F{\mathbb{F}}

Given a field \F, and a linear space V over \F given by the tensor product

V = T_1 \otimes T_2 \otimes \cdots \otimes T_n \text{.}

The prover has a tensor T \in V and wants to prove some linear functional f(T) = s.

Verifiers perspective

Verifier has a sum s_0.

In the first round it receives a (typically short) vector \vec v_1. And verifies \vec w_1 \cdot \vec v_1 = s_0 for some public vector \vec w_1. It then sends a random challenge r_1. It sets s_1 \leftarrow \vec s_1(r_1) \cdot \vec v_1 for some public vector valued function \vec s_1(r_1).

This repeats for several rounds and verifier is left with $s_n

What does this prove to the verifier?

s_n = \vec s_n(r_n) \cdot \vec v_n and \vec w_n \cdot \vec v_n = s_{n-1}