Suyash Bagad
Research Team
The State of Multi-Linear PCS
Journal Club
Apr 3 2025
Recall Sumcheck
- Honest prover starts by computing \(v = \sum_{X \in \{0,1\}^\mu}f(x_1, x_2, \dots, x_\mu)\)
\(g_1(\textcolor{orange}{X_1}) := \sum_{x_2\dots}f(\textcolor{orange}{X_1},x_2, \dots, x_m)\)
\(g_2(\textcolor{orange}{X_2}) := \sum_{x_3\dots}f(\textcolor{green}{r_1}, \textcolor{orange}{X_2}, x_3, \dots, x_m)\)
\(v \stackrel{?}{=} g_1(0) + g_1(1)\)
\(g_1(\textcolor{green}{r_1}) \stackrel{?}{=} g_2(0) + g_2(1)\)
\(g_3(\textcolor{orange}{X_3}) := \sum_{x_4\dots}f(\textcolor{green}{r_1}, \textcolor{green}{r_2}, \textcolor{orange}{X_3}, x_4, \dots, x_m)\)
\(g_\mu(\textcolor{orange}{X_\mu}) := f(\textcolor{green}{r_1}, \textcolor{green}{r_2}, \dots, \textcolor{green}{r_{\mu-1}}, \textcolor{orange}{X_\mu})\)
\(g_2(\textcolor{green}{r_2}) \stackrel{?}{=} g_3(0) + g_3(1)\)
\(g_{\mu-1}(\textcolor{green}{r_{\mu-1}}) \stackrel{?}{=} g_\mu(0) + g_\mu(1)\)
\(g_{\mu}(\textcolor{green}{r_{\mu}}) \stackrel{?}{=} f(\textcolor{green}{r_1}, \textcolor{green}{r_2}, \dots, \textcolor{green}{r_\mu})\)
Prover \(\mathcal{P}\)
Verifier \(\mathcal{V}\)
\(g_1\)
\(r_1\)
\(g_2\)
\(g_3\)
\(g_\mu\)
\(r_{\mu-1}\)
\(r_2\)
\(\vdots\)
\(\vdots\)
\(\vdots\)
Need MLE evaluation proof
Gemini PCS
- Main idea: \(\textsf{MLE} \iff \textsf{Univariate} \) equivalence
f_{\textsf{multi}}(X_1, \dots, X_m) = \sum_{\textcolor{grey}{i=0}}^{\textcolor{grey}{N-1}} \textsf{eq}((X_1, \dots, X_m), (\textcolor{grey}{i_1, \dots, i_m}))
\cdot a_{\textcolor{grey}{i}}
f_{\textsf{uni}}(x) = \sum_{\textcolor{grey}{i=0}}^{\textcolor{grey}{N-1}} x^{i}
\cdot a_{\textcolor{grey}{i}}
a_1
a_2
a_3
a_4
a_5
a_6
a_7
a_8
a_9
a_{10}
a_{11}
a_{12}
a_{13}
a_{14}
a_{15}
a_{16}
Gemini PCS
- Main idea: \(\textsf{MLE} \iff \textsf{Univariate} \) equivalence
a_1
a_2
a_3
a_4
a_5
a_6
a_7
a_8
a_9
a_{10}
a_{11}
a_{12}
a_{13}
a_{14}
a_{15}
a_{16}
f_{\textsf{multi}}(X_1, \dots, X_m)
a_1
a_2
a_3
a_4
a_5
a_6
a_7
a_8
a_9
a_{10}
a_{11}
a_{12}
a_{13}
a_{14}
a_{15}
a_{16}
f_{\textsf{uni}}(x)
Gemini PCS
- Main idea: \(\textsf{MLE} \iff \textsf{Univariate} \) equivalence
a_1
a_2
a_3
a_4
a_5
a_6
a_7
a_8
a_9
a_{10}
a_{11}
a_{12}
a_{13}
a_{14}
a_{15}
a_{16}
f_{\textsf{multi}}(X_1, \dots, X_m)
f_{\textsf{uni}}(x)
a_1
a_2
a_3
a_4
a_5
a_6
a_7
a_8
a_9
a_{10}
a_{11}
a_{12}
a_{13}
a_{14}
a_{15}
a_{16}
\cdot \ (1 - r_1)
\cdot \ (r_1)
f_{\textsf{multi}}(\textcolor{red}{r_1}, X_2, \dots, X_m)
f^{(1)}_{\textsf{uni}}(x)
\textcolor{red}{(1-r_1)}\cdot
+ \ \ \textcolor{red}{r_1}\cdot
f_{\textcolor{orange}{\textsf{odd}}}(x)
f_{\textcolor{lightgreen}{\textsf{even}}}(x)
\equiv
Gemini PCS
- Main idea: \(\textsf{MLE} \iff \textsf{Univariate} \) equivalence
a_1
a_2
a_3
a_4
a_5
a_6
a_7
a_8
a_9
a_{10}
a_{11}
a_{12}
a_{13}
a_{14}
a_{15}
a_{16}
f_{\textsf{multi}}(X_1, \dots, X_m)
f_{\textsf{uni}}(x)
a_1
a_2
a_3
a_4
a_5
a_6
a_7
a_8
a_9
a_{10}
a_{11}
a_{12}
a_{13}
a_{14}
a_{15}
a_{16}
\cdot \ (1 - r_1)
\cdot \ (r_1)
f_{\textsf{multi}}(\textcolor{red}{r_1}, X_2, \dots, X_m)
f^{(1)}_{\textsf{uni}}(x)
\begin{gather*}
f^{(1)}_{\textsf{uni}}(x)
=
\textcolor{red}{(1-r_1)} \cdot
f_{\textcolor{orange}{\textsf{odd}}}(x) +
\textcolor{red}{r_1} \cdot
f_{\textcolor{lightgreen}{\textsf{even}}}(x) \\[10pt]
f_{\textcolor{orange}{\textsf{odd}}}(x^2)
=
\frac{f(x) + f(-x)}{2}
\quad
f_{\textcolor{lightgreen}{\textsf{even}}}(x^2)
=
\frac{f(x) - f(-x)}{2x}
\end{gather*}
a_1
a_2
a_3
a_4
a_5
a_6
a_7
a_8
a_9
a_{10}
a_{11}
a_{12}
a_{13}
a_{14}
a_{15}
a_{16}
Gemini PCS
- Main idea: \(\textsf{MLE} \iff \textsf{Univariate} \) equivalence
a_1
a_2
a_3
a_4
a_5
a_6
a_7
a_8
a_9
a_{10}
a_{11}
a_{12}
a_{13}
a_{14}
a_{15}
a_{16}
f_{\textsf{multi}}(X_1, \dots, X_m)
f_{\textsf{uni}}(x)
a_1
a_2
a_3
a_4
a_5
a_6
a_7
a_8
a_9
a_{10}
a_{11}
a_{12}
a_{13}
a_{14}
a_{15}
a_{16}
\cdot \ (1 - r_1)
\cdot \ (r_1)
f_{\textsf{multi}}(\textcolor{red}{r_1}, X_2, \dots, X_m)
f^{(1)}_{\textsf{uni}}(x)
f^{(2)}_{\textsf{uni}}(x)
f^{(3)}_{\textsf{uni}}(x)
f_{\textsf{multi}}(\textcolor{red}{r_1, r_2}, X_3, \dots, X_m)
f_{\textsf{multi}}(\textcolor{red}{r_1, r_2, r_3}, X_4, \dots, X_m)
Gemini PCS
- Main idea: \(\textsf{MLE} \iff \textsf{Univariate} \) equivalence
f_{\textsf{uni}}(x)
f^{(1)}_{\textsf{uni}}(x)
f^{(2)}_{\textsf{uni}}(x)
f^{(3)}_{\textsf{uni}}(x)
- Uses KZG for univariate evaluation proofs
- For each round \(i \in \{1, 2, \dots, m\}\), need to open:
- \(f^{(i)}_{\textsf{uni}}(x)\) at \(x = \textcolor{yellow}{\beta}^2\)
- \(f^{(i-1)}_{\textsf{uni}}(x)\) at \(x = \textcolor{yellow}{\beta}\)
- \(f^{(i-1)}_{\textsf{uni}}(x)\) at \(x = -\textcolor{yellow}{\beta}\)
\begin{aligned}
f^{(i)}_{\textsf{uni}}(\textcolor{yellow}{\beta}^2)
&=
\textcolor{red}{(1-r_1)} \cdot
f_{\textcolor{orange}{\textsf{odd}}}(\textcolor{yellow}{\beta}^2) +
\textcolor{red}{r_1} \cdot
f_{\textcolor{lightgreen}{\textsf{even}}}(\textcolor{yellow}{\beta}^2) \\[10pt]
&=
\textcolor{red}{(1-r_1)} \cdot
\frac{f^{(i-1)}_{\textsf{uni}}(\textcolor{yellow}{\beta}) + f^{(i-1)}_{\textsf{uni}}(-\textcolor{yellow}{\beta})}{2} +
\textcolor{red}{r_1} \cdot
\frac{f^{(i-1)}_{\textsf{uni}}(\textcolor{yellow}{\beta}) - f^{(i-1)}_{\textsf{uni}}(-\textcolor{yellow}{\beta})}{2\textcolor{yellow}{\beta}}
\end{aligned}
HyperKZG
- For each round \(i \in \{1, 2, \dots, m\}\), need to open:
- \(f^{(i)}_{\textsf{uni}}(x)\) at \(x = \textcolor{yellow}{\beta}^2\)
- \(f^{(i-1)}_{\textsf{uni}}(x)\) at \(x = \textcolor{yellow}{\beta}\)
- \(f^{(i-1)}_{\textsf{uni}}(x)\) at \(x = -\textcolor{yellow}{\beta}\)
- Simplify to open all polynomials at all points and use batched KZG
f^{(0)}_{\textsf{uni}}(x), \ f^{(1)}_{\textsf{uni}}(x), \ \dots ,
f^{(m-1)}_{\textsf{uni}}(x)
\textcolor{yellow}{\beta^2}
\textcolor{yellow}{\beta}
\textcolor{yellow}{-\beta}
- Using SHPLONK can reduce 3 quotients to 2 (i.e., \(2N\) scalar mults. instead of \(3N\))
Q_1(X)
Q_2(X)
Q_3(X)
Mercury
\vdots
- Gemini does \(\text{log}(N)\) steps of univariate evaluation proofs
- Thus, multi-linear evaluation proofs are sizes \(\text{log}(N)\)
- KZG gives \(\mathcal{O}(1)\) proofs for univariate polynomials.
- Can we get constant-sized proofs for MLE polynomials?
- Starting point: Squash first \(t\) rounds of Gemini as a single proof
f_{\textsf{uni}}(x)
f^{(1)}_{\textsf{uni}}(x)
f^{(2)}_{\textsf{uni}}(x)
f^{(m-1)}_{\textsf{uni}}(x)
\mathcal{O}(1)
\mathcal{O}(m-t) \approx \mathcal{O}(1)
f_{\textsf{multi}}(\textcolor{red}{\underbrace{r_1, r_2, \dots, r_t}_{u_1}}, \underbrace{X_{t+1} \dots, X_m}_{u_2})
=
\sum_{0 \le i < 2^t} \textsf{eq}(\textcolor{red}{u_1}, i) \cdot f(\textcolor{red}{u_1} \ \| \ i)
Mercury
\vdots
f_{\textsf{uni}}(x)
f^{(1)}_{\textsf{uni}}(x)
f^{(2)}_{\textsf{uni}}(x)
f^{(m-1)}_{\textsf{uni}}(x)
f_{\textsf{multi}}(\textcolor{red}{\underbrace{r_1, r_2, \dots, r_t}_{u_1}}, \underbrace{X_{t+1} \dots, X_m}_{u_2})
=
\sum_{0 \le i < 2^t} \textsf{eq}(\textcolor{red}{u_1}, i) \cdot f(\textcolor{red}{u_1} \ \| \ i)
f \equiv
\begin{bmatrix}
a_0 & a_1 & \dots & a_{2^t-1} & a_{2^t} & \dots & a_{2^{t + 1}-1} & \dots & a_{2^{m-t}} & \dots & a_{2^m-1}
\end{bmatrix}
=:
\begin{bmatrix}
f_1(x) \\
f_2(x) \\
\vdots \\
f_k(x)
\end{bmatrix}
f \equiv
\begin{bmatrix}
a_0 & a_1 & \dots & a_{2^t-1} \\
a_{2^t} & \dots & \dots & a_{2^{t + 1}-1} \\
\vdots & \vdots & \vdots & \vdots \\
a_{2^{m-t}} & \dots & \dots & a_{2^m-1}
\end{bmatrix}
Gemini \(\longrightarrow\) Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round 1
Round 2
Round \(t\)
Round \(n\)
Multi-variate sumcheck
Uni-variate relations
f(\textcolor{grey}{x}) = f_e(\textcolor{grey}{x^2}) + \textcolor{grey}{x} \cdot f_o(\textcolor{grey}{x^2})
\begin{aligned}
f^{(\textcolor{red}{1})}(\textcolor{grey}{x}) &= (1 - \textcolor{red}{z_1}) \cdot f_e(\textcolor{grey}{x}) + \textcolor{red}{z_1} \cdot f_o(\textcolor{grey}{x})
\end{aligned}
Commitment
\big[f(\cdot)\big]
\big[f^{(\textcolor{red}{1})}(\cdot)\big]
\begin{aligned}
f^{(\textcolor{red}{2})}(\textcolor{grey}{x}) &= (1 - \textcolor{red}{z_2}) \cdot f^{(\textcolor{red}{1})}_e(\textcolor{grey}{x}) + \textcolor{red}{z_2} \cdot f^{(\textcolor{red}{1})}_o(\textcolor{grey}{x})
\end{aligned}
\big[f^{(\textcolor{red}{2})}(\cdot)\big]
\vdots
\vdots
\begin{aligned}
f^{(\textcolor{red}{t})}(\textcolor{grey}{x}) &= (1 - \textcolor{red}{z_t}) \cdot f^{(\textcolor{red}{t-1})}_e(\textcolor{grey}{x}) + \textcolor{red}{z_t} \cdot f^{(\textcolor{red}{t-1})}_o(\textcolor{grey}{x})
\end{aligned}
\big[f^{(\textcolor{red}{t})}(\cdot)\big]
\vdots
\vdots
\begin{aligned}
f^{(\textcolor{red}{n})}(\textcolor{grey}{x}) &= (1 - \textcolor{red}{z_n}) \cdot f^{(\textcolor{red}{n-1})}_e(\textcolor{grey}{x}) + \textcolor{red}{z_n} \cdot f^{(\textcolor{red}{n-1})}_o(\textcolor{grey}{x})
\end{aligned}
\big[f^{(\textcolor{red}{n})}(\cdot)\big]
Gemini \(\longrightarrow\) Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round 1
Round 2
Round \(t\)
Round \(n\)
Multi-variate sumcheck
Commitment
\big[f(\cdot)\big]
\big[f^{(\textcolor{red}{1})}(\cdot)\big]
\big[f^{(\textcolor{red}{2})}(\cdot)\big]
\vdots
\big[f^{(\textcolor{red}{t})}(\cdot)\big]
\vdots
\big[f^{(\textcolor{red}{n})}(\cdot)\big]
Gemini \(\longrightarrow\) Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
- We need to prove two things:
- Jump to state \(t\) was correct
- MLE eval at \(t\) (over remaining variables)
- How to use univariate relations to prove?
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 & a_1 & a_2 & a_3 &
a_4 & a_5 & a_6 & a_7 &
a_8 & a_9 & a_{10} & a_{11} &
a_{12} & a_{13} & a_{14} & a_{15}
\end{bmatrix}
\begin{bmatrix}
a_0 & a_1 & a_2 & a_3 &
a_4 & a_5 & a_6 & a_7 &
a_8 & a_9 & a_{10} & a_{11} &
a_{12} & a_{13} & a_{14} & a_{15}
\end{bmatrix}
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 & a_1 & a_2 & a_3 &
a_4 & a_5 & a_6 & a_7 &
a_8 & a_9 & a_{10} & a_{11} &
a_{12} & a_{13} & a_{14} & a_{15}
\end{bmatrix}
\begin{bmatrix}
a_0 & a_1 & a_2 & a_3 &
a_4 & a_5 & a_6 & a_7 &
a_8 & a_9 & a_{10} & a_{11} &
a_{12} & a_{13} & a_{14} & a_{15}
\end{bmatrix}
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 & a_2 &
a_4 & a_6 &
a_8 & a_{10} &
a_{12} & a_{14}
\end{bmatrix}
\begin{bmatrix}
a_1 & a_3 &
a_5 & a_7 &
a_9 & a_{11} &
a_{13} & a_{15}
\end{bmatrix}
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 & a_2 &
a_4 & a_6 &
a_8 & a_{10} &
a_{12} & a_{14}
\end{bmatrix}
\begin{bmatrix}
a_1 & a_3 &
a_5 & a_7 &
a_9 & a_{11} &
a_{13} & a_{15}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
\begin{bmatrix}
a_1 & a_3 &
a_5 & a_7 &
a_9 & a_{11} &
a_{13} & a_{15}
\end{bmatrix}
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 & a_2 &
a_4 & a_6 &
a_8 & a_{10} &
a_{12} & a_{14}
\end{bmatrix}
\begin{bmatrix}
a_1 & a_3 &
a_5 & a_7 &
a_9 & a_{11} &
a_{13} & a_{15}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
\begin{bmatrix}
a_0 & a_2 &
a_4 & a_6 &
a_8 & a_{10} &
a_{12} & a_{14}
\end{bmatrix}
\begin{bmatrix}
a_1 & a_3 &
a_5 & a_7 &
a_9 & a_{11} &
a_{13} & a_{15}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 & a_2 &
a_4 & a_6 &
a_8 & a_{10} &
a_{12} & a_{14}
\end{bmatrix}
\begin{bmatrix}
a_1 & a_3 &
a_5 & a_7 &
a_9 & a_{11} &
a_{13} & a_{15}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
\begin{bmatrix}
a_0 & a_2 &
a_4 & a_6 &
a_8 & a_{10} &
a_{12} & a_{14}
\end{bmatrix}
\begin{bmatrix}
a_1 & a_3 &
a_5 & a_7 &
a_9 & a_{11} &
a_{13} & a_{15}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 &
a_4 &
a_8 &
a_{12}
\end{bmatrix}
\begin{bmatrix}
a_1 &
a_5 &
a_9 &
a_{13}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
\begin{bmatrix}
a_2 &
a_6 &
a_{10} &
a_{14}
\end{bmatrix}
\begin{bmatrix}
a_3 &
a_7 &
a_{11} &
a_{15}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 &
a_4 &
a_8 &
a_{12}
\end{bmatrix}
\begin{bmatrix}
a_1 &
a_5 &
a_9 &
a_{13}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
\begin{bmatrix}
a_2 &
a_6 &
a_{10} &
a_{14}
\end{bmatrix}
\begin{bmatrix}
a_3 &
a_7 &
a_{11} &
a_{15}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 &
a_4 &
a_8 &
a_{12}
\end{bmatrix}
\begin{bmatrix}
a_1 &
a_5 &
a_9 &
a_{13}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
\begin{bmatrix}
a_2 &
a_6 &
a_{10} &
a_{14}
\end{bmatrix}
\begin{bmatrix}
a_3 &
a_7 &
a_{11} &
a_{15}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
New univariates
f_0(\textcolor{grey}{x})
f_2(\textcolor{grey}{x})
f_2(\textcolor{grey}{x})
f_3(\textcolor{grey}{x})
(X_1, \dots, X_t)
(X_{t+1}, \dots, X_n)
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 &
a_4 &
a_8 &
a_{12}
\end{bmatrix}
\begin{bmatrix}
a_1 &
a_5 &
a_9 &
a_{13}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
\begin{bmatrix}
a_2 &
a_6 &
a_{10} &
a_{14}
\end{bmatrix}
\begin{bmatrix}
a_3 &
a_7 &
a_{11} &
a_{15}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
New univariates
f_0(\textcolor{grey}{x})
f_1(\textcolor{grey}{x})
f_2(\textcolor{grey}{x})
f_3(\textcolor{grey}{x})
(X_1, \dots, X_t)
(X_{t+1}, \dots, X_n)
h(\textcolor{grey}{x})= (1-\textcolor{red}{z_1})(1-\textcolor{red}{z_2})\dots (1-\textcolor{red}{z_t}) \cdot f_0(\textcolor{grey}{x}) \ +
(1-\textcolor{red}{z_1})(1-\textcolor{red}{z_2})\dots (\textcolor{red}{z_t}) \cdot f_1(\textcolor{grey}{x}) \ +
(\textcolor{red}{z_1})(\textcolor{red}{z_2})\dots (\textcolor{red}{z_t}) \cdot f_{b-1}(\textcolor{grey}{x}).
\vdots
\iff
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 &
a_4 &
a_8 &
a_{12}
\end{bmatrix}
\begin{bmatrix}
a_1 &
a_5 &
a_9 &
a_{13}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
\begin{bmatrix}
a_2 &
a_6 &
a_{10} &
a_{14}
\end{bmatrix}
\begin{bmatrix}
a_3 &
a_7 &
a_{11} &
a_{15}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
New univariates
f_0(\textcolor{grey}{x})
f_1(\textcolor{grey}{x})
f_2(\textcolor{grey}{x})
f_3(\textcolor{grey}{x})
h(\textcolor{grey}{x})=
\iff
\sum_{i=0}^{b-1} \textsf{eq}(\textcolor{red}{(z_1, \dots, z_t)}, i) \cdot f_i(\textcolor{grey}{x})
(X_1, \dots, X_t)
(X_{t+1}, \dots, X_n)
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 &
a_4 &
a_8 &
a_{12}
\end{bmatrix}
\begin{bmatrix}
a_1 &
a_5 &
a_9 &
a_{13}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
\begin{bmatrix}
a_2 &
a_6 &
a_{10} &
a_{14}
\end{bmatrix}
\begin{bmatrix}
a_3 &
a_7 &
a_{11} &
a_{15}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
New univariates
f_0(\textcolor{grey}{x})
f_1(\textcolor{grey}{x})
f_2(\textcolor{grey}{x})
f_3(\textcolor{grey}{x})
h(\textcolor{grey}{x})=
\iff
\sum_{i=0}^{b-1} \textsf{eq}(\textcolor{red}{(z_1, \dots, z_t)}, i) \cdot f_i(\textcolor{grey}{x})
(X_1, \dots, X_t)
(X_{t+1}, \dots, X_n)
\longleftrightarrow
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 &
a_4 &
a_8 &
a_{12}
\end{bmatrix}
\begin{bmatrix}
a_1 &
a_5 &
a_9 &
a_{13}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
\begin{bmatrix}
a_2 &
a_6 &
a_{10} &
a_{14}
\end{bmatrix}
\begin{bmatrix}
a_3 &
a_7 &
a_{11} &
a_{15}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
New univariates
f_0(\textcolor{grey}{x})
f_1(\textcolor{grey}{x})
f_2(\textcolor{grey}{x})
f_3(\textcolor{grey}{x})
h(\textcolor{grey}{x})=
\iff
\sum_{i=0}^{b-1} \textsf{eq}(\textcolor{red}{(z_1, \dots, z_t)}, i) \cdot f_i(\textcolor{grey}{x})
(X_1, \dots, X_t)
(X_{t+1}, \dots, X_n)
\longleftrightarrow
f_0(\textcolor{red}{\alpha})
f_1(\textcolor{red}{\alpha})
f_2(\textcolor{red}{\alpha})
f_3(\textcolor{red}{\alpha})
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 &
a_4 &
a_8 &
a_{12}
\end{bmatrix}
\begin{bmatrix}
a_1 &
a_5 &
a_9 &
a_{13}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
\begin{bmatrix}
a_2 &
a_6 &
a_{10} &
a_{14}
\end{bmatrix}
\begin{bmatrix}
a_3 &
a_7 &
a_{11} &
a_{15}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
New univariates
f_0(\textcolor{grey}{x})
f_1(\textcolor{grey}{x})
f_2(\textcolor{grey}{x})
f_3(\textcolor{grey}{x})
h(\textcolor{grey}{x})=
\iff
\sum_{i=0}^{b-1} \textsf{eq}(\textcolor{red}{(z_1, \dots, z_t)}, i) \cdot f_i(\textcolor{grey}{x})
(X_1, \dots, X_t)
(X_{t+1}, \dots, X_n)
\longleftrightarrow
f_0(\textcolor{red}{\alpha})
f_1(\textcolor{red}{\alpha})
f_2(\textcolor{red}{\alpha})
f_3(\textcolor{red}{\alpha})
g(\textcolor{grey}{x}) =
+ \ \textcolor{grey}{x}
+ \ \textcolor{grey}{x^2}
+ \ \textcolor{grey}{x^3}
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 &
a_4 &
a_8 &
a_{12}
\end{bmatrix}
\begin{bmatrix}
a_1 &
a_5 &
a_9 &
a_{13}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
\begin{bmatrix}
a_2 &
a_6 &
a_{10} &
a_{14}
\end{bmatrix}
\begin{bmatrix}
a_3 &
a_7 &
a_{11} &
a_{15}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
New univariates
f_0(\textcolor{grey}{x})
f_1(\textcolor{grey}{x})
f_2(\textcolor{grey}{x})
f_3(\textcolor{grey}{x})
h(\textcolor{grey}{x})=
\iff
\sum_{i=0}^{b-1} \textsf{eq}(\textcolor{red}{(z_1, \dots, z_t)}, i) \cdot f_i(\textcolor{grey}{x})
(X_{t+1}, \dots, X_n)
\longleftrightarrow
g(\textcolor{grey}{x}) =
\sum_{i=0}^{b-1} \textcolor{grey}{x^i} \cdot f_i(\textcolor{red}{\alpha})
(X_1, \dots, X_t)
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 &
a_4 &
a_8 &
a_{12}
\end{bmatrix}
\begin{bmatrix}
a_1 &
a_5 &
a_9 &
a_{13}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
\begin{bmatrix}
a_2 &
a_6 &
a_{10} &
a_{14}
\end{bmatrix}
\begin{bmatrix}
a_3 &
a_7 &
a_{11} &
a_{15}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
New univariates
f_0(\textcolor{grey}{x})
f_1(\textcolor{grey}{x})
f_2(\textcolor{grey}{x})
f_3(\textcolor{grey}{x})
h(\textcolor{grey}{x})=
\iff
\sum_{i=0}^{b-1} \textsf{eq}(\textcolor{red}{(z_1, \dots, z_t)}, i) \cdot f_i(\textcolor{grey}{x})
(X_{t+1}, \dots, X_n)
\longleftrightarrow
g(\textcolor{grey}{x}) =
\sum_{i=0}^{b-1} \textcolor{grey}{x^i} \cdot f_i(\textcolor{red}{\alpha})
(X_1, \dots, X_t)
\longleftrightarrow
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
\begin{bmatrix}
a_0 &
a_4 &
a_8 &
a_{12}
\end{bmatrix}
\begin{bmatrix}
a_1 &
a_5 &
a_9 &
a_{13}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
\begin{bmatrix}
a_2 &
a_6 &
a_{10} &
a_{14}
\end{bmatrix}
\begin{bmatrix}
a_3 &
a_7 &
a_{11} &
a_{15}
\end{bmatrix}
(1 - \textcolor{red}{z_1}) \ \cdot
(\textcolor{red}{z_1}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(1 - \textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
(\textcolor{red}{z_2}) \ \cdot
New univariates
f_0(\textcolor{grey}{x})
f_1(\textcolor{grey}{x})
f_2(\textcolor{grey}{x})
f_3(\textcolor{grey}{x})
h(\textcolor{grey}{x})=
\iff
\sum_{i=0}^{b-1} \textsf{eq}(\textcolor{red}{(z_1, \dots, z_t)}, i) \cdot f_i(\textcolor{grey}{x})
g(\textcolor{grey}{x}) =
\sum_{i=0}^{b-1} \textcolor{grey}{x^i} \cdot f_i(\textcolor{red}{\alpha})
h(\textcolor{red}{\alpha}) = g_{\textsf{multi}}(\textcolor{red}{(z_1, \dots, z_t)})
How does \(g(x)\) relate to original \(f(x)?\)
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
h(\textcolor{grey}{x})=
\sum_{i=0}^{b-1} \textsf{eq}(\textcolor{red}{(z_1, \dots, z_t)}, i) \cdot f_i(\textcolor{grey}{x})
g(\textcolor{grey}{x}) =
\sum_{i=0}^{b-1} \textcolor{grey}{x^i} \cdot f_i(\textcolor{red}{\alpha})
h(\textcolor{red}{\alpha}) = g_{\textsf{multi}}(\textcolor{red}{(z_1, \dots, z_t)})
How does \(g(x)\) relate to original \(f(x)?\)
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
h(\textcolor{grey}{x})=
\sum_{i=0}^{b-1} \textsf{eq}(\textcolor{red}{(z_1, \dots, z_t)}, i) \cdot f_i(\textcolor{grey}{x})
g(\textcolor{grey}{x}) =
\sum_{i=0}^{b-1} \textcolor{grey}{x^i} \cdot f_i(\textcolor{red}{\alpha})
h(\textcolor{red}{\alpha}) = g_{\textsf{multi}}(\textcolor{red}{(z_1, \dots, z_t)})
How does \(g(x)\) relate to original \(f(x)?\)
\therefore \ g(\textcolor{grey}{x}) := f(\textcolor{grey}{x}) \ \ \textsf{mod} \ \ (\textcolor{grey}{x^3} - \textcolor{red}{\alpha})
g(\textcolor{grey}{x}) = f_0(\textcolor{red}{\alpha}) \ +
\textcolor{grey}{x} f_1(\textcolor{red}{\alpha}) \ +
\textcolor{grey}{x^2} f_2(\textcolor{red}{\alpha}) \ +
\textcolor{grey}{x^3} f_3(\textcolor{red}{\alpha}).
= \textcolor{grey}{x^0} \ \cdot
\begin{bmatrix}
a_0 + \
\textcolor{red}{\alpha} a_4 + \
\textcolor{red}{\alpha^2} a_8 + \
\textcolor{red}{\alpha^3} a_{12}
\end{bmatrix}
\begin{bmatrix}
a_1 + \
\textcolor{red}{\alpha} a_5 + \
\textcolor{red}{\alpha^2} a_9 + \
\textcolor{red}{\alpha^3} a_{13}
\end{bmatrix}
\textcolor{grey}{x^1} \ \cdot
\begin{bmatrix}
a_2 + \
\textcolor{red}{\alpha} a_6 + \
\textcolor{red}{\alpha^2} a_{10} + \
\textcolor{red}{\alpha^3} a_{14}
\end{bmatrix}
\textcolor{grey}{x^2} \ \cdot
\begin{bmatrix}
a_3 + \
\textcolor{red}{\alpha} a_7 + \
\textcolor{red}{\alpha^2} a_{11} + \
\textcolor{red}{\alpha^3} a_{15}
\end{bmatrix}
\textcolor{grey}{x^3} \ \cdot
Mercury
F(X_1, X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ X_2, \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots \ \dots \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, X_n)
F(\textcolor{red}{z_1}, \ \textcolor{red}{z_2}, \ \dots, \textcolor{red}{z_t}, \ \dots, \ \textcolor{red}{z_n})
\vdots
\vdots
\vdots
\vdots
Round \(n\)
Multi-variate sumcheck
Round 1
h(\textcolor{grey}{x})=
\sum_{i=0}^{b-1} \textsf{eq}(\textcolor{red}{(z_1, \dots, z_t)}, i) \cdot f_i(\textcolor{grey}{x})
g(\textcolor{grey}{x}) =
\sum_{i=0}^{b-1} \textcolor{grey}{x^i} \cdot f_i(\textcolor{red}{\alpha})
h(\textcolor{red}{\alpha}) = g_{\textsf{multi}}(\textcolor{red}{(z_1, \dots, z_t)})
\therefore \ g(\textcolor{grey}{x}) := f(\textcolor{grey}{x}) \ \ \textsf{mod} \ \ (\textcolor{grey}{x^3} - \textcolor{red}{\alpha})
Plan is:
- Eval \(h(\textcolor{red}{\alpha})\) using KZG
- Eval \(g_{\textsf{multi}}(\textcolor{red}{z_1, \dots, z_t})\) using Gemini (size \(b\))
- Eval \(h_{\textsf{multi}}(\textcolor{red}{z_{t+1}, \dots, z_n})\) using Gemini (size \(n/b\))
- Prove \(g(\textcolor{grey}{x}) \longleftrightarrow f(\textcolor{grey}{x})\) using univariate division
Mercury
Plan is:
- Eval \(h(\textcolor{red}{\alpha})\) using KZG
- Eval \(g_{\textsf{multi}}(\textcolor{red}{z_1, \dots, z_t})\) using Gemini (size \(b\))
- Eval \(h_{\textsf{multi}}(\textcolor{red}{z_{t+1}, \dots, z_n})\) using Gemini (size \(n/b\))
- Prove \(g(\textcolor{grey}{x}) \longleftrightarrow f(\textcolor{grey}{x})\) using univariate division
g_{\textsf{multi}}(\textcolor{red}{z_1, \dots, z_t}) := \sum_{i=0}^{b-1} \textsf{eq}((\textcolor{red}{z_1, \dots, z_t}), i) \cdot g_i
= \big\langle \textsf{eq}_{\textsf{uni}} \ , \ g_{\textsf{uni}} \big\rangle
p(\textcolor{grey}{x}) \cdot q\left(\textcolor{grey}{\frac{1}{x}}\right)
=
(p_0 + p_1\textcolor{grey}{x} + p_1\textcolor{grey}{x^2} + \dots + p_d\textcolor{grey}{x^d})
\cdot
\left(q_0 + q_1\textcolor{grey}{\frac{1}{x}} + p_1\textcolor{grey}{\frac{1}{x^2}} + \dots + p_d\textcolor{grey}{\frac{1}{x^d}}\right)
= \langle p, q \rangle + \textcolor{grey}{\frac{1}{x}} \cdot S_1\left(\textcolor{grey}{\frac{1}{x}}\right) + \textcolor{grey}{x} \cdot S_2(\textcolor{grey}{x})
Mercury
Plan is:
- Eval \(h(\textcolor{red}{\alpha})\) using KZG
- Eval \(g_{\textsf{multi}}(\textcolor{red}{z_1, \dots, z_t})\) using Gemini (size \(b\))
- Eval \(h_{\textsf{multi}}(\textcolor{red}{z_{t+1}, \dots, z_n})\) using Gemini (size \(n/b\))
- Prove \(g(\textcolor{grey}{x}) \longleftrightarrow f(\textcolor{grey}{x})\) using univariate division
f(\textcolor{grey}{x}) = (\textcolor{grey}{x^b} - \textcolor{red}{\alpha}) \cdot q(\textcolor{grey}{x}) + g(\textcolor{grey}{x})
\(\implies\) Univariate KZG!
Mercury
Plan is:
- Eval \(h(\textcolor{red}{\alpha})\) using KZG
- Eval \(g_{\textsf{multi}}(\textcolor{red}{z_1, \dots, z_t})\) using Gemini (size \(b\))
- Eval \(h_{\textsf{multi}}(\textcolor{red}{z_{t+1}, \dots, z_n})\) using Gemini (size \(n/b\))
- Prove \(g(\textcolor{grey}{x}) \longleftrightarrow f(\textcolor{grey}{x})\) using univariate division
Mercury \(\equiv\) Constant-sized MLE evaluation proofs.
Multi-Linear PCS
By Suyash Bagad
