Sumcheck in GPU
Sumcheck Primer
- Given a polynomial \(g: \mathbb{F}^\mu \rightarrow \mathbb{F}\) and \(X = \{x_i\}_{i \in [\mu]}\) compute the sum
\begin{aligned}
H = \sum_{X \in B_\mu} g(x_1, x_2, \dots, x_\mu)
\end{aligned}
- Naively, a verifier would require \(|B_\mu|=2^\mu\) evaluations of \(g(.)\)
- Sumcheck protocol requires \(\mathcal{O}(\mu + \lambda)\) verifier work
- Here \(\lambda\) is the cost to evaluate \(g(.)\) at some \(r \in \mathbb{F}^{\mu}\)
- Prover's work is \(\mathcal{O}(2^\mu)\), i.e. linear in no of constraints
\(g_1(\textcolor{orange}{X_1}) := \sum_{x_2\dots}g(\textcolor{orange}{X_1},x_2, \dots, x_m)\)
\(g_2(\textcolor{orange}{X_2}) := \sum_{x_3\dots}g(\textcolor{green}{r_1}, \textcolor{orange}{X_2}, x_3, \dots, x_m)\)
\(v \stackrel{?}{=} g_1(0) + g_1(1)\)
\(g_1(\textcolor{green}{r_1}) \stackrel{?}{=} g_2(0) + g_2(1)\)
\(g_3(\textcolor{orange}{X_3}) := \sum_{x_4\dots}g(\textcolor{green}{r_1}, \textcolor{green}{r_2}, \textcolor{orange}{X_3}, x_4, \dots, x_m)\)
\(g_\mu(\textcolor{orange}{X_\mu}) := g(\textcolor{green}{r_1}, \textcolor{green}{r_2}, \dots, \textcolor{green}{r_{\mu-1}}, \textcolor{orange}{X_\mu})\)
\(g_2(\textcolor{green}{r_2}) \stackrel{?}{=} g_3(0) + g_3(1)\)
\(g_{\mu-1}(\textcolor{green}{r_{\mu-1}}) \stackrel{?}{=} g_\mu(0) + g_\mu(1)\)
\(g_{\mu}(\textcolor{green}{r_{\mu}}) \stackrel{?}{=} g(\textcolor{green}{r_1}, \textcolor{green}{r_2}, \dots, \textcolor{green}{r_\mu})\)
Prover \(\mathcal{P}\)
Verifier \(\mathcal{V}\)
\(g_1\)
\(r_1\)
\(g_2\)
\(g_3\)
\(g_\mu\)
\(r_{\mu-1}\)
\(r_2\)
\(\vdots\)
\(\vdots\)
\(\vdots\)
Sumcheck for MLE Polynomials
- Given a vector \(\vec{f}\in \mathbb{F}^{2^\mu},\) we can write its (unique) multi-linear extension \(f(\mathbf{X})\) as:
f(\textcolor{lightgrey}{0, 0, \dots, 0, 0}) \leftarrow \textcolor{orange}{\vec{f}[0]}
f(\textcolor{lightgrey}{0, 0, \dots, 0,} \textcolor{skyblue}{1}) \leftarrow \textcolor{orange}{\vec{f}[1]}
f(\textcolor{lightgrey}{0, 0, \dots, } \textcolor{skyblue}{1} \textcolor{lightgrey}{,0}) \leftarrow \textcolor{orange}{\vec{f}[2]}
f(\textcolor{skyblue}{1, 1, \dots, 1} \textcolor{lightgrey}{,0}) \leftarrow \textcolor{orange}{\vec{f}[2^\mu-2]}
f(\textcolor{skyblue}{1, 1, \dots, 1, 1}) \leftarrow \textcolor{orange}{\vec{f}[2^\mu-1]}
\begin{aligned}
f(\mathbf{X}) := & \
\textcolor{lightgrey}{(1-X_\mu)(1-X_{\mu-1})\dots}\textcolor{lightgrey}{(1-X_2)}\textcolor{lightgrey}{(1-X_1)}\textcolor{orange}{\vec{f}[0]} \ + \\
& \ \textcolor{lightgrey}{(1-X_\mu)(1-X_{\mu-1})\dots}\textcolor{lightgrey}{(1-X_2)}\textcolor{skyblue}{X_1}\textcolor{orange}{\vec{f}[1]} \ + \\
& \ \textcolor{lightgrey}{(1-X_\mu)(1-X_{\mu-1})\dots}\textcolor{skyblue}{X_2}\textcolor{lightgrey}{(1-X_1)}\textcolor{orange}{\vec{f}[2]} \ + \\
& \ \vdots \\
& \ \textcolor{skyblue}{X_\mu X_{\mu-1} \dots}\textcolor{skyblue}{X_2}\textcolor{lightgrey}{(1-X_1)}\textcolor{orange}{\vec{f}[2^\mu-2]} \ + \\
& \ \textcolor{skyblue}{X_\mu X_{\mu-1} \dots}\textcolor{skyblue}{X_2}\textcolor{skyblue}{X_1}\textcolor{orange}{\vec{f}[2^\mu-1]}. \\
\end{aligned}
\implies
\vdots
- MLEs makes sumcheck prover's computation faster and parallelisable
- Computing round polynomials is easy
\implies
\begin{aligned}
r_1(X_\mu) = (1-X_\mu) \Big( \textcolor{lightgreen}{\vec{f}[0] + \vec{f}[1] + \dots + \vec{f}[2^{\mu-1}-1]} \Big)
\end{aligned}
+ \ X_\mu \Big( \textcolor{orange}{\vec{f}[2^{\mu-1}] + \dots + \vec{f}[2^{\mu}-1]} \Big)
Sums of halves \(\rightarrow\) easy!
- Updating original polynomial with challenge \(\alpha_1\) is tricky
f(\alpha_1\textcolor{lightgrey}{, 0, \dots, 0, 0}) \leftarrow (1-\alpha_1)\textcolor{lightgreen}{\vec{f}[0]} + \alpha_1\textcolor{orange}{\vec{f}[2^{\mu-1}]}
f(\alpha_1\textcolor{lightgrey}{, 0, \dots, 0, }\textcolor{skyblue}{1}) \leftarrow (1-\alpha_1)\textcolor{lightgreen}{\vec{f}[1]} + \alpha_1\textcolor{orange}{\vec{f}[2^{\mu-1}+1]}
f(\alpha_1\textcolor{lightgrey}{, 0, \dots, }\textcolor{skyblue}{1} \textcolor{lightgrey}{, 0}) \leftarrow (1-\alpha_1)\textcolor{lightgreen}{\vec{f}[2]} + \alpha_1\textcolor{orange}{\vec{f}[2^{\mu-1}+2]}
Sumcheck for MLE Polynomials
f(\mathbf{X})
Round computation can be parallelised
Round \(i+1\) depends on round \(i\) challenge
Need to process all terms to compute challenge
f_{1,o}
f_{1,e}
\sum
\sum
\textsf{hash}
\alpha_1
f_{2,o}
f_{2,e}
\sum
\sum
\textsf{hash}
\alpha_2
f_{3,o}
f_{3,e}
\sum
\sum
\textsf{hash}
\alpha_3
f_{4,o}
f_{4,e}
\textsf{hash}
\alpha_4
Sumcheck for R1CS
\begin{bmatrix}
\\
& \bar{A} & \\
\\
\end{bmatrix}
\hspace{-6pt}
\begin{bmatrix}
\\
\vec{z}\\
\\
\end{bmatrix}
\circ
\begin{bmatrix}
\\
& \bar{B} & \\
\\
\end{bmatrix}
\hspace{-6pt}
\begin{bmatrix}
\\
\vec{z}\\
\\
\end{bmatrix}
=
\begin{bmatrix}
\\
& \bar{C} & \\
\\
\end{bmatrix}
\hspace{-6pt}
\begin{bmatrix}
\\
\vec{z}\\
\\
\end{bmatrix}
\underbrace{\hspace{2.6cm}}
2^\mu
F(x) := \Bigg(\sum_{y \in \mathbb{B}_\mu}A(x,y)z(y)\Bigg)\Bigg(\sum_{y \in \mathbb{B}_\mu}B(x,y)z(y)\Bigg) - \sum_{y \in \mathbb{B}_\mu}C(x,y)z(y)
\sum_{x\in \mathbb{B}_\mu}\gamma_x F(x) = 0
\implies F(x) = 0 \quad \forall x \in \mathbb{B}_\mu
We need to prove that \(F\) is 0 on all points of \(\mathbb{B}_\mu\)
G(x) := \left(\Bigg(\sum_{y \in \mathbb{B}_\mu}A(x,y)z(y)\Bigg)\Bigg(\sum_{y \in \mathbb{B}_\mu}B(x,y)z(y)\Bigg) - \sum_{y \in \mathbb{B}_\mu}C(x,y)z(y)\right)
\textcolor{orange}{\textsf{eq}(x, \tau)}
G(x) := a(x)b(x)\textsf{eq}(x) - c(x)\textsf{eq}(x)
- We need sumcheck on sums of products of MLE polynomials
- Trivial to extend sumcheck for one polynomial to sums of products
A: \mathbb{F}^{2 \mu} \rightarrow \mathbb{F}
B: \mathbb{F}^{2 \mu} \rightarrow \mathbb{F}
C: \mathbb{F}^{2 \mu} \rightarrow \mathbb{F}
z: \mathbb{F}^{\mu} \rightarrow \mathbb{F}
\textsf{combine}_G(a,b,c,d) := abd - cd
f_1(\mathbf{X})
f_2(\mathbf{X})
f_3(\mathbf{X})
f_4(\mathbf{X})
f_5(\mathbf{X})
f_{1,o}
f_{1,e}
f_{2,o}
f_{2,e}
f_{3,o}
f_{3,e}
f_{4,o}
f_{4,e}
f_{5,o}
f_{5,e}
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
r_i(X)
\textsf{hash}
\alpha_i
f_1(\mathbf{X})
f_2(\mathbf{X})
f_3(\mathbf{X})
f_4(\mathbf{X})
f_5(\mathbf{X})
f_{1,o}
f_{1,e}
f_{2,o}
f_{2,e}
f_{3,o}
f_{3,e}
f_{4,o}
f_{4,e}
f_{5,o}
f_{5,e}
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
r_i(X)
\textsf{hash}
\alpha_i
L_{(1-\alpha, \alpha)}
L_{(1-\alpha, \alpha)}
L_{(1-\alpha, \alpha)}
L_{(1-\alpha, \alpha)}
L_{(1-\alpha, \alpha)}
(1-\alpha_i)
(\alpha_i)
+
f_1(\mathbf{X})
f_2(\mathbf{X})
f_3(\mathbf{X})
f_4(\mathbf{X})
f_5(\mathbf{X})
f_{1,o}
f_{1,e}
f_{2,o}
f_{2,e}
f_{3,o}
f_{3,e}
f_{4,o}
f_{4,e}
f_{5,o}
f_{5,e}
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
r_i(X)
\textsf{hash}
\alpha_i
L_{(1-\alpha, \alpha)}
L_{(1-\alpha, \alpha)}
L_{(1-\alpha, \alpha)}
L_{(1-\alpha, \alpha)}
f_1(\mathbf{X})
f_2(\mathbf{X})
f_3(\mathbf{X})
f_4(\mathbf{X})
f_5(\mathbf{X})
f_{1,o}
f_{1,e}
f_{2,o}
f_{2,e}
f_{3,o}
f_{3,e}
f_{4,o}
f_{4,e}
f_{5,o}
f_{5,e}
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
r_i(X)
\textsf{hash}
\alpha_i
L_{(1-\alpha, \alpha)}
L_{(1-\alpha, \alpha)}
L_{(1-\alpha, \alpha)}
f_1(\mathbf{X})
f_2(\mathbf{X})
f_3(\mathbf{X})
f_4(\mathbf{X})
f_5(\mathbf{X})
f_{1,o}
f_{1,e}
f_{2,o}
f_{2,e}
f_{3,o}
f_{3,e}
f_{4,o}
f_{4,e}
f_{5,o}
f_{5,e}
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
r_i(X)
\textsf{hash}
\alpha_i
f'_1(\mathbf{X})
f'_2(\mathbf{X})
f'_3(\mathbf{X})
f'_4(\mathbf{X})
f'_5(\mathbf{X})
f_{1,o}
f_{1,e}
f_{2,o}
f_{2,e}
f_{3,o}
f_{3,e}
f_{4,o}
f_{4,e}
f_{5,o}
f_{5,e}
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
r_i(X)
\textsf{hash}
\alpha_i
f'_1(\mathbf{X})
f'_2(\mathbf{X})
f'_3(\mathbf{X})
f'_4(\mathbf{X})
f'_5(\mathbf{X})
f_{1,o}
f_{1,e}
f_{2,o}
f_{2,e}
f_{3,o}
f_{3,e}
f_{4,o}
f_{4,e}
f_{5,o}
f_{5,e}
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
\sum
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
r_{i+1}(X)
\textsf{hash}
\alpha_{i+1}
L2 Cache
\underbrace{\hspace{4.25cm}}
\ell
Sumcheck in GPU
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
L2 Cache
\underbrace{\hspace{4.25cm}}
\ell
Sumcheck in GPU
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
L2 Cache
\underbrace{\hspace{4.25cm}}
\ell
Sumcheck in GPU
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
L2 Cache
\underbrace{\hspace{4.25cm}}
\ell
Sumcheck in GPU
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
L2 Cache
\underbrace{\hspace{4.25cm}}
\ell
Sumcheck in GPU
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
\begin{aligned}
\textsf{sumcheck}_{2^{21}}
\begin{pmatrix}
& & \\
& \\
\end{pmatrix}
= &\
\textsf{sumcheck}_{2^{19}}(\quad)
+
\beta \cdot \textsf{sumcheck}_{2^{19}}(\quad) + \\
&\
\beta^2 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
+
\beta^3 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
\end{aligned}
*This isn't quite what we need, we need to show that independent sumchecks sum to a certain sum.
L2 Cache
r_1^{(1)}(X)
\alpha_1^{(1)}
Sumcheck in GPU
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
\begin{aligned}
\textsf{sumcheck}_{2^{21}}
\begin{pmatrix}
& & \\
& \\
\end{pmatrix}
= &\
\textsf{sumcheck}_{2^{19}}(\quad)
+
\beta \cdot \textsf{sumcheck}_{2^{19}}(\quad) + \\
&\
\beta^2 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
+
\beta^3 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
\end{aligned}
L2 Cache
Sumcheck in GPU
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
\begin{aligned}
\textsf{sumcheck}_{2^{21}}
\begin{pmatrix}
& & \\
& \\
\end{pmatrix}
= &\
\textsf{sumcheck}_{2^{19}}(\quad)
+
\beta \cdot \textsf{sumcheck}_{2^{19}}(\quad) + \\
&\
\beta^2 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
+
\beta^3 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
\end{aligned}
L2 Cache
r_1^{(2)}(X)
\alpha_1^{(2)}
Sumcheck in GPU
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
\begin{aligned}
\textsf{sumcheck}_{2^{21}}
\begin{pmatrix}
& & \\
& \\
\end{pmatrix}
= &\
\textsf{sumcheck}_{2^{19}}(\quad)
+
\beta \cdot \textsf{sumcheck}_{2^{19}}(\quad) + \\
&\
\beta^2 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
+
\beta^3 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
\end{aligned}
L2 Cache
Sumcheck in GPU
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
\begin{aligned}
\textsf{sumcheck}_{2^{21}}
\begin{pmatrix}
& & \\
& \\
\end{pmatrix}
= &\
\textsf{sumcheck}_{2^{19}}(\quad)
+
\beta \cdot \textsf{sumcheck}_{2^{19}}(\quad) + \\
&\
\beta^2 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
+
\beta^3 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
\end{aligned}
L2 Cache
r_1^{(3)}(X)
\alpha_1^{(3)}
Sumcheck in GPU
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
\begin{aligned}
\textsf{sumcheck}_{2^{21}}
\begin{pmatrix}
& & \\
& \\
\end{pmatrix}
= &\
\textsf{sumcheck}_{2^{19}}(\quad)
+
\beta \cdot \textsf{sumcheck}_{2^{19}}(\quad) + \\
&\
\beta^2 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
+
\beta^3 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
\end{aligned}
L2 Cache
Not the most optimal use of L2 cache
Verifier generates \((k + n)\) challenges
Sumcheck in GPU
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
\begin{aligned}
\textsf{sumcheck}_{2^{21}}
\begin{pmatrix}
& & \\
& \\
\end{pmatrix}
= &\
\textsf{sumcheck}_{2^{19}}(\quad)
+
\beta \cdot \textsf{sumcheck}_{2^{19}}(\quad) + \\
&\
\beta^2 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
+
\beta^3 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
\end{aligned}
L2 Cache
\underbrace{\hspace{4.25cm}}
\ell
Sumcheck in GPU
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
\begin{aligned}
\textsf{sumcheck}_{2^{21}}
\begin{pmatrix}
& & \\
& \\
\end{pmatrix}
= &\
\textsf{sumcheck}_{2^{19}}(\quad)
+
\beta \cdot \textsf{sumcheck}_{2^{19}}(\quad) + \\
&\
\beta^2 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
+
\beta^3 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
\end{aligned}
L2 Cache
\underbrace{\hspace{4.25cm}}
\ell
Sumcheck in GPU
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
\begin{aligned}
\textsf{sumcheck}_{2^{21}}
\begin{pmatrix}
& & \\
& \\
\end{pmatrix}
= &\
\textsf{sumcheck}_{2^{19}}(\quad)
+
\beta \cdot \textsf{sumcheck}_{2^{19}}(\quad) + \\
&\
\beta^2 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
+
\beta^3 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
\end{aligned}
L2 Cache
\underbrace{\hspace{4.25cm}}
\ell
r_1^{(2)}(X)
\alpha_1^{(2)}
r_1^{(1)}(X)
\alpha_1^{(1)}
Sumcheck in GPU
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
\begin{aligned}
\textsf{sumcheck}_{2^{21}}
\begin{pmatrix}
& & \\
& \\
\end{pmatrix}
= &\
\textsf{sumcheck}_{2^{19}}(\quad)
+
\beta \cdot \textsf{sumcheck}_{2^{19}}(\quad) + \\
&\
\beta^2 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
+
\beta^3 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
\end{aligned}
L2 Cache
\underbrace{\hspace{4.25cm}}
\ell
r_1^{(2)}(X)
\alpha_1^{(2)}
r_1^{(1)}(X)
\alpha_1^{(1)}
Sumcheck in GPU
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
\begin{aligned}
\textsf{sumcheck}_{2^{21}}
\begin{pmatrix}
& & \\
& \\
\end{pmatrix}
= &\
\textsf{sumcheck}_{2^{19}}(\quad)
+
\beta \cdot \textsf{sumcheck}_{2^{19}}(\quad) + \\
&\
\beta^2 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
+
\beta^3 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
\end{aligned}
L2 Cache
\underbrace{\hspace{4.25cm}}
\ell
Sumcheck in GPU
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
\begin{aligned}
\textsf{sumcheck}_{2^{21}}
\begin{pmatrix}
& & \\
& \\
\end{pmatrix}
= &\
\textsf{sumcheck}_{2^{19}}(\quad)
+
\beta \cdot \textsf{sumcheck}_{2^{19}}(\quad) + \\
&\
\beta^2 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
+
\beta^3 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
\end{aligned}
L2 Cache
\underbrace{\hspace{4.25cm}}
\ell
r_1^{(4)}(X)
\alpha_1^{(4)}
r_1^{(3)}(X)
\alpha_1^{(3)}
r_1^{(5)}(X)
\alpha_1^{(5)}
Relocation of data within L2 cache
Optimal usage of L2 cache
Sumcheck in GPU
Verifier generates \((2k + n)\) challenges
- Main hurdle: L2 cache size is just 96MB
- Can fit an R1CS sumcheck instance of size \(\le 2^{19}\)
- For larger instances, we need to read and write to memory to update the state
- Alternatively, we can break a large sumcheck instance into several smaller instances
\begin{aligned}
\textsf{sumcheck}_{2^{21}}
\begin{pmatrix}
& & \\
& \\
\end{pmatrix}
= &\
\textsf{sumcheck}_{2^{19}}(\quad)
+
\beta \cdot \textsf{sumcheck}_{2^{19}}(\quad) + \\
&\
\beta^2 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
+
\beta^3 \cdot \textsf{sumcheck}_{2^{19}}(\quad)
\end{aligned}
Path Forward
-
Idea 1: Independent sumchecks
- Still need to show \(\sum_j c_j = c\) - additional overhead
- Better ideas to show that?
-
Idea 2: Multi-variate round polynomials
- First round polynomial:
- \(r_1(X) = \sum_{x_j \in \{0,1\}, j \neq 1}f(X, x_2, x_3, \dots, x_\mu)\)
- What if round polynomials were bi-variate polynomials:
- \(r_1(X, Y) = \sum_{x_j \in \{0,1\}, j \neq 1, 2}f(X, Y, x_3, x_4, \dots, x_\mu)\)
- First round polynomial:
-
Idea 3: Bulletproofs-like folding
- Convert a sumcheck problem to a bulletproofs problem (lot of MSMs)
- Run \(k\) bulletproofs rounds until sumcheck size is \(\le 2^l\)
-
Idea 4: Ternary hypercube
- Will help reduce rounds in sumcheck but is that useful to us?
f_1(\mathbf{X})
f_2(\mathbf{X})
f_3(\mathbf{X})
f_4(\mathbf{X})
f_5(\mathbf{X})
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
L(1,0)
L(0,1)
L(-1,2)
L(-2,3)
L(-3,4)
L(-4,5)
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
\textsf{combine}
r_i(X)
\textsf{hash}
\alpha_i
\sum
\sum
\sum
\sum
\sum
\sum
Sumcheck in GPU
By Suyash Bagad
Sumcheck in GPU
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