GKR-based Lookups
Rational Sumcheck
c=x∈H∑q(X)p(X)
\begin{aligned}
c = \sum_{x\in H} \frac{\textcolor{lightgreen}{p(X)}}{\textcolor{orange}{q(X)}}
\end{aligned}
- Polynomial functions:
- f(X)=c0+c1X+…cnXn
- Rational functions:
- r(X)=q(X)p(X)
- Generally, a rational function is not a polynomial function
- Therefore, we cannot directly apply the sumcheck protocol
- There is a workaround though:
Rational Sumcheck
- Define a polynomial f(X) such that
∀x∈H,f(x):=q(x)p(x).
\begin{aligned}
\forall x\in H, \quad f(x) := \frac{\textcolor{lightgreen}{p(x)}}{\textcolor{orange}{q(x)}}.
\end{aligned}
- Then we can compute g(X) and h(X) such that:
f(X)=Xg(X)+∣H∣σ
\begin{aligned}
f(X) = X\textcolor{red}{g(X)} + \frac{\sigma}{|H|}
\end{aligned}
f(X)⋅q(X)−p(X)=h(X)vH(X)
\begin{aligned}
f(X) \cdot q(X) - p(X) = \textcolor{red}{h(X)}v_H(X)
\end{aligned}
⟵ Classic Aurora sumcheck
⟵ Quotient check
- Prover cost: O(N⋅log(N)) where N=∣H∣
- Can we do better?
Sumcheck
x1
x_1
x2
x_2
x3
x_3
x1
x_1
x2
x_2
x3
x_3
- Round polynomial: ri(X)=flo(1−X)+fhiX
- Challenge: αi=H(ri)
Sumcheck
x1
x_1
x2
x_2
x3
x_3
x1
x_1
x2
x_2
- Round polynomial: ri(X)=flo(1−X)+fhiX
- Challenge: αi=H(ri)
Sumcheck
x1
x_1
x2
x_2
x3
x_3
x1
x_1
x2
x_2
x1
x_1
x2
x_2
- Round polynomial: ri(X)=flo(1−X)+fhiX
- Challenge: αi=H(ri)
Sumcheck
x1
x_1
x2
x_2
x3
x_3
x1
x_1
x2
x_2
x1
x_1
- Round polynomial: ri(X)=flo(1−X)+fhiX
- Challenge: αi=H(ri)
Sumcheck
x1
x_1
x2
x_2
x3
x_3
x1
x_1
x2
x_2
x1
x_1
- Round polynomial: ri(X)=flo(1−X)+fhiX
- Challenge: αi=H(ri)
Sumcheck
x1
x_1
x2
x_2
x3
x_3
x1
x_1
x2
x_2
x1
x_1
- Round polynomial: ri(X)=flo(1−X)+fhiX
- Challenge: αi=H(ri)
23
2^3
22
2^2
21
2^1
Dimensionality reduction!
Rethinking Rational Sumcheck
q1p1
\begin{aligned}
\frac{\textcolor{orange}{p_1}}{\textcolor{lightgreen}{q_1}}
\end{aligned}
q2p2
\begin{aligned}
\frac{\textcolor{orange}{p_2}}{\textcolor{lightgreen}{q_2}}
\end{aligned}
q3p3
\begin{aligned}
\frac{\textcolor{orange}{p_3}}{\textcolor{lightgreen}{q_3}}
\end{aligned}
q4p4
\begin{aligned}
\frac{\textcolor{orange}{p_4}}{\textcolor{lightgreen}{q_4}}
\end{aligned}
q5p5
\begin{aligned}
\frac{\textcolor{orange}{p_5}}{\textcolor{lightgreen}{q_5}}
\end{aligned}
q6p6
\begin{aligned}
\frac{\textcolor{orange}{p_6}}{\textcolor{lightgreen}{q_6}}
\end{aligned}
q7p7
\begin{aligned}
\frac{\textcolor{orange}{p_7}}{\textcolor{lightgreen}{q_7}}
\end{aligned}
q8p8
\begin{aligned}
\frac{\textcolor{orange}{p_8}}{\textcolor{lightgreen}{q_8}}
\end{aligned}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
q1q2p1q2+p2q1
\begin{aligned}
\frac{\textcolor{orange}{p_1}\textcolor{lightgreen}{q_2} + \textcolor{orange}{p_2}\textcolor{lightgreen}{q_1}}{\textcolor{lightgreen}{q_1q_2}}
\end{aligned}
q3q4p3q4+p4q3
\begin{aligned}
\frac{\textcolor{orange}{p_3}\textcolor{lightgreen}{q_4} + \textcolor{orange}{p_4}\textcolor{lightgreen}{q_3}}{\textcolor{lightgreen}{q_3q_4}}
\end{aligned}
q5q6p5q6+p6q5
\begin{aligned}
\frac{\textcolor{orange}{p_5}\textcolor{lightgreen}{q_6} + \textcolor{orange}{p_6}\textcolor{lightgreen}{q_5}}{\textcolor{lightgreen}{q_5q_6}}
\end{aligned}
q7q8p7q8+p8q7
\begin{aligned}
\frac{\textcolor{orange}{p_7}\textcolor{lightgreen}{q_8} + \textcolor{orange}{p_8}\textcolor{lightgreen}{q_7}}{\textcolor{lightgreen}{q_7q_8}}
\end{aligned}
q1q2q3q4p1q2q3q4+p2q1q3q4+p3q1q2q4+p4q1q2q3
\begin{aligned}
\frac{\textcolor{orange}{p_1}\textcolor{lightgreen}{q_2q_3q_4} +
\textcolor{orange}{p_2}\textcolor{lightgreen}{q_1q_3q_4} +
\textcolor{orange}{p_3}\textcolor{lightgreen}{q_1q_2q_4} +
\textcolor{orange}{p_4}\textcolor{lightgreen}{q_1q_2q_3}}{
\textcolor{lightgreen}{q_1q_2q_3q_4}}
\end{aligned}
q5q6q7q8p5q6q7q8+p6q5q7q8+p7q5q6q8+p8q5q6q7
\begin{aligned}
\frac{\textcolor{orange}{p_5}\textcolor{lightgreen}{q_6q_7q_8} +
\textcolor{orange}{p_6}\textcolor{lightgreen}{q_5q_7q_8} +
\textcolor{orange}{p_7}\textcolor{lightgreen}{q_5q_6q_8} +
\textcolor{orange}{p_8}\textcolor{lightgreen}{q_5q_6q_7}}{
\textcolor{lightgreen}{q_5q_6q_7q_8}}
\end{aligned}
q1q2q3q4q5q6q7q8p1q2q3…q8+p2q1q3…q8+p3q1q2…q8+p4q1q2…q8+p5q1q2…q8+p6q1q2…q8+p7q1q2…q8+p8q1q2…q8
\begin{aligned}
\frac{
\textcolor{orange}{p_1}\textcolor{lightgreen}{q_2q_3\dots q_8} +
\textcolor{orange}{p_2}\textcolor{lightgreen}{q_1q_3 \dots q_8} +
\textcolor{orange}{p_3}\textcolor{lightgreen}{q_1q_2 \dots q_8} +
\textcolor{orange}{p_4}\textcolor{lightgreen}{q_1q_2 \dots q_8} +
\textcolor{orange}{p_5}\textcolor{lightgreen}{q_1q_2\dots q_8} +
\textcolor{orange}{p_6}\textcolor{lightgreen}{q_1q_2 \dots q_8} +
\textcolor{orange}{p_7}\textcolor{lightgreen}{q_1q_2 \dots q_8} +
\textcolor{orange}{p_8}\textcolor{lightgreen}{q_1q_2 \dots q_8}
}{
\textcolor{lightgreen}{q_1q_2q_3q_4q_5q_6q_7q_8}}
\end{aligned}
23
2^3
22
2^2
21
2^1
GKR Circuit
∗
*
∗
*
∗
*
∗
*
∗
*
∗
*
∗
*
∗
*
(a1b1)
(a_1\quad b_1)
(a2b2)
(a_2\quad b_2)
(a3b3)
(a_3\quad b_3)
(a4b4)
(a_4\quad b_4)
(a5b5)
(a_5\quad b_5)
(a6b6)
(a_6\quad b_6)
(a7b7)
(a_7\quad b_7)
(a8b8)
(a_8\quad b_8)
∗
*
∗
*
∗
*
∗
*
∗
*
∗
*
∗
*
+/× gates
∗
*
:((p1,q1), (p2,q2))⟶(p1q2+p2q1)
: \left(
(\textcolor{orange}{p_1}, \textcolor{lightgreen}{q_1}), \
(\textcolor{orange}{p_2}, \textcolor{lightgreen}{q_2})
\right)
\longrightarrow
(\textcolor{orange}{p_1}\textcolor{lightgreen}{q_2} + \textcolor{orange}{p_2}\textcolor{lightgreen}{q_1})
⟵numerator
\longleftarrow \text{numerator}
#
\#
:((p1,q1), (p2,q2))⟶(q2q1)
: \left(
(\textcolor{orange}{p_1}, \textcolor{lightgreen}{q_1}), \
(\textcolor{orange}{p_2}, \textcolor{lightgreen}{q_2})
\right)
\longrightarrow
(\textcolor{lightgreen}{q_2}\textcolor{lightgreen}{q_1})
⟵denominator
\longleftarrow \text{denominator}
Recall logUp
lhs== (β+f1)(β+f2)(β+f3)…(β+f18)(β+f19) i=1∏n(β+fi)
\begin{aligned}
\textsf{lhs} =&\ (\textcolor{brown}{\beta} + f_1)(\textcolor{brown}{\beta} + f_2)(\textcolor{brown}{\beta} + f_3)
\dots(\textcolor{brown}{\beta} + f_{18})(\textcolor{brown}{\beta} + f_{19}) \\
=&\ \prod_{i=1}^{n} (\textcolor{brown}{\beta} + f_i)
\end{aligned}
rhs== (β+t1)4(β+t2)2(β+t3)4(β+t4)2(β+t5)1(β+t6)3(β+t7)3 i=1∏N(β+ti)mi
\begin{aligned}
\textsf{rhs} =&\ (\textcolor{brown}{\beta} + t_1)^4
(\textcolor{brown}{\beta} + t_2)^2
(\textcolor{brown}{\beta} + t_3)^4
(\textcolor{brown}{\beta} + t_4)^2
(\textcolor{brown}{\beta} + t_5)^1
(\textcolor{brown}{\beta} + t_6)^3
(\textcolor{brown}{\beta} + t_7)^3\\
=&\ \prod_{i=1}^{N} (\textcolor{brown}{\beta} + t_i)^{m_i}
\end{aligned}
⟹∏j=1n(β+fj)=∏i=1N(β+ti)mi
\implies
\prod_{j=1}^{n} (\textcolor{brown}{\beta} + f_j) =
\prod_{i=1}^{N} (\textcolor{brown}{\beta} + t_i)^{m_i}
⟹∑j=1nlog(β+fj)=∑i=1Nmi⋅log(β+ti)
\implies
\sum_{j=1}^{n} \text{log}(\textcolor{brown}{\beta} + f_j) =
\sum_{i=1}^{N} m_i \cdot \text{log}(\textcolor{brown}{\beta} + t_i)
⟹j=1∑n(β+fj)1=i=1∑N(β+ti)mi
\implies
\begin{aligned}
\sum_{j=1}^{n} \frac{1}{(\textcolor{brown}{\beta} + f_j)} =
\sum_{i=1}^{N} \frac{m_i}{(\textcolor{brown}{\beta} + t_i)}
\end{aligned}
\underbrace{\hspace{13.4cm}}_{}
\underbrace{\hspace{4.9cm}}_{}
Table
Values
(β+t1)
(\textcolor{brown}{\beta} + t_1)
(β+t2)
(\textcolor{brown}{\beta} + t_2)
(β+t3)
(\textcolor{brown}{\beta} + t_3)
(β+t4)
(\textcolor{brown}{\beta} + t_4)
(β+t5)
(\textcolor{brown}{\beta} + t_5)
(β+t6)
(\textcolor{brown}{\beta} + t_6)
(β+t7)
(\textcolor{brown}{\beta} + t_7)
(β+f1)
(\textcolor{brown}{\beta} + f_1)
(β+f2)
(\textcolor{brown}{\beta} + f_2)
(β+f3)
(\textcolor{brown}{\beta} + f_3)
(β+f4)
(\textcolor{brown}{\beta} + f_4)
(β+f5)
(\textcolor{brown}{\beta} + f_5)
(β+f6)
(\textcolor{brown}{\beta} + f_6)
(β+f7)
(\textcolor{brown}{\beta} + f_7)
(β+f8)
(\textcolor{brown}{\beta} + f_8)
(β+f9)
(\textcolor{brown}{\beta} + f_9)
(β+f10)
(\textcolor{brown}{\beta} + f_{10})
(β+f11)
(\textcolor{brown}{\beta} + f_{11})
(β+f12)
(\textcolor{brown}{\beta} + f_{12})
(β+f13)
(\textcolor{brown}{\beta} + f_{13})
(β+f14)
(\textcolor{brown}{\beta} + f_{14})
(β+f15)
(\textcolor{brown}{\beta} + f_{15})
(β+f16)
(\textcolor{brown}{\beta} + f_{16})
(β+f17)
(\textcolor{brown}{\beta} + f_{17})
(β+f18)
(\textcolor{brown}{\beta} + f_{18})
(β+f19)
(\textcolor{brown}{\beta} + f_{19})
Recall logUp
⟹j=1∑n(β+fj)1=i=1∑N(β+ti)mi
\implies
\begin{aligned}
\sum_{j=1}^{n} \frac{1}{(\textcolor{brown}{\beta} + f_j)} =
\sum_{i=1}^{N} \frac{m_i}{(\textcolor{brown}{\beta} + t_i)}
\end{aligned}
Table
Values
(β+t1)
(\textcolor{brown}{\beta} + t_1)
(β+t2)
(\textcolor{brown}{\beta} + t_2)
(β+t3)
(\textcolor{brown}{\beta} + t_3)
(β+t4)
(\textcolor{brown}{\beta} + t_4)
(β+t5)
(\textcolor{brown}{\beta} + t_5)
(β+t6)
(\textcolor{brown}{\beta} + t_6)
(β+t7)
(\textcolor{brown}{\beta} + t_7)
(β+f1)
(\textcolor{brown}{\beta} + f_1)
(β+f2)
(\textcolor{brown}{\beta} + f_2)
(β+f3)
(\textcolor{brown}{\beta} + f_3)
(β+f4)
(\textcolor{brown}{\beta} + f_4)
(β+f5)
(\textcolor{brown}{\beta} + f_5)
(β+f6)
(\textcolor{brown}{\beta} + f_6)
(β+f7)
(\textcolor{brown}{\beta} + f_7)
(β+f8)
(\textcolor{brown}{\beta} + f_8)
(β+f9)
(\textcolor{brown}{\beta} + f_9)
(β+f10)
(\textcolor{brown}{\beta} + f_{10})
(β+f11)
(\textcolor{brown}{\beta} + f_{11})
(β+f12)
(\textcolor{brown}{\beta} + f_{12})
(β+f13)
(\textcolor{brown}{\beta} + f_{13})
(β+f14)
(\textcolor{brown}{\beta} + f_{14})
(β+f15)
(\textcolor{brown}{\beta} + f_{15})
(β+f16)
(\textcolor{brown}{\beta} + f_{16})
(β+f17)
(\textcolor{brown}{\beta} + f_{17})
(β+f18)
(\textcolor{brown}{\beta} + f_{18})
(β+f19)
(\textcolor{brown}{\beta} + f_{19})
- In most cases: N≫n, so we can write:
⟹j=1∑n(β+fj)1=j=1,mij=0∑n(β+tij)mij
\implies
\begin{aligned}
\sum_{j=1}^{n} \frac{1}{(\textcolor{brown}{\beta} + f_j)} =
\sum_{j=1, m_{i_j} \neq 0}^{n} \frac{m_{i_j}}{(\textcolor{brown}{\beta} + t_{i_j})}
\end{aligned}
⟹x∈Hn∑((β+t(x))m(x)−(β+f(x))1)=0
\implies
\begin{aligned}
\sum_{\vec{x}\in H_n} \left( \frac{m(\vec{x})}{(\textcolor{brown}{\beta} + t(\vec{x}))} - \frac{1}{(\textcolor{brown}{\beta} + f(\vec{x}))} \right) = 0
\end{aligned}
⟹x∈Hn∑(m(x)h(x)−h1(x))=0
\implies
\begin{aligned}
\sum_{\vec{x}\in H_n} \left( m(\vec{x})\textcolor{red}{h(\vec{x})} - \textcolor{red}{h_1(\vec{x})} \right) = 0
\end{aligned}
- Prover has to send helpers h(X),h1(X)
Recall logUp
Table
Values
(β+t1)
(\textcolor{brown}{\beta} + t_1)
(β+t2)
(\textcolor{brown}{\beta} + t_2)
(β+t3)
(\textcolor{brown}{\beta} + t_3)
(β+t4)
(\textcolor{brown}{\beta} + t_4)
(β+t5)
(\textcolor{brown}{\beta} + t_5)
(β+t6)
(\textcolor{brown}{\beta} + t_6)
(β+t7)
(\textcolor{brown}{\beta} + t_7)
(β+f1)
(\textcolor{brown}{\beta} + f_1)
(β+f2)
(\textcolor{brown}{\beta} + f_2)
(β+f3)
(\textcolor{brown}{\beta} + f_3)
(β+f4)
(\textcolor{brown}{\beta} + f_4)
(β+f5)
(\textcolor{brown}{\beta} + f_5)
(β+f6)
(\textcolor{brown}{\beta} + f_6)
(β+f7)
(\textcolor{brown}{\beta} + f_7)
(β+f8)
(\textcolor{brown}{\beta} + f_8)
(β+f9)
(\textcolor{brown}{\beta} + f_9)
(β+f10)
(\textcolor{brown}{\beta} + f_{10})
(β+f11)
(\textcolor{brown}{\beta} + f_{11})
(β+f12)
(\textcolor{brown}{\beta} + f_{12})
(β+f13)
(\textcolor{brown}{\beta} + f_{13})
(β+f14)
(\textcolor{brown}{\beta} + f_{14})
(β+f15)
(\textcolor{brown}{\beta} + f_{15})
(β+f16)
(\textcolor{brown}{\beta} + f_{16})
(β+f17)
(\textcolor{brown}{\beta} + f_{17})
(β+f18)
(\textcolor{brown}{\beta} + f_{18})
(β+f19)
(\textcolor{brown}{\beta} + f_{19})
⟹x∈Hn∑(m(x)h(x)−h1(x))=0
\implies
\begin{aligned}
\sum_{\vec{x}\in H_n} \left( m(\vec{x})\textcolor{red}{h(\vec{x})} - \textcolor{red}{h_1(\vec{x})} \right) = 0
\end{aligned}
Recall logUp
Table
(β+t1)
(\textcolor{brown}{\beta} + t_1)
(β+t2)
(\textcolor{brown}{\beta} + t_2)
(β+t3)
(\textcolor{brown}{\beta} + t_3)
(β+t4)
(\textcolor{brown}{\beta} + t_4)
(β+t5)
(\textcolor{brown}{\beta} + t_5)
(β+t6)
(\textcolor{brown}{\beta} + t_6)
(β+t7)
(\textcolor{brown}{\beta} + t_7)
(β+f1)
(\textcolor{brown}{\beta} + f_1)
(β+f2)
(\textcolor{brown}{\beta} + f_2)
(β+f3)
(\textcolor{brown}{\beta} + f_3)
(β+f4)
(\textcolor{brown}{\beta} + f_4)
(β+f5)
(\textcolor{brown}{\beta} + f_5)
(β+f6)
(\textcolor{brown}{\beta} + f_6)
(β+f7)
(\textcolor{brown}{\beta} + f_7)
(β+f8)
(\textcolor{brown}{\beta} + f_8)
(β+f9)
(\textcolor{brown}{\beta} + f_9)
(β+f10)
(\textcolor{brown}{\beta} + f_{10})
(β+f11)
(\textcolor{brown}{\beta} + f_{11})
(β+f12)
(\textcolor{brown}{\beta} + f_{12})
(β+f13)
(\textcolor{brown}{\beta} + f_{13})
(β+f14)
(\textcolor{brown}{\beta} + f_{14})
(β+f15)
(\textcolor{brown}{\beta} + f_{15})
(β+f16)
(\textcolor{brown}{\beta} + f_{16})
(β+f17)
(\textcolor{brown}{\beta} + f_{17})
(β+f18)
(\textcolor{brown}{\beta} + f_{18})
(β+f19)
(\textcolor{brown}{\beta} + f_{19})
⟹x∈Hn∑(m(x)h(x)−h1(x))=0
\implies
\begin{aligned}
\sum_{\vec{x}\in H_n} \left( m(\vec{x})\textcolor{red}{h(\vec{x})} - \textcolor{red}{h_1(\vec{x})} \right) = 0
\end{aligned}
Values
h1(x)+h2(x)
\textcolor{red}{h_1(\vec{x})} +
\textcolor{red}{h_2(\vec{x})}
h1(x)+h2(x)+h3(x)
\textcolor{red}{h_1(\vec{x})} +
\textcolor{red}{h_2(\vec{x})} +
\textcolor{red}{h_3(\vec{x})}
h1(x)+h2(x)+h3(x)+h4(x)
\textcolor{red}{h_1(\vec{x})}+
\textcolor{red}{h_2(\vec{x})}+
\textcolor{red}{h_3(\vec{x})}+
\textcolor{red}{h_4(\vec{x})}
⟹x∈Hn∑(m(x)h(x)−j=1∑Mhj(x))=0
\implies
\begin{aligned}
\sum_{\vec{x}\in H_n} \left( m(\vec{x})\textcolor{red}{h(\vec{x})} - \textcolor{red}{\sum_{j=1}^{M}h_j(\vec{x})} \right) = 0
\end{aligned}
- Prover has to send helpers m+1 helpers
- Idea: Use GKR to do rational sumcheck!
GKR-logUp
⟹x∈Hn∑((β+t(x))m(x)−j=1∑M(β+fj(x))1)=0
\implies
\begin{aligned}
\sum_{\vec{x}\in H_n} \left( \frac{m(\vec{x})}{(\textcolor{brown}{\beta} + t(\vec{x}))} - \sum_{j=1}^{M} \frac{1}{(\textcolor{brown}{\beta} + f_j(\vec{x}))} \right) = 0
\end{aligned}
⟹(β+t(0))m(0)+(β+t(1))m(1)+⋯+(β+t(2n−1))m(2n−1)+
\implies
\begin{aligned}
\frac{m(0)}{(\textcolor{brown}{\beta} + t(0))} +
\frac{m(1)}{(\textcolor{brown}{\beta} + t(1))} +
\dots +
\frac{m(2^n-1)}{(\textcolor{brown}{\beta} + t(2^n-1))} +
\end{aligned}
(β+f1(0))−1+(β+f1(1))−1+⋯+(β+f1(2n−1))−1+
\begin{aligned}
\frac{-1}{(\textcolor{brown}{\beta} + f_1(0))} +
\frac{-1}{(\textcolor{brown}{\beta} + f_1(1))} +
\dots +
\frac{-1}{(\textcolor{brown}{\beta} + f_1(2^n-1))} +
\end{aligned}
(β+f2(0))−1+(β+f2(1))−1+⋯+(β+f2(2n−1))−1+
\begin{aligned}
\frac{-1}{(\textcolor{brown}{\beta} + f_2(0))} +
\frac{-1}{(\textcolor{brown}{\beta} + f_2(1))} +
\dots +
\frac{-1}{(\textcolor{brown}{\beta} + f_2(2^n-1))} +
\end{aligned}
(β+fM(0))−1+(β+fM(1))−1+⋯+(β+fM(2n−1))−1.
\begin{aligned}
\frac{-1}{(\textcolor{brown}{\beta} + f_M(0))} +
\frac{-1}{(\textcolor{brown}{\beta} + f_M(1))} +
\dots +
\frac{-1}{(\textcolor{brown}{\beta} + f_M(2^n-1))}.
\end{aligned}
⋮
\vdots
\underbrace{\hspace{7cm}}{}
2n(1+M) terms
2^n(1 + M) \text{ terms}
2n+k rational sumcheck
2^{n+k} \text{ rational sumcheck}
⟹
\implies
Summary
- Prover has only 1 extra commitment: the multiplicities polynomial m(x)
- Cost of rational sumcheck is O(N⋅M) due to GKR trick
- Possible research directions:
- Where else do we use rational sumcheck? Marlin,…
- Power circuits (https://eprint.iacr.org/2023/1611.pdf)
- Plonk permutation argument with GKR-logUp
inclusion list committee
\textsf{inclusion list committee}
(txs1, S1)
\left(\textsf{txs}_1, \ S_1 \right)
(txs2, S2)
\left(\textsf{txs}_2, \ S_2 \right)
(txs3, S3)
\left(\textsf{txs}_3, \ S_3 \right)
(txs4, S4)
\left(\textsf{txs}_4, \ S_4 \right)
(txs5, S5)
\left(\textsf{txs}_5, \ S_5 \right)
(txs6, S6)
\left(\textsf{txs}_6, \ S_6 \right)
(txs7, S7)
\left(\textsf{txs}_7, \ S_7 \right)
(txsn, Sn)
\left(\textsf{txs}_n, \ S_n \right)
⋮
\vdots
proposer
\textsf{proposer}
txs1txs2txs3txs4txs5txs6txs7⋮txsn
\begin{bmatrix}
\textsf{txs}_1 \\
\textsf{txs}_2 \\
\textsf{txs}_3 \\
\textsf{txs}_4 \\
\textsf{txs}_5 \\
\textsf{txs}_6 \\
\textsf{txs}_7 \\
\vdots \\
\textsf{txs}_n
\end{bmatrix}
inclusion list committee
\textsf{inclusion list committee}
(txs1, S1)
\left(\textsf{txs}_1, \ S_1 \right)
(txs2, S2)
\left(\textsf{txs}_2, \ S_2 \right)
(txs3, S3)
\left(\textsf{txs}_3, \ S_3 \right)
(txs4, S4)
\left(\textsf{txs}_4, \ S_4 \right)
(txs5, S5)
\left(\textsf{txs}_5, \ S_5 \right)
(txs6, S6)
\left(\textsf{txs}_6, \ S_6 \right)
(txs7, S7)
\left(\textsf{txs}_7, \ S_7 \right)
(txsn, Sn)
\left(\textsf{txs}_n, \ S_n \right)
⋮
\vdots
proposer
\textsf{proposer}
txs1txs2txs3txs4txs5txs6txs7⋮txsn
\begin{bmatrix}
\textsf{txs}_1 \\
\textsf{txs}_2 \\
\textsf{txs}_3 \\
\textsf{txs}_4 \\
\textsf{txs}_5 \\
\textsf{txs}_6 \\
\textsf{txs}_7 \\
\vdots \\
\textsf{txs}_n
\end{bmatrix}
proposer
\textsf{proposer}
txs1txs2txs3txs4txs5txs6txs7⋮txsn
\begin{bmatrix}
\textsf{txs}_1 \\
\textsf{txs}_2 \\
\textsf{txs}_3 \\
\textsf{txs}_4 \\
\textsf{txs}_5 \\
\textsf{txs}_6 \\
\textsf{txs}_7 \\
\vdots \\
\textsf{txs}_n
\end{bmatrix}
inclusion list committee
\textsf{inclusion list committee}
(txs2, S2)
\left(\textsf{txs}_2, \ S_2 \right)
(txs3, S3)
\left(\textsf{txs}_3, \ S_3 \right)
(txs4, S4)
\left(\textsf{txs}_4, \ S_4 \right)
(txs5, S5)
\left(\textsf{txs}_5, \ S_5 \right)
(txs6, S6)
\left(\textsf{txs}_6, \ S_6 \right)
(txs7, S7)
\left(\textsf{txs}_7, \ S_7 \right)
(txsn, Sn)
\left(\textsf{txs}_n, \ S_n \right)
⋮
\vdots
(txs1, S1)
\left(\textsf{txs}_1, \ S_1 \right)
K1
K_1
proposer
\textsf{proposer}
txs1txs2txs3txs4txs5txs6txs7⋮txsn
\begin{bmatrix}
\textsf{txs}_1 \\
\textsf{txs}_2 \\
\textsf{txs}_3 \\
\textsf{txs}_4 \\
\textsf{txs}_5 \\
\textsf{txs}_6 \\
\textsf{txs}_7 \\
\vdots \\
\textsf{txs}_n
\end{bmatrix}
inclusion list committee
\textsf{inclusion list committee}
(txs2, S2)
\left(\textsf{txs}_2, \ S_2 \right)
(txs3, S3)
\left(\textsf{txs}_3, \ S_3 \right)
(txs4, S4)
\left(\textsf{txs}_4, \ S_4 \right)
(txs5, S5)
\left(\textsf{txs}_5, \ S_5 \right)
(txs6, S6)
\left(\textsf{txs}_6, \ S_6 \right)
(txs7, S7)
\left(\textsf{txs}_7, \ S_7 \right)
(txsn, Sn)
\left(\textsf{txs}_n, \ S_n \right)
⋮
\vdots
(txs1, S1)
\left(\textsf{txs}_1, \ S_1 \right)
K1
K_1
proposer
\textsf{proposer}
txs1txs2txs3txs4txs5txs6txs7⋮txsn
\begin{bmatrix}
\textsf{txs}_1 \\
\textsf{txs}_2 \\
\textsf{txs}_3 \\
\textsf{txs}_4 \\
\textsf{txs}_5 \\
\textsf{txs}_6 \\
\textsf{txs}_7 \\
\vdots \\
\textsf{txs}_n
\end{bmatrix}
inclusion list committee
\textsf{inclusion list committee}
(txs2, S2)
\left(\textsf{txs}_2, \ S_2 \right)
(txs3, S3)
\left(\textsf{txs}_3, \ S_3 \right)
(txs4, S4)
\left(\textsf{txs}_4, \ S_4 \right)
(txs5, S5)
\left(\textsf{txs}_5, \ S_5 \right)
(txs6, S6)
\left(\textsf{txs}_6, \ S_6 \right)
(txs7, S7)
\left(\textsf{txs}_7, \ S_7 \right)
(txsn, Sn)
\left(\textsf{txs}_n, \ S_n \right)
⋮
\vdots
(txs1, S1)
\left(\textsf{txs}_1, \ S_1 \right)
K1
K_1
K2
K_2
proposer
\textsf{proposer}
txs1txs2txs3txs4txs5txs6txs7⋮txsn
\begin{bmatrix}
\textsf{txs}_1 \\
\textsf{txs}_2 \\
\textsf{txs}_3 \\
\textsf{txs}_4 \\
\textsf{txs}_5 \\
\textsf{txs}_6 \\
\textsf{txs}_7 \\
\vdots \\
\textsf{txs}_n
\end{bmatrix}
inclusion list committee
\textsf{inclusion list committee}
(txs2, S2)
\left(\textsf{txs}_2, \ S_2 \right)
(txs3, S3)
\left(\textsf{txs}_3, \ S_3 \right)
(txs4, S4)
\left(\textsf{txs}_4, \ S_4 \right)
(txs5, S5)
\left(\textsf{txs}_5, \ S_5 \right)
(txs6, S6)
\left(\textsf{txs}_6, \ S_6 \right)
(txs7, S7)
\left(\textsf{txs}_7, \ S_7 \right)
(txsn, Sn)
\left(\textsf{txs}_n, \ S_n \right)
⋮
\vdots
(txs1, S1)
\left(\textsf{txs}_1, \ S_1 \right)
K1
K_1
K2
K_2
⋮
\vdots
proposer
\textsf{proposer}
txs1txs2txs3txs4txs5txs6txs7⋮txsn
\begin{bmatrix}
\textsf{txs}_1 \\
\textsf{txs}_2 \\
\textsf{txs}_3 \\
\textsf{txs}_4 \\
\textsf{txs}_5 \\
\textsf{txs}_6 \\
\textsf{txs}_7 \\
\vdots \\
\textsf{txs}_n
\end{bmatrix}
inclusion list committee
\textsf{inclusion list committee}
(txs2, R2)
\left(\textsf{txs}_2, \ R_2 \right)
(txs3, R3)
\left(\textsf{txs}_3, \ R_3 \right)
(txs4, R4)
\left(\textsf{txs}_4, \ R_4 \right)
(txs5, R5)
\left(\textsf{txs}_5, \ R_5 \right)
(txs6, R6)
\left(\textsf{txs}_6, \ R_6 \right)
(txs7, R7)
\left(\textsf{txs}_7, \ R_7 \right)
(txsn, Rn)
\left(\textsf{txs}_n, \ R_n \right)
⋮
\vdots
(txs1, R1)
\left(\textsf{txs}_1, \ R_1 \right)
K1
K_1
K2
K_2
⋮
\vdots
K3
K_3
K4
K_4
K5
K_5
K6
K_6
K7
K_7
Kn
K_n
GKR-based Lookups
GKR-based Lookups
By Suyash Bagad
GKR-based Lookups
- 173
