GKR-based Lookups

Rational Sumcheck

\begin{aligned} c = \sum_{x\in H} \frac{\textcolor{lightgreen}{p(X)}}{\textcolor{orange}{q(X)}} \end{aligned}

Polynomial functions:
- \(f(X) = c_0 + c_1X + \dots c_nX^n\)
Rational functions:
- \(r(X) = \frac{\textcolor{lightgreen}{p(X)}}{\textcolor{orange}{q(X)}}\)
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

\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:

\begin{aligned} f(X) = X\textcolor{red}{g(X)} + \frac{\sigma}{|H|} \end{aligned}

\begin{aligned} f(X) \cdot q(X) - p(X) = \textcolor{red}{h(X)}v_H(X) \end{aligned}

\(\longleftarrow\) Classic Aurora sumcheck

\(\longleftarrow\) Quotient check

Prover cost: \(\mathcal{O}(N\cdot \text{log}(N))\) where \(N = |H|\)
Can we do better?

Sumcheck

x_1

x_2

x_3

x_1

x_2

x_3

Round polynomial: \(r_i(X) = \textcolor{lightgreen}{f_{\text{lo}}}(1-X) + \textcolor{red}{f_{\text{hi}}}X\)
Challenge: \(\alpha_i = H(r_i)\)

Sumcheck

x_1

x_2

x_3

x_1

x_2

Round polynomial: \(r_i(X) = \textcolor{lightgreen}{f_{\text{lo}}}(1-X) + \textcolor{red}{f_{\text{hi}}}X\)
Challenge: \(\alpha_i = H(r_i)\)

Sumcheck

x_1

x_2

x_3

x_1

x_2

x_1

x_2

Round polynomial: \(r_i(X) = \textcolor{lightgreen}{f_{\text{lo}}}(1-X) + \textcolor{red}{f_{\text{hi}}}X\)
Challenge: \(\alpha_i = H(r_i)\)

Sumcheck

x_1

x_2

x_3

x_1

x_2

x_1

Round polynomial: \(r_i(X) = \textcolor{lightgreen}{f_{\text{lo}}}(1-X) + \textcolor{red}{f_{\text{hi}}}X\)
Challenge: \(\alpha_i = H(r_i)\)

Sumcheck

x_1

x_2

x_3

x_1

x_2

x_1

Round polynomial: \(r_i(X) = \textcolor{lightgreen}{f_{\text{lo}}}(1-X) + \textcolor{red}{f_{\text{hi}}}X\)
Challenge: \(\alpha_i = H(r_i)\)

Sumcheck

x_1

x_2

x_3

x_1

x_2

x_1

Round polynomial: \(r_i(X) = \textcolor{lightgreen}{f_{\text{lo}}}(1-X) + \textcolor{red}{f_{\text{hi}}}X\)
Challenge: \(\alpha_i = H(r_i)\)

2^3

2^2

2^1

Dimensionality reduction!

Rethinking Rational Sumcheck

\begin{aligned} \frac{\textcolor{orange}{p_1}}{\textcolor{lightgreen}{q_1}} \end{aligned}

\begin{aligned} \frac{\textcolor{orange}{p_2}}{\textcolor{lightgreen}{q_2}} \end{aligned}

\begin{aligned} \frac{\textcolor{orange}{p_3}}{\textcolor{lightgreen}{q_3}} \end{aligned}

\begin{aligned} \frac{\textcolor{orange}{p_4}}{\textcolor{lightgreen}{q_4}} \end{aligned}

\begin{aligned} \frac{\textcolor{orange}{p_5}}{\textcolor{lightgreen}{q_5}} \end{aligned}

\begin{aligned} \frac{\textcolor{orange}{p_6}}{\textcolor{lightgreen}{q_6}} \end{aligned}

\begin{aligned} \frac{\textcolor{orange}{p_7}}{\textcolor{lightgreen}{q_7}} \end{aligned}

\begin{aligned} \frac{\textcolor{orange}{p_8}}{\textcolor{lightgreen}{q_8}} \end{aligned}

\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}

\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}

\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}

\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}

\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}

\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}

\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}

2^3

2^2

2^1

GKR Circuit

(a_1\quad b_1)

(a_2\quad b_2)

(a_3\quad b_3)

(a_4\quad b_4)

(a_5\quad b_5)

(a_6\quad b_6)

(a_7\quad b_7)

(a_8\quad b_8)

\(+/\times\) gates

: \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})

\longleftarrow \text{numerator}

: \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})

\longleftarrow \text{denominator}

Recall logUp

\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}

\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}

\implies \prod_{j=1}^{n} (\textcolor{brown}{\beta} + f_j) = \prod_{i=1}^{N} (\textcolor{brown}{\beta} + t_i)^{m_i}

\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)

\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

(\textcolor{brown}{\beta} + t_1)

(\textcolor{brown}{\beta} + t_2)

(\textcolor{brown}{\beta} + t_3)

(\textcolor{brown}{\beta} + t_4)

(\textcolor{brown}{\beta} + t_5)

(\textcolor{brown}{\beta} + t_6)

(\textcolor{brown}{\beta} + t_7)

(\textcolor{brown}{\beta} + f_1)

(\textcolor{brown}{\beta} + f_2)

(\textcolor{brown}{\beta} + f_3)

(\textcolor{brown}{\beta} + f_4)

(\textcolor{brown}{\beta} + f_5)

(\textcolor{brown}{\beta} + f_6)

(\textcolor{brown}{\beta} + f_7)

(\textcolor{brown}{\beta} + f_8)

(\textcolor{brown}{\beta} + f_9)

(\textcolor{brown}{\beta} + f_{10})

(\textcolor{brown}{\beta} + f_{11})

(\textcolor{brown}{\beta} + f_{12})

(\textcolor{brown}{\beta} + f_{13})

(\textcolor{brown}{\beta} + f_{14})

(\textcolor{brown}{\beta} + f_{15})

(\textcolor{brown}{\beta} + f_{16})

(\textcolor{brown}{\beta} + f_{17})

(\textcolor{brown}{\beta} + f_{18})

(\textcolor{brown}{\beta} + f_{19})

Recall logUp

\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

(\textcolor{brown}{\beta} + t_1)

(\textcolor{brown}{\beta} + t_2)

(\textcolor{brown}{\beta} + t_3)

(\textcolor{brown}{\beta} + t_4)

(\textcolor{brown}{\beta} + t_5)

(\textcolor{brown}{\beta} + t_6)

(\textcolor{brown}{\beta} + t_7)

(\textcolor{brown}{\beta} + f_1)

(\textcolor{brown}{\beta} + f_2)

(\textcolor{brown}{\beta} + f_3)

(\textcolor{brown}{\beta} + f_4)

(\textcolor{brown}{\beta} + f_5)

(\textcolor{brown}{\beta} + f_6)

(\textcolor{brown}{\beta} + f_7)

(\textcolor{brown}{\beta} + f_8)

(\textcolor{brown}{\beta} + f_9)

(\textcolor{brown}{\beta} + f_{10})

(\textcolor{brown}{\beta} + f_{11})

(\textcolor{brown}{\beta} + f_{12})

(\textcolor{brown}{\beta} + f_{13})

(\textcolor{brown}{\beta} + f_{14})

(\textcolor{brown}{\beta} + f_{15})

(\textcolor{brown}{\beta} + f_{16})

(\textcolor{brown}{\beta} + f_{17})

(\textcolor{brown}{\beta} + f_{18})

(\textcolor{brown}{\beta} + f_{19})

In most cases: \(N \gg n\), so we can write:

\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}

\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}

\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 \(h(X), h_1(X)\)

Recall logUp

Table

Values

(\textcolor{brown}{\beta} + t_1)

(\textcolor{brown}{\beta} + t_2)

(\textcolor{brown}{\beta} + t_3)

(\textcolor{brown}{\beta} + t_4)

(\textcolor{brown}{\beta} + t_5)

(\textcolor{brown}{\beta} + t_6)

(\textcolor{brown}{\beta} + t_7)

(\textcolor{brown}{\beta} + f_1)

(\textcolor{brown}{\beta} + f_2)

(\textcolor{brown}{\beta} + f_3)

(\textcolor{brown}{\beta} + f_4)

(\textcolor{brown}{\beta} + f_5)

(\textcolor{brown}{\beta} + f_6)

(\textcolor{brown}{\beta} + f_7)

(\textcolor{brown}{\beta} + f_8)

(\textcolor{brown}{\beta} + f_9)

(\textcolor{brown}{\beta} + f_{10})

(\textcolor{brown}{\beta} + f_{11})

(\textcolor{brown}{\beta} + f_{12})

(\textcolor{brown}{\beta} + f_{13})

(\textcolor{brown}{\beta} + f_{14})

(\textcolor{brown}{\beta} + f_{15})

(\textcolor{brown}{\beta} + f_{16})

(\textcolor{brown}{\beta} + f_{17})

(\textcolor{brown}{\beta} + f_{18})

(\textcolor{brown}{\beta} + f_{19})

\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

(\textcolor{brown}{\beta} + t_1)

(\textcolor{brown}{\beta} + t_2)

(\textcolor{brown}{\beta} + t_3)

(\textcolor{brown}{\beta} + t_4)

(\textcolor{brown}{\beta} + t_5)

(\textcolor{brown}{\beta} + t_6)

(\textcolor{brown}{\beta} + t_7)

(\textcolor{brown}{\beta} + f_1)

(\textcolor{brown}{\beta} + f_2)

(\textcolor{brown}{\beta} + f_3)

(\textcolor{brown}{\beta} + f_4)

(\textcolor{brown}{\beta} + f_5)

(\textcolor{brown}{\beta} + f_6)

(\textcolor{brown}{\beta} + f_7)

(\textcolor{brown}{\beta} + f_8)

(\textcolor{brown}{\beta} + f_9)

(\textcolor{brown}{\beta} + f_{10})

(\textcolor{brown}{\beta} + f_{11})

(\textcolor{brown}{\beta} + f_{12})

(\textcolor{brown}{\beta} + f_{13})

(\textcolor{brown}{\beta} + f_{14})

(\textcolor{brown}{\beta} + f_{15})

(\textcolor{brown}{\beta} + f_{16})

(\textcolor{brown}{\beta} + f_{17})

(\textcolor{brown}{\beta} + f_{18})

(\textcolor{brown}{\beta} + f_{19})

\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

\textcolor{red}{h_1(\vec{x})} + \textcolor{red}{h_2(\vec{x})}

\textcolor{red}{h_1(\vec{x})} + \textcolor{red}{h_2(\vec{x})} + \textcolor{red}{h_3(\vec{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})}

\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

\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}

\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}

\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}

\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}

\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}}{}

2^n(1 + M) \text{ terms}

2^{n+k} \text{ rational sumcheck}

\implies

Summary

Prover has only 1 extra commitment: the multiplicities polynomial \(m(\vec{x})\)
Cost of rational sumcheck is \(\mathcal{O}(N\cdot M)\) due to GKR trick
Possible research directions:
- Where else do we use rational sumcheck? Marlin\(, \dots\)
- Power circuits (https://eprint.iacr.org/2023/1611.pdf)
- Plonk permutation argument with GKR-logUp