Suyash Bagad
Research Team
Karatsuba Trick in
Small Fields Sumcheck
Sumcheck in Small Fields - Algo \(3\)
r_i(0) = \sum_{\mathbf{X}} \textcolor{blue}{a}(\alpha_1, \dots, \alpha_{i-1}, 0, \mathbf{X})\cdot
\textcolor{red}{b}(\alpha_1, \dots, \alpha_{i-1}, 0, \mathbf{X})
r_1(c) = \sum_{\mathbf{X}} \textcolor{blue}{a}(c, \mathbf{X})\cdot
\textcolor{red}{b}(c, \mathbf{X})
= \sum_{\mathbf{X}}
\Big(
\bar{c} \cdot \textcolor{blue}{a}(\textcolor{green}{0}, \mathbf{X}) +
c \cdot \textcolor{blue}{a}(\textcolor{green}{1}, \mathbf{X})
\Big)
\Big(
\bar{c} \cdot \textcolor{red}{b}(\textcolor{green}{0}, \mathbf{X}) +
c \cdot \textcolor{red}{b}(\textcolor{green}{1}, \mathbf{X})
\Big)
= \sum_{\mathbf{X}}
\Big(
\bar{c}^2\cdot \textcolor{blue}{a}(\textcolor{green}{0}, \mathbf{X})\textcolor{red}{b}(\textcolor{green}{0}, \mathbf{X})
+c^2\cdot \textcolor{blue}{a}(\textcolor{green}{1}, \mathbf{X})\textcolor{red}{b}(\textcolor{green}{1}, \mathbf{X})
+\bar{c}c \cdot\textcolor{blue}{a}(\textcolor{green}{0}, \mathbf{X})\textcolor{red}{b}(\textcolor{green}{1}, \mathbf{X})
+\bar{c}c \cdot\textcolor{blue}{a}(\textcolor{green}{1}, \mathbf{X})\textcolor{red}{b}(\textcolor{green}{0}, \mathbf{X})
\Big)
\underbrace{\hspace{3.6cm}}
\underbrace{\hspace{3.6cm}}
\underbrace{\hspace{3.6cm}}
\underbrace{\hspace{3.6cm}}
Pre-compute with \(4\times \frac{n}{2} \ \textsf{bb}\)
+ \bar{c}c \cdot\textcolor{blue}{a}(\textcolor{green}{0}, \mathbf{X})\textcolor{red}{b}(\textcolor{green}{0}, \mathbf{X})
+\bar{c}c \cdot\textcolor{blue}{a}(\textcolor{green}{1}, \mathbf{X})\textcolor{red}{b}(\textcolor{green}{1}, \mathbf{X})
-\bar{c}c \cdot\textcolor{blue}{a}(\textcolor{green}{0}, \mathbf{X})\textcolor{red}{b}(\textcolor{green}{0}, \mathbf{X})
-\bar{c}c \cdot\textcolor{blue}{a}(\textcolor{green}{1}, \mathbf{X})\textcolor{red}{b}(\textcolor{green}{1}, \mathbf{X})
= \sum_{\mathbf{X}}
\Big(
(\bar{c}^2 - \bar{c}c)\cdot \textcolor{blue}{a}(\textcolor{green}{0}, \mathbf{X})\textcolor{red}{b}(\textcolor{green}{0}, \mathbf{X})
+(\bar{c}^2 - \bar{c}c)\cdot \textcolor{blue}{a}(\textcolor{green}{1}, \mathbf{X})\textcolor{red}{b}(\textcolor{green}{1}, \mathbf{X})
+\bar{c}c \cdot \left( \textcolor{blue}{a}(\textcolor{green}{0},\mathbf{X}) + \textcolor{blue}{a}(\textcolor{green}{1}, \mathbf{X}) \right)
\left( \textcolor{red}{b}(\textcolor{green}{0},\mathbf{X}) + \textcolor{red}{b}(\textcolor{green}{1}, \mathbf{X}) \right)
\Big)
\underbrace{\hspace{3.6cm}}
\underbrace{\hspace{3.6cm}}
\underbrace{\hspace{9.6cm}}
Pre-compute with \(3\times \frac{n}{2} \ \textsf{bb}\)
\begin{bmatrix}
0\cdot 0 & 0\cdot 1 \\
1\cdot 0 & 1\cdot 1
\end{bmatrix}
\(\implies\) Karatsuba trick!
Sumcheck in Small Fields - Algo \(3\)
r_i(0) = \sum_{\mathbf{X}} \textcolor{blue}{a}(\alpha_1, \dots, \alpha_{i-1}, 0, \mathbf{X})\cdot
\textcolor{red}{b}(\alpha_1, \dots, \alpha_{i-1}, 0, \mathbf{X})
r_2(c) = \sum_{\mathbf{X}} \textcolor{blue}{a}(\alpha_1, c, \mathbf{X})\cdot
\textcolor{red}{b}(\alpha_1, c, \mathbf{X})
= \sum_{\mathbf{X}}
\Big(
L_{1, \vec{\alpha}_1} \cdot \textcolor{blue}{a}(\textcolor{green}{0}, c, \mathbf{X}) +
L_{2, \vec{\alpha}_1} \cdot \textcolor{blue}{a}(\textcolor{green}{1}, c, \mathbf{X})
\Big)
\Big(
L_{1, \vec{\alpha}_1} \cdot \textcolor{red}{b}(\textcolor{green}{0}, c, \mathbf{X}) +
L_{2, \vec{\alpha}_1} \cdot \textcolor{red}{b}(\textcolor{green}{1}, c, \mathbf{X})
\Big)
= \sum_{\mathbf{X}}
\Big(
L_{1, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{blue}{a}(\textcolor{green}{0, 0}, \mathbf{X}) + c \cdot \textcolor{blue}{a}(\textcolor{green}{0, 1}, \mathbf{X})) +
L_{2, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{blue}{a}(\textcolor{green}{1, 0}, \mathbf{X}) + c \cdot \textcolor{blue}{a}(\textcolor{green}{1, 1}, \mathbf{X}))
\Big)
\Big(
L_{1, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{red}{b}(\textcolor{green}{0, 0}, \mathbf{X}) + c \cdot \textcolor{red}{b}(\textcolor{green}{0, 1}, \mathbf{X})) +
L_{2, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{red}{b}(\textcolor{green}{1, 0}, \mathbf{X}) + c \cdot \textcolor{red}{b}(\textcolor{green}{1, 1}, \mathbf{X}))
\Big)
\(\implies\) Pre-compute with \(4^2 \times \frac{n}{4} \ \textsf{bb}\)
= \sum_{\mathbf{X}}
L_{1, \vec{\alpha}_1}^2
\Big(
(\bar{c}^2 - \bar{c}c)\cdot \textcolor{blue}{a}(\textcolor{green}{0, 0}, \mathbf{X})\textcolor{red}{b}(\textcolor{green}{0, 0}, \mathbf{X})
+(\bar{c}^2 - \bar{c}c)\cdot \textcolor{blue}{a}(\textcolor{green}{0, 1}, \mathbf{X})\textcolor{red}{b}(\textcolor{green}{0, 1}, \mathbf{X})
+\bar{c}c \cdot \left( \textcolor{blue}{a}(\textcolor{green}{0, 0},\mathbf{X}) + \textcolor{blue}{a}(\textcolor{green}{0, 1}, \mathbf{X}) \right)
\left( \textcolor{red}{b}(\textcolor{green}{0, 0},\mathbf{X}) + \textcolor{red}{b}(\textcolor{green}{0, 1}, \mathbf{X}) \right)
\Big)
L_{2, \vec{\alpha}_1}^2
\Big(
(\bar{c}^2 - \bar{c}c)\cdot \textcolor{blue}{a}(\textcolor{green}{1, 0}, \mathbf{X})\textcolor{red}{b}(\textcolor{green}{1, 0}, \mathbf{X})
+(\bar{c}^2 - \bar{c}c)\cdot \textcolor{blue}{a}(\textcolor{green}{1, 1}, \mathbf{X})\textcolor{red}{b}(\textcolor{green}{1, 1}, \mathbf{X})
+\bar{c}c \cdot \left( \textcolor{blue}{a}(\textcolor{green}{1, 0},\mathbf{X}) + \textcolor{blue}{a}(\textcolor{green}{1, 1}, \mathbf{X}) \right)
\left( \textcolor{red}{b}(\textcolor{green}{1, 0},\mathbf{X}) + \textcolor{red}{b}(\textcolor{green}{1, 1}, \mathbf{X}) \right)
\Big)
\Big(
\bar{c}c \cdot \left( \textcolor{blue}{a}(\textcolor{green}{0, 0},\mathbf{X}) + \textcolor{blue}{a}(\textcolor{green}{1, 0}, \mathbf{X}) \right)
\left( \textcolor{red}{b}(\textcolor{green}{0, 0},\mathbf{X}) + \textcolor{red}{b}(\textcolor{green}{1, 0}, \mathbf{X}) \right)
\Big)
\Big(
\bar{c}c \cdot \left( \textcolor{blue}{a}(\textcolor{green}{0, 1},\mathbf{X}) + \textcolor{blue}{a}(\textcolor{green}{1, 1}, \mathbf{X}) \right)
\left( \textcolor{red}{b}(\textcolor{green}{0, 1},\mathbf{X}) + \textcolor{red}{b}(\textcolor{green}{1, 1}, \mathbf{X}) \right)
\Big)
\Big(
\bar{c}c \cdot \left( \textcolor{blue}{a}(\textcolor{green}{0, 1},\mathbf{X}) + \textcolor{blue}{a}(\textcolor{green}{1, 0}, \mathbf{X}) \right)
\left( \textcolor{red}{b}(\textcolor{green}{0, 1},\mathbf{X}) + \textcolor{red}{b}(\textcolor{green}{1, 0}, \mathbf{X}) \right)
\Big)
\Big(
\bar{c}c \cdot \left( \textcolor{blue}{a}(\textcolor{green}{0, 0},\mathbf{X}) + \textcolor{blue}{a}(\textcolor{green}{1, 1}, \mathbf{X}) \right)
\left( \textcolor{red}{b}(\textcolor{green}{0, 0},\mathbf{X}) + \textcolor{red}{b}(\textcolor{green}{1, 1}, \mathbf{X}) \right)
\Big)
Sumcheck in Small Fields - Algo \(3\)
r_i(0) = \sum_{\mathbf{X}} \textcolor{blue}{a}(\alpha_1, \dots, \alpha_{i-1}, 0, \mathbf{X})\cdot
\textcolor{red}{b}(\alpha_1, \dots, \alpha_{i-1}, 0, \mathbf{X})
r_2(c) = \sum_{\mathbf{X}} \textcolor{blue}{a}(\alpha_1, c, \mathbf{X})\cdot
\textcolor{red}{b}(\alpha_1, c, \mathbf{X})
= \sum_{\mathbf{X}}
\Big(
L_{1, \vec{\alpha}_1} \cdot \textcolor{blue}{a}(\textcolor{green}{0}, c, \mathbf{X}) +
L_{2, \vec{\alpha}_1} \cdot \textcolor{blue}{a}(\textcolor{green}{1}, c, \mathbf{X})
\Big)
\Big(
L_{1, \vec{\alpha}_1} \cdot \textcolor{red}{b}(\textcolor{green}{0}, c, \mathbf{X}) +
L_{2, \vec{\alpha}_1} \cdot \textcolor{red}{b}(\textcolor{green}{1}, c, \mathbf{X})
\Big)
= \sum_{\mathbf{X}}
\Big(
L_{1, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{blue}{a}(\textcolor{green}{0, 0}, \mathbf{X}) + c \cdot \textcolor{blue}{a}(\textcolor{green}{0, 1}, \mathbf{X})) +
L_{2, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{blue}{a}(\textcolor{green}{1, 0}, \mathbf{X}) + c \cdot \textcolor{blue}{a}(\textcolor{green}{1, 1}, \mathbf{X}))
\Big)
\Big(
L_{1, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{red}{b}(\textcolor{green}{0, 0}, \mathbf{X}) + c \cdot \textcolor{red}{b}(\textcolor{green}{0, 1}, \mathbf{X})) +
L_{2, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{red}{b}(\textcolor{green}{1, 0}, \mathbf{X}) + c \cdot \textcolor{red}{b}(\textcolor{green}{1, 1}, \mathbf{X}))
\Big)
\begin{bmatrix}
0\cdot 0 & 0\cdot 1 & 0\cdot 2 & 0\cdot 3 \\
1\cdot 0 & 1\cdot 1 & 1\cdot 2 & 1\cdot 3 \\
2\cdot 0 & 2\cdot 1 & 2\cdot 2 & 2\cdot 3 \\
3\cdot 0 & 3\cdot 1 & 3\cdot 2 & 3\cdot 3 \\
\end{bmatrix}
Sumcheck in Small Fields - Algo \(3\)
r_i(0) = \sum_{\mathbf{X}} \textcolor{blue}{a}(\alpha_1, \dots, \alpha_{i-1}, 0, \mathbf{X})\cdot
\textcolor{red}{b}(\alpha_1, \dots, \alpha_{i-1}, 0, \mathbf{X})
r_2(c) = \sum_{\mathbf{X}} \textcolor{blue}{a}(\alpha_1, c, \mathbf{X})\cdot
\textcolor{red}{b}(\alpha_1, c, \mathbf{X})
= \sum_{\mathbf{X}}
\Big(
L_{1, \vec{\alpha}_1} \cdot \textcolor{blue}{a}(\textcolor{green}{0}, c, \mathbf{X}) +
L_{2, \vec{\alpha}_1} \cdot \textcolor{blue}{a}(\textcolor{green}{1}, c, \mathbf{X})
\Big)
\Big(
L_{1, \vec{\alpha}_1} \cdot \textcolor{red}{b}(\textcolor{green}{0}, c, \mathbf{X}) +
L_{2, \vec{\alpha}_1} \cdot \textcolor{red}{b}(\textcolor{green}{1}, c, \mathbf{X})
\Big)
= \sum_{\mathbf{X}}
\Big(
L_{1, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{blue}{a}(\textcolor{green}{0, 0}, \mathbf{X}) + c \cdot \textcolor{blue}{a}(\textcolor{green}{0, 1}, \mathbf{X})) +
L_{2, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{blue}{a}(\textcolor{green}{1, 0}, \mathbf{X}) + c \cdot \textcolor{blue}{a}(\textcolor{green}{1, 1}, \mathbf{X}))
\Big)
\Big(
L_{1, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{red}{b}(\textcolor{green}{0, 0}, \mathbf{X}) + c \cdot \textcolor{red}{b}(\textcolor{green}{0, 1}, \mathbf{X})) +
L_{2, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{red}{b}(\textcolor{green}{1, 0}, \mathbf{X}) + c \cdot \textcolor{red}{b}(\textcolor{green}{1, 1}, \mathbf{X}))
\Big)
\begin{bmatrix}
0\cdot 0 & 0\cdot 1 & 0\cdot 2 & 0\cdot 3 \\
1\cdot 0 & 1\cdot 1 & 1\cdot 2 & 1\cdot 3 \\
2\cdot 0 & 2\cdot 1 & 2\cdot 2 & 2\cdot 3 \\
3\cdot 0 & 3\cdot 1 & 3\cdot 2 & 3\cdot 3 \\
\end{bmatrix}
Sumcheck in Small Fields - Algo \(3\)
r_i(0) = \sum_{\mathbf{X}} \textcolor{blue}{a}(\alpha_1, \dots, \alpha_{i-1}, 0, \mathbf{X})\cdot
\textcolor{red}{b}(\alpha_1, \dots, \alpha_{i-1}, 0, \mathbf{X})
r_2(c) = \sum_{\mathbf{X}} \textcolor{blue}{a}(\alpha_1, c, \mathbf{X})\cdot
\textcolor{red}{b}(\alpha_1, c, \mathbf{X})
= \sum_{\mathbf{X}}
\Big(
L_{1, \vec{\alpha}_1} \cdot \textcolor{blue}{a}(\textcolor{green}{0}, c, \mathbf{X}) +
L_{2, \vec{\alpha}_1} \cdot \textcolor{blue}{a}(\textcolor{green}{1}, c, \mathbf{X})
\Big)
\Big(
L_{1, \vec{\alpha}_1} \cdot \textcolor{red}{b}(\textcolor{green}{0}, c, \mathbf{X}) +
L_{2, \vec{\alpha}_1} \cdot \textcolor{red}{b}(\textcolor{green}{1}, c, \mathbf{X})
\Big)
= \sum_{\mathbf{X}}
\Big(
L_{1, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{blue}{a}(\textcolor{green}{0, 0}, \mathbf{X}) + c \cdot \textcolor{blue}{a}(\textcolor{green}{0, 1}, \mathbf{X})) +
L_{2, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{blue}{a}(\textcolor{green}{1, 0}, \mathbf{X}) + c \cdot \textcolor{blue}{a}(\textcolor{green}{1, 1}, \mathbf{X}))
\Big)
\Big(
L_{1, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{red}{b}(\textcolor{green}{0, 0}, \mathbf{X}) + c \cdot \textcolor{red}{b}(\textcolor{green}{0, 1}, \mathbf{X})) +
L_{2, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{red}{b}(\textcolor{green}{1, 0}, \mathbf{X}) + c \cdot \textcolor{red}{b}(\textcolor{green}{1, 1}, \mathbf{X}))
\Big)
\begin{bmatrix}
0\cdot 0 & 0\cdot 1 & 0\cdot 2 & 0\cdot 3 \\
1\cdot 0 & 1\cdot 1 & 1\cdot 2 & 1\cdot 3 \\
2\cdot 0 & 2\cdot 1 & 2\cdot 2 & 2\cdot 3 \\
3\cdot 0 & 3\cdot 1 & 3\cdot 2 & 3\cdot 3 \\
\end{bmatrix}
Sumcheck in Small Fields - Algo \(3\)
r_i(0) = \sum_{\mathbf{X}} \textcolor{blue}{a}(\alpha_1, \dots, \alpha_{i-1}, 0, \mathbf{X})\cdot
\textcolor{red}{b}(\alpha_1, \dots, \alpha_{i-1}, 0, \mathbf{X})
r_2(c) = \sum_{\mathbf{X}} \textcolor{blue}{a}(\alpha_1, c, \mathbf{X})\cdot
\textcolor{red}{b}(\alpha_1, c, \mathbf{X})
= \sum_{\mathbf{X}}
\Big(
L_{1, \vec{\alpha}_1} \cdot \textcolor{blue}{a}(\textcolor{green}{0}, c, \mathbf{X}) +
L_{2, \vec{\alpha}_1} \cdot \textcolor{blue}{a}(\textcolor{green}{1}, c, \mathbf{X})
\Big)
\Big(
L_{1, \vec{\alpha}_1} \cdot \textcolor{red}{b}(\textcolor{green}{0}, c, \mathbf{X}) +
L_{2, \vec{\alpha}_1} \cdot \textcolor{red}{b}(\textcolor{green}{1}, c, \mathbf{X})
\Big)
= \sum_{\mathbf{X}}
\Big(
L_{1, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{blue}{a}(\textcolor{green}{0, 0}, \mathbf{X}) + c \cdot \textcolor{blue}{a}(\textcolor{green}{0, 1}, \mathbf{X})) +
L_{2, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{blue}{a}(\textcolor{green}{1, 0}, \mathbf{X}) + c \cdot \textcolor{blue}{a}(\textcolor{green}{1, 1}, \mathbf{X}))
\Big)
\Big(
L_{1, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{red}{b}(\textcolor{green}{0, 0}, \mathbf{X}) + c \cdot \textcolor{red}{b}(\textcolor{green}{0, 1}, \mathbf{X})) +
L_{2, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{red}{b}(\textcolor{green}{1, 0}, \mathbf{X}) + c \cdot \textcolor{red}{b}(\textcolor{green}{1, 1}, \mathbf{X}))
\Big)
\begin{bmatrix}
0\cdot 0 & 0\cdot 1 & 0\cdot 2 & 0\cdot 3 \\
1\cdot 0 & 1\cdot 1 & 1\cdot 2 & 1\cdot 3 \\
2\cdot 0 & 2\cdot 1 & 2\cdot 2 & 2\cdot 3 \\
3\cdot 0 & 3\cdot 1 & 3\cdot 2 & 3\cdot 3 \\
\end{bmatrix}
Sumcheck in Small Fields - Algo \(3\)
r_i(0) = \sum_{\mathbf{X}} \textcolor{blue}{a}(\alpha_1, \dots, \alpha_{i-1}, 0, \mathbf{X})\cdot
\textcolor{red}{b}(\alpha_1, \dots, \alpha_{i-1}, 0, \mathbf{X})
r_2(c) = \sum_{\mathbf{X}} \textcolor{blue}{a}(\alpha_1, c, \mathbf{X})\cdot
\textcolor{red}{b}(\alpha_1, c, \mathbf{X})
= \sum_{\mathbf{X}}
\Big(
L_{1, \vec{\alpha}_1} \cdot \textcolor{blue}{a}(\textcolor{green}{0}, c, \mathbf{X}) +
L_{2, \vec{\alpha}_1} \cdot \textcolor{blue}{a}(\textcolor{green}{1}, c, \mathbf{X})
\Big)
\Big(
L_{1, \vec{\alpha}_1} \cdot \textcolor{red}{b}(\textcolor{green}{0}, c, \mathbf{X}) +
L_{2, \vec{\alpha}_1} \cdot \textcolor{red}{b}(\textcolor{green}{1}, c, \mathbf{X})
\Big)
= \sum_{\mathbf{X}}
\Big(
L_{1, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{blue}{a}(\textcolor{green}{0, 0}, \mathbf{X}) + c \cdot \textcolor{blue}{a}(\textcolor{green}{0, 1}, \mathbf{X})) +
L_{2, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{blue}{a}(\textcolor{green}{1, 0}, \mathbf{X}) + c \cdot \textcolor{blue}{a}(\textcolor{green}{1, 1}, \mathbf{X}))
\Big)
\Big(
L_{1, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{red}{b}(\textcolor{green}{0, 0}, \mathbf{X}) + c \cdot \textcolor{red}{b}(\textcolor{green}{0, 1}, \mathbf{X})) +
L_{2, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{red}{b}(\textcolor{green}{1, 0}, \mathbf{X}) + c \cdot \textcolor{red}{b}(\textcolor{green}{1, 1}, \mathbf{X}))
\Big)
\begin{bmatrix}
0\cdot 0 & 0\cdot 1 & 0\cdot 2 & 0\cdot 3 \\
1\cdot 0 & 1\cdot 1 & 1\cdot 2 & 1\cdot 3 \\
2\cdot 0 & 2\cdot 1 & 2\cdot 2 & 2\cdot 3 \\
3\cdot 0 & 3\cdot 1 & 3\cdot 2 & 3\cdot 3 \\
\end{bmatrix}
Sumcheck in Small Fields - Algo \(3\)
r_i(0) = \sum_{\mathbf{X}} \textcolor{blue}{a}(\alpha_1, \dots, \alpha_{i-1}, 0, \mathbf{X})\cdot
\textcolor{red}{b}(\alpha_1, \dots, \alpha_{i-1}, 0, \mathbf{X})
r_2(c) = \sum_{\mathbf{X}} \textcolor{blue}{a}(\alpha_1, c, \mathbf{X})\cdot
\textcolor{red}{b}(\alpha_1, c, \mathbf{X})
= \sum_{\mathbf{X}}
\Big(
L_{1, \vec{\alpha}_1} \cdot \textcolor{blue}{a}(\textcolor{green}{0}, c, \mathbf{X}) +
L_{2, \vec{\alpha}_1} \cdot \textcolor{blue}{a}(\textcolor{green}{1}, c, \mathbf{X})
\Big)
\Big(
L_{1, \vec{\alpha}_1} \cdot \textcolor{red}{b}(\textcolor{green}{0}, c, \mathbf{X}) +
L_{2, \vec{\alpha}_1} \cdot \textcolor{red}{b}(\textcolor{green}{1}, c, \mathbf{X})
\Big)
= \sum_{\mathbf{X}}
\Big(
L_{1, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{blue}{a}(\textcolor{green}{0, 0}, \mathbf{X}) + c \cdot \textcolor{blue}{a}(\textcolor{green}{0, 1}, \mathbf{X})) +
L_{2, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{blue}{a}(\textcolor{green}{1, 0}, \mathbf{X}) + c \cdot \textcolor{blue}{a}(\textcolor{green}{1, 1}, \mathbf{X}))
\Big)
\Big(
L_{1, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{red}{b}(\textcolor{green}{0, 0}, \mathbf{X}) + c \cdot \textcolor{red}{b}(\textcolor{green}{0, 1}, \mathbf{X})) +
L_{2, \vec{\alpha}_1} \cdot (\bar{c} \cdot \textcolor{red}{b}(\textcolor{green}{1, 0}, \mathbf{X}) + c \cdot \textcolor{red}{b}(\textcolor{green}{1, 1}, \mathbf{X}))
\Big)
\begin{bmatrix}
0\cdot 0 & 0\cdot 1 & 0\cdot 2 & 0\cdot 3 \\
1\cdot 0 & 1\cdot 1 & 1\cdot 2 & 1\cdot 3 \\
2\cdot 0 & 2\cdot 1 & 2\cdot 2 & 2\cdot 3 \\
3\cdot 0 & 3\cdot 1 & 3\cdot 2 & 3\cdot 3 \\
\end{bmatrix}
Karatsuba Trick by Yuval
By Suyash Bagad
Karatsuba Trick by Yuval
- 81
