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[book] Fix updates to Sinsemilla writeup.

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ying tong 2021-07-23 20:34:16 +08:00 committed by GitHub
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book/src/design/circuit/gadgets/sinsemilla.md
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@ -81,7 +81,7 @@ In other words, $z_{n-i} = \sum\limits_{h=0}^{i-1} 2^{kh} \cdot m_{h+1}$.
> In order to support chaining multiple field elements without a gap, we will use a slightly more complicated expression for $m_{i+1}$ that effectively forces $\mathbf{z}_n$ to zero for the last step of each element, as indicated by $q_{S2}$. This allows the cell that would have been $\mathbf{z}_n$ to be used to reinitialize the running sum for the next element.
### Generator lookup table
The
The Sinsemilla circuit makes use of $2^10$ pre-computed random generators. These are loaded into a lookup table:
$$
\begin{array}{|c|c|c|}
\hline
@ -110,6 +110,7 @@ $$
2' & x'_{A,2} & x_{P[m'_3]} & z'_2 & \lambda'_{1,2} & \lambda'_{2,2} & 1 & 1 & 0 & 0 \\\hline
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & 1 & 1 & 0 & 0 \\\hline
n-1' & x'_{A,n-1} & x_{P[m'_n]} & z'_{n-1} & \lambda'_{1,n-1} & \lambda'_{2,n-1} & 1 & 0 & 1 & 0 \\\hline
n' & x'_{A,n} & & & y_{A,n} & & 0 & 0 & 0 & 0 \\\hline
\end{array}
$$
@ -140,10 +141,10 @@ $$
\begin{array}{|c|l|}
\hline
\text{Degree} & \text{Constraint} \\\hline
4 & q_{S4} \cdot (2 \cdot y_Q - Y_{A,0}) = 0 \\\hline
2 & q_{S4} \cdot (2 \cdot y_Q - Y_{A,0}) = 0 \\\hline
5 & q_{S1,i} \Rightarrow (m_{i+1},\, x_{P,i},\, y_{P,i}) \in \mathcal{P} \\\hline
3 & q_{S1,i} \cdot \big(\lambda_{2,i}^2 - (x_{A,i+1} + x_{R,i} + x_{A,i})\big) \\\hline
6 & q_{S1,i} \cdot \left(2 \cdot \lambda_{2,i} \cdot (x_{A,i} - x_{A,i+1}) - (Y_{A,i} + (1 - q_{S3,i}) \cdot Y_{A,i+1} + 2 \cdot q_{S3,i} \cdot y_{A,n})\right) = 0 \\\hline
4 & q_{S1,i} \cdot \left(2 \cdot \lambda_{2,i} \cdot (x_{A,i} - x_{A,i+1}) - (Y_{A,i} + (1 - q_{S3,i}) \cdot Y_{A,i+1} + 2 \cdot q_{S3,i} \cdot y_{A,n})\right) = 0 \\\hline
\end{array}
$$