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book: Move Sinsemilla constraints table to bottom of page
This moves it below the rationale for the lookup being degree-6.
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@ -138,17 +138,6 @@ The Halo 2 circuit API can automatically substitute $y_{P,i}$, $x_{R,i}$, $y_{A,
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Note that each term of the last constraint is multiplied by $4$ relative to the constraint program given earlier. This is a small optimization that avoids divisions by $2$.
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$$
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\begin{array}{|c|l|}
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\hline
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\text{Degree} & \text{Constraint} \\\hline
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4 & q_{S4} \cdot (2 \cdot y_Q - Y_{A,0}) = 0 \\\hline
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6 & q_{S1,i} \Rightarrow (m_{i+1},\, x_{P,i},\, y_{P,i}) \in \mathcal{P} \\\hline
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3 & q_{S1,i} \cdot \big(\lambda_{2,i}^2 - (x_{A,i+1} + x_{R,i} + x_{A,i})\big) \\\hline
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5 & q_{S1,i} \cdot \left(4 \cdot \lambda_{2,i} \cdot (x_{A,i} - x_{A,i+1}) - (2 \cdot Y_{A,i} + (2 - q_{S3,i}) \cdot Y_{A,i+1} + 2 \cdot q_{S3,i} \cdot y_{A,n})\right) = 0 \\\hline
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\end{array}
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$$
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By gating the lookup expression on $q_{S1}$, we avoid the need to fill in unused cells with dummy values to pass the lookup argument. The optimized lookup value (using a default index of $0$) is:
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$$
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@ -160,3 +149,14 @@ $$
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$$
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This increases the degree of the lookup argument to $6$.
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$$
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\begin{array}{|c|l|}
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\hline
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\text{Degree} & \text{Constraint} \\\hline
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4 & q_{S4} \cdot (2 \cdot y_Q - Y_{A,0}) = 0 \\\hline
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6 & q_{S1,i} \Rightarrow (m_{i+1},\, x_{P,i},\, y_{P,i}) \in \mathcal{P} \\\hline
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3 & q_{S1,i} \cdot \big(\lambda_{2,i}^2 - (x_{A,i+1} + x_{R,i} + x_{A,i})\big) \\\hline
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5 & q_{S1,i} \cdot \left(4 \cdot \lambda_{2,i} \cdot (x_{A,i} - x_{A,i+1}) - (2 \cdot Y_{A,i} + (2 - q_{S3,i}) \cdot Y_{A,i+1} + 2 \cdot q_{S3,i} \cdot y_{A,n})\right) = 0 \\\hline
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\end{array}
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$$
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