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Add weil reciprocity example, weil pairing properties, small typo in hypernova notes

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arnaucube
2023-06-21 08:57:41 +02:00
parent a0c0742815
commit cc9bb99ac5
4 changed files with 42 additions and 11 deletions
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10
notes_hypernova.tex
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@@ -349,17 +349,15 @@ $$\theta_{k,j} = \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \wide
And in \emph{step 5}, $V$ checks
% TODO check orange gamma^j...
\begin{align*}
c &= \left(\sum_{j \in [t]} \gamma^j \cdot e_1 \cdot \sigma_{1,j}
~\textcolor{orange}{+ \gamma^{t+j} \cdot e_1 \cdot \sigma_{2,j}}\right)\\
&+ \gamma^{2t+1} \cdot e_2 \cdot \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \theta_j \right)
+ \textcolor{cyan}{\gamma^{2t+2} \cdot e_2 \cdot \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \theta_j \right)}
~\textcolor{orange}{+ \gamma^{t+j} \cdot e_2 \cdot \sigma_{2,j}}\right)\\
&+ \gamma^{2t+1} \cdot e_3 \cdot \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \theta_j \right)
+ \textcolor{cyan}{\gamma^{2t+2} \cdot e_4 \cdot \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \theta_j \right)}
\end{align*}
where
% TODO check e_4
$e_1 \leftarrow \widetilde{eq}(r_{1,x}, r_x'),~ e_2 \leftarrow \widetilde{eq}(r_{2,x}, r_x')$, $e_3 \leftarrow \widetilde{eq}(\beta, r_x'),~ e_4 \leftarrow \widetilde{eq}(\beta', r_x')$ (note: wip, pending check for $\beta, \beta'$ used in step 3).
$e_1 \leftarrow \widetilde{eq}(r_{1,x}, r_x'),~ e_2 \leftarrow \widetilde{eq}(r_{2,x}, r_x')$, $e_3, e_4 \leftarrow \widetilde{eq}(\beta, r_x')$.
\vspace{0.5cm}