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weil-pairing.tex add rational functions & divisors section

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arnaucube
2023-06-20 10:29:04 +02:00
parent f1079b0a74
commit a0c0742815
4 changed files with 115 additions and 14 deletions
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notes_nova.tex
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@@ -109,7 +109,7 @@ where R1CS set $E=0,~u=1$.
\begin{align*}
Az \circ Bz &= A z_1 \circ B z_1 + r(A z_1 \circ B z_2 + A z_2 \circ B z_1) + r^2 (A z_2 \circ B z_2)\\
&= (u_1 C z_1 + E_1) + r (A z_1 \circ B z_2 + A z_2 \circ B z_1) + r^2 (u_2 C z_2 + E_2)\\
&= u_1 C z_1 + \underbrace{E_1 + r(A z_1 \circ B z_2 + A z_2 \circ B z_1) + r^2 E_2}_\text{E} + r^1 u_2 C z_2\\
&= u_1 C z_1 + \underbrace{E_1 + r(A z_1 \circ B z_2 + A z_2 \circ B z_1) + r^2 E_2}_\text{E} + r^2 u_2 C z_2\\
&= u_1 C z_1 + r^2 u_2 C z_2 + E\\
&= (u_1 + r u_2) \cdot C \cdot (z_1 + r z_2) + E\\
&= uCz + E
@@ -151,7 +151,7 @@ Let $Z_1 = (W_1, x_1, u_1)$ and $Z_2 = (W_2, x_2, u_2)$.
% \paragraph{Protocol}
\begin{enumerate}
\item P send $\overline{T} = Com(T, r_T)$,\\
where $T=A z_1 \circ B z_1 + A z_2 \circ B z_2 - u_1 C z_2 - u_2 C z_2$\\
where $T=A z_1 \circ B z_1 + A z_2 \circ B z_2 - u_1 C z_1 - u_2 C z_2$\\
and rand $r_T \in \mathbb{F}$
\item V sample random challenge $r \in \mathbb{F}$
\item V, P output the folded instance $\varphi = (\overline{E}, u, \overline{W}, x)$