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zwW{7DrzL{6f8}pIzp~P$f+dC)!!Xj_4eicY@Xe?y2<|T1%H^g-2d7qEbW=^cE7-gH zOg+K-k6^z-34+pW57-r V`}%RIs%ojJt8ht48QYj}{Rc1B(3t=L diff --git a/notes_hypernova.tex b/notes_hypernova.tex index 839fed9..188df69 100644 --- a/notes_hypernova.tex +++ b/notes_hypernova.tex @@ -163,7 +163,7 @@ Let $s= \log m,~ s'= \log n$. where $\sigma_j,~\theta_j$ are the checks from LCCCS and CCCS respectively with $x=r_x'$. \item V: $e_1 \leftarrow \widetilde{eq}(r_x, r_x')$, $e_2 \leftarrow \widetilde{eq}(\beta, r_x')$\\ check: - $$c = \left( \sum_{j \in [t]} \gamma^j e_1 \sigma_j + \gamma^{t+1} e_2 \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \sigma \right) \right)$$ + $$c = \left(\sum_{j \in [t]} \gamma^j \cdot e_1 \cdot \sigma_j \right) + \gamma^{t+1} \cdot e_2 \cdot \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \theta_j \right)$$ which should be equivalent to the $g(x)$ computed by $V,P$ in the sum-check protocol. \item $V \rightarrow P: \rho \in^R \mathbb{F}$ \item $V, P$: output the folded LCCCS instance $(C', u', \mathsf{x}', r_x', v_1', \ldots, v_t')$, where $\forall i \in [t]$: @@ -173,7 +173,11 @@ Let $s= \log m,~ s'= \log n$. \mathsf{x}' &\leftarrow \mathsf{x}_1 + \rho \cdot \mathsf{x}_2\\ v_i' &\leftarrow \sigma_i + \rho \cdot \theta_i \end{align*} - \item $P$: output folded witness: $\widetilde{w}' \leftarrow \widetilde{w}_1 + \rho \cdot \widetilde{w}_2$. + \item $P$: output folded witness and the folded $r_w'$ (random value used for the witness commitment $C$): + \begin{align*} + \widetilde{w}' &\leftarrow \widetilde{w}_1 + \rho \cdot \widetilde{w}_2\\ + r_w' &\leftarrow r_{w_1} + \rho \cdot r_{w_2} + \end{align*} \end{enumerate} @@ -285,7 +289,116 @@ where $e_1 = \widetilde{eq}(r_x, r_x')$ and $e_2=\widetilde{eq}(\beta, r_x')$. Which is the check that $V$ performs at step $5$. +\subsection{Multifolding for multiple instances} +The multifolding of multiple LCCCS \& CCCS instances is not shown in the HyperNova paper, but Srinath Setty gave an overview in the PSE HyperNova presentation. This section unfolds it. + +We're going to do this example with parameters \textcolor{orange}{LCCCS: $\mu = 2$}, \textcolor{cyan}{CCCS: $\nu = 2$}, which means that we have 2 LCCCS instances and 2 CCCS instances. + +Assume we have 4 $z$ vectors, $z_1,~ \textcolor{orange}{z_2}$ for the two LCCCS instances, and $z_3,~ \textcolor{cyan}{z_4}$ for the two CCCS instances, where $z_1,~z_3$ are the vectors that we already had in the example with $\mu=1,\nu=1$, and $z_2,~z_4$ are the extra ones that we're adding now. + +In \emph{step 3} of the multifolding with more than one LCCCS and more than one CCCS instances, we have: + +\begin{align*} + g(x) &:= \left( \sum_{j \in [t]} \gamma^j \cdot L_{1,j}(x) + \textcolor{orange}{\gamma^{t+j} \cdot L_{2,j}(x)} \right) ++ \gamma^{2t+1} \cdot Q_1(x) + \textcolor{cyan}{\gamma^{2t+2} \cdot Q_2(x)} \\ + &L_{1,j}(x) := \widetilde{eq}(r_{1,x}, x) \cdot \left( + \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_1(y) +\right)\\ + &\textcolor{orange}{L_{2,j}(x)} := \widetilde{eq}(\textcolor{orange}{r_{2,x}}, x) \cdot \left( + \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(x, y) \cdot \textcolor{orange}{\widetilde{z}_2(y)} +\right)\\ + &Q_1(x) := \widetilde{eq}(\beta, x) \cdot \left( +\sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \left( \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(x, y) \cdot \widetilde{z}_3(y) \right)\right)\\ + &\textcolor{cyan}{Q_2(x)} := \widetilde{eq}(\textcolor{cyan}{\beta'}, x) \cdot \left( + \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \left( \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(x, y) \cdot \textcolor{cyan}{\widetilde{z}_4(y)} \right)\right) +\end{align*} + +\framebox{\begin{minipage}{4.3 in} +A generic definition of $g(x)$ for $\mu>1~\nu>1$, would be + +$$ +g(x) := \left( \sum_{i \in [\mu]} \left( \sum_{j \in [t]} \gamma^{i \cdot t+j} \cdot L_{i,j}(x) \right) \right) ++ \left( \sum_{i \in [\nu]} \gamma^{\mu \cdot t + i} \cdot Q_i(x) \right) +$$ +\end{minipage}} + +Recall, the original $g(x)$ definition was +$$\textcolor{gray}{g(x) := \left( \sum_{j \in [t]} \gamma^j \cdot L_j(x) \right) + \gamma^{t+1} \cdot Q(x)}$$ + + + +\vspace{0.5cm} +In \emph{step 4}, $P \rightarrow V$: +$(\{\sigma_{1,j}\}, \textcolor{orange}{\{\sigma_{2,j}\}}, \{\theta_{1,j}\}, \textcolor{cyan}{\{\theta_{2,j}\}}),~ \text{where} ~\forall j \in [t]$, + +$$\sigma_{1,j} = \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \widetilde{z}_1(y)$$ +$$\textcolor{orange}{\sigma_{2,j}} = \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \textcolor{orange}{\widetilde{z}_2(y)}$$ +$$\theta_{1,j} = \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \widetilde{z}_3(y)$$ +$$\textcolor{cyan}{\theta_{2,j}} = \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \textcolor{cyan}{\widetilde{z}_4(y)}$$ + +\framebox{\begin{minipage}{4.3 in} + so in a generic way,\\ +$P \rightarrow V$: +$(\{\sigma_{i,j}\}, \{\theta_{k,j}\}),~ \text{where} ~\forall~ j \in [t],~ \forall~ i \in [\mu],~ \forall~ k \in [\nu]$ +where +$$\sigma_{i,j} = \sum_{y \in \{0,1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \widetilde{z}_i(y)$$ +$$\theta_{k,j} = \sum_{y \in \{0, 1\}^{s'}} \widetilde{M}_j(r_x', y) \cdot \widetilde{z}_{\mu+k}(y)$$ +\end{minipage}} + +\vspace{1cm} + +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)} +\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). + +\vspace{0.5cm} + +\framebox{\begin{minipage}{4.3 in} +A generic definition of the check would be +$$ +c = \sum_{i \in [\mu]} \left(\sum_{j \in [t]} \gamma^{i \cdot t + j} \cdot e_i \cdot \sigma_{i,j} \right) \\ ++ \sum_{k \in [\nu]} \gamma^{\mu \cdot t+k} \cdot e_k \cdot \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \theta_{k,j} \right) +$$ +\end{minipage}} + +where the original check was\\ +$\textcolor{gray}{c = \left(\sum_{j \in [t]} \gamma^j \cdot e_1 \cdot \sigma_j \right) + \gamma^{t+1} \cdot e_2 \cdot \left( \sum_{i=1}^q c_i \cdot \prod_{j \in S_i} \theta_j \right)}$ + +% TODO +% Pending questions: +% - \beta & \beta' can be the same? or related somehow like \beta'=\beta^2 ? + +\vspace{0.5cm} + +And for the \emph{step 7}, +\begin{align*} + C' &\leftarrow C_1 + \rho \cdot C_2 + \rho^2 C_3 + \rho^3 C_4 + \ldots = \sum_{i \in [\mu + \nu]} \rho^i \cdot C_i \\ + u' &\leftarrow \sum_{i \in [\mu]} \rho^i \cdot u_i + \sum_{i \in [\nu]} \rho^{\mu + i-1} \cdot 1\\ + \mathsf{x}' &\leftarrow \sum_{i \in [\mu+\nu]} \rho^i \cdot \mathsf{x}_i\\ + v_i' &\leftarrow \sum_{i \in [\mu]} \rho^i \cdot \sigma_i + \sum_{i \in [\nu]} \rho^{\mu + i-1} \cdot \theta_i\\ +\end{align*} + +and \emph{step 8}, +\begin{align*} + \widetilde{w}' &\leftarrow \sum_{i \in [\mu+\nu]} \rho^i\cdot \widetilde{w}_i\\ + r_w' &\leftarrow \sum_{i \in [\mu+\nu]} \rho^i \cdot r_{w_i}\\ +\end{align*} + + +Note that over all the multifolding for $\mu >1$ and $\nu>1$, we can easily parallelize most of the computation. + +\vspace{2cm} %%%%%% APPENDIX \appendix