Usually while reading papers I take handwritten notes, this document contains some of them re-written to $LaTeX$.
Usually while reading papers I take handwritten notes, this document contains some of them re-written to $LaTeX$.
The notes are not complete, don't include all the steps neither all the proofs.
The notes are not complete, don't include all the steps neither all the proofs.
Thanks to \href{https://twitter.com/levs57}{Levs57}, \href{https://twitter.com/nibnalin}{Nalin Bhardwaj} and \href{https://twitter.com/cperezz19}{Carlos Pérez} for clarifications on the Nova paper.
\end{abstract}
\end{abstract}
\tableofcontents
\tableofcontents
@ -72,7 +74,15 @@
\subsection{R1CS modification}
\subsection{R1CS modification}
Want: merge 2 instances of R1CS with the same matrices into a single one. Each instance has $z_i =(W_i,~ x_i)$ (public witness, private values resp.).
\paragraph{R1CS}
R1CS instance: $(A, B, C, io, m, n)$, where $io$ denotes the public input and output, $A, B, C \in\mathbb{F}^{m \times n}$, with $m \geq |io|+1$.
R1CS is satisfied by a witness $w \in\mathbb{F}^{m-|io|-1}$ such that
$$Az \circ Bz = Cz$$
where $z=(io, 1, w)$.
\vspace{0.5cm}
\textbf{Want}: merge 2 instances of R1CS with the same matrices into a single one. Each instance has $z_i =(W_i,~ x_i)$ (public witness, private values resp.).
\paragraph{traditional R1CS}
\paragraph{traditional R1CS}
Merged instance with $z=z_1+ r z_2$, for rand $r$. But, since R1CS is not linear $\longrightarrow$ can not apply.
Merged instance with $z=z_1+ r z_2$, for rand $r$. But, since R1CS is not linear $\longrightarrow$ can not apply.
