From fcb0aba0c2692b296da16205289e7b827cf77048 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Sun, 5 Mar 2023 16:30:53 +0100 Subject: [PATCH] FRI notes: simplify proving, add verification steps --- notes_fri.pdf | Bin 222146 -> 219590 bytes notes_fri.tex | 181 ++++++++++++++++++++++++++++++++++--------------- notes_nova.pdf | Bin 192527 -> 192549 bytes notes_nova.tex | 2 +- 4 files changed, 129 insertions(+), 54 deletions(-) diff --git a/notes_fri.pdf b/notes_fri.pdf index d3553526f672a53333db0ecb399188b7710e87cf..4e3598ce7ff3211c67ebc1e8899712033a633750 100644 GIT binary patch delta 115565 zcmZs>Q*fXU@TDEwwkNi2I}_WslL_8%V%wP5wry)-V`4km-@m@vueNIUrt4hvRo|Sa zyU)NV;z%PR>JMdcNd^{14!9q4OT%k$?Cc4s=-});97&`o)WEWqoa+WBO3#VrjfTz7 z0WWlE%CLvMXucfEX{&|cBya{1PAatWI~A(eM|~sG$iy?L^=Li;3Xp=W;r$IqZ1}}P z)`Ia$s)yX$u-Bv{RR{D`^qFM&yQUf^em@`1 zcjoV-L$$dO7QiHBUnB>u+%3^rEZV$GpIv>P=`Biowc8P;Gq(rPAFg)0(VFhP5Dk9 zTwlT`6ne149r>J)(OY=%Fx&5ih#~JDnW9eu+(<*)&CdTw2bwIhSp7l8Ie@38`X7|0 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zlvJeA57Q3EsLVWLg;Cj43RD8EPirdg{&-qf`TWJxhRXhir%l!LM^DdH%QcdTCe`1ufmbH{QB^-I zdkU&Q=RGB=wHKc1G_-Ads@G6|-uIN!@ba6dw1ziZo-!JCK6%P&`1RFOPLrl_PYs&1 z=i|P`nMmn%q;n_I^*7R8i^R^*WFQwAY>tdDZ|qg1{5~?d7nx$-bbn+viOl_oEEFP( zekE1{I>+c7qjSvvICP!$fp-pFVsweoB}SLnkS0rSW2yB$`44||UO)<$o>>DN3^Fk@ MI57$(B}Gq03K67r82|tP diff --git a/notes_fri.tex b/notes_fri.tex index 15ada21..e6c02d4 100644 --- a/notes_fri.tex +++ b/notes_fri.tex @@ -32,11 +32,11 @@ \maketitle \begin{abstract} - Notes taken from \href{https://sites.google.com/site/vincenzoiovinoit/}{Vincenzo Iovino} explainations and while reading about FRI \cite{fri}, \cite{cryptoeprint:2022/1216}. + Notes taken from \href{https://sites.google.com/site/vincenzoiovinoit/}{Vincenzo Iovino} explainations about FRI \cite{fri}, \cite{cryptoeprint:2022/1216}. - Usually while reading papers I take handwritten notes, this document contains some of them re-written to $LaTeX$. + These notes are for self-consumption, 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. + An implementation of FRI can be found at \href{https://github.com/arnaucube/fri-commitment}{https://github.com/arnaucube/fri-commitment}. \end{abstract} \tableofcontents @@ -117,64 +117,139 @@ eg. for $f(x)=x^4+x^3+x^2+x+1$, = x^4 + x^2 + 1 &+ x^3 + x \end{align*} -\begin{enumerate} - \item V sends to P some $\alpha_0 \in \mathbb{F}$. - Let - \begin{equation}\tag{$A_0$} - f_0(x) = f_0^L(x^2) + x f_0^R(x^2) - \end{equation} - \item P sends - \begin{equation}\tag{$B_0$} - f_1(x) = f_0^L(x) + \alpha_0 f_0^R(x) - \end{equation} - to V. +% \begin{enumerate} +% \item V sends to P some $\alpha_0 \in \mathbb{F}$. +% Let +% \begin{equation}\tag{$A_0$} +% f_0(x) = f_0^L(x^2) + x f_0^R(x^2) +% \end{equation} +% \item P sends +% \begin{equation}\tag{$B_0$} +% f_1(x) = f_0^L(x) + \alpha_0 f_0^R(x) +% \end{equation} +% to V. +% +% (remember that "sends" in IOP model is that P commits to it) +% \item V sends to P some $\alpha_1 \in \mathbb{F}$. +% Let +% \begin{equation}\tag{$A_1$} +% f_1(x) = f_1^L(x^2) + x f_1^R(x^2) +% \end{equation} +% \item P sends +% \begin{equation}\tag{$B_1$} +% f_2(x) = f_1^L(x) + \alpha_1 f_1^R(x) +% \end{equation} +% to V. +% \item Keep repeating the process, eg. let +% \begin{equation}\tag{$A_2$} +% f_2(x) = f_2^L(x^2) + x f_2^R(x^2) +% \end{equation} +% until $f_i^L,~ f_i^R$ are constant (degree 0 polynomials). +% \item Once $f_i^L,~ f_i^R$ are constant, P sends them to V. +% \end{enumerate} +% +% Notice that at each step, $deg(f_i)$ halves. + + +\vspace{30px} + +\paragraph{Proof generation} +P starts from $f(x)$, and for $i=0$ sets $f_0(x)=f(x)$. - (remember that "sends" in IOP model is that P commits to it) - \item V sends to P some $\alpha_1 \in \mathbb{F}$. - Let - \begin{equation}\tag{$A_1$} - f_1(x) = f_1^L(x^2) + x f_1^R(x^2) - \end{equation} - \item P sends - \begin{equation}\tag{$B_1$} - f_2(x) = f_1^L(x) + \alpha_1 f_1^R(x) - \end{equation} - to V. - \item Keep repeating the process, eg. let - \begin{equation}\tag{$A_2$} - f_2(x) = f_2^L(x^2) + x f_2^R(x^2) +\begin{enumerate} + \item $\forall~i \in \{0, log(d)\}$, with $d = deg~f(x)$,\\ + P computes $f_i^L(x),~ f_i^R(x)$ for which + \begin{equation}\tag{eq. $A_i$} + f_i(x) = f_i^L(x^2) + x f_i^R(x^2) \end{equation} - until $f_i^L,~ f_i^R$ are constant (degree 0 polynomials). - \item Once $f_i^L,~ f_i^R$ are constant, P sends them to V. + holds. + \item V sends challenge $\alpha_i$ + \item P commits to the random linear combination $f_{i+1}$, for + \begin{equation}\tag{eq. $B_i$} + f_{i+1}(x) = f_i^L(x) + \alpha_i f_i^R(x) + \end{equation} + \item P sets $f_i(x) := f_{i+1}(x)$ and starts again the iteration. \end{enumerate} - Notice that at each step, $deg(f_i)$ halves. -\paragraph{Query phase} +This is done until the last step, where $f_i^L(x),~ f_i^R(x)$ are constant (degree 0 polynomials). For which P does not commit but gives their values directly to V. -\begin{enumerate} - \item V sends rand $z \in \mathbb{F}$ to P - \item P sends $\{ f_i(z^{2^i}), f_i(- z^{2^i}) \}$ to V.\\ +\paragraph{Data sent from P to V} +\begin{itemize} + \item[] Commitments: $\{Comm(f_i)\}_0^{log(d)}$\\ + {\scriptsize eg. $\{Comm(f_0),~ Comm(f_1),~ Comm(f_2),~ ...,~ Comm(f_{log(d)})\}$ } + \item[] Openings: $\{ f_i(z^{2^i}),~f_i(-(z^{2^i})) \}_0^{log(d)}$\\ + for a challenge $z \in \mathbb{F}$ set by V\\ {\scriptsize eg. $f_0(z),~ f_0(-z),~ f_1(z^2),~ f_1(-z^2),~ f_2(z^4),~ f_2(-z^4),~ f_3(z^8),~ f_3(-z^8),~ \ldots$} - \item V checks $f_i(a)=f_i^L(a^2) + a f_i^R(a^2)$ for $a=\{z, -z\}$ - $$ - \begin{pmatrix} - 1 & z\\ - 1 & -z - \end{pmatrix} - \begin{pmatrix} - f_i^L(z^2)\\ - f_i^R(z^2) - \end{pmatrix} - = - \begin{pmatrix} - f_i(z)\\ - f_i(-z) - \end{pmatrix} - $$ -\end{enumerate} + \item[] Constant values of last iteration: $\{f_k^L,~f_k^R\}$, for $k=log(d)$ +\end{itemize} + +\paragraph{Verification} + +V receives: +\begin{align*} + \text{Commitments:}~ &Comm(f_i),~ \forall i \in \{0, log(d)\}\\ + \text{Openings:}~ &\{o_i, o_i'\}=\{ f_i(z^{2^i}),~f_i(-(z^{2^i})) \},~ \forall i \in \{0, log(d)\}\\ + \text{Constant vals:}~ &\{f_k^L,~f_k^R\} +\end{align*} + +\vspace{20px} + +For all $i \in \{0, log(d)\}$, V knows the openings at $z^{2^i}$ and $-(z^{2^i})$ for $Comm(f_i(x))$, which are $o_i=f_i(z^{2^i})$ and $o_i'=f_i(-(z^{2^i}))$ respectively. + +V, from (eq. $A_i$), knows that +$$f_i(x)=f_i^L(x^2) + x f_i^R(x^2)$$ +should hold, thus +$$f_i(z)=f_i^L(z^2) + z f_i^R(z^2)$$ +where $f_i(z)$ is known, but $f_i^L(z^2),~f_i^R(z^2)$ are unknown. +But, V also knows the value for $f_i(-z)$, which can be represented as +$$f_i(-z)=f_i^L(z^2) - z f_i^R(z^2)$$ +(note that when replacing $x$ by $-z$, it loses the negative in the power, not in the linear combination). + +Thus, we have the system of independent linear equations +\begin{align*} % TODO add braces on left + f_i(z)&=f_i^L(z^2) + z f_i^R(z^2)\\ + f_i(-z)&=f_i^L(z^2) - z f_i^R(z^2) +\end{align*} +for which V will find the value of $f_i^L(z^{2^i}),~f_i^R(z^{2^i})$. +Equivalently it can be represented by +$$ +\begin{pmatrix} + 1 & z\\ + 1 & -z +\end{pmatrix} +\begin{pmatrix} + f_i^L(z^2)\\ + f_i^R(z^2) +\end{pmatrix} += +\begin{pmatrix} + f_i(z)\\ + f_i(-z) +\end{pmatrix} +$$ +where V will find the values of $f_i^L(z^{2^i}),~f_i^R(z^{2^i})$ being +\begin{align*} + f_i^L(z^{2^i})=\frac{f_i(z) + f_i(-z)}{2}\\ + f_i^R(z^{2^i})=\frac{f_i(z) - f_i(-z)}{2z}\\ +\end{align*} + +Once, V has computed $f_i^L(z^{2^i}),~f_i^R(z^{2^i})$, can use them to compute the linear combination of +$$ +f_{i+1}(z^2) = f_i^L(z^2) + \alpha_i f_i^R(z^2) +$$ +obtaining then $f_{i+1}(z^2)$. This comes from (eq. $B_i$). + +Now, V checks that the obtained $f_{i+1}(z^2)$ is equal to the received opening $o_{i+1}=f_{i+1}(z^2)$ from the commitment done by P. +V checks also the commitment of $Comm(f_{i+1}(x))$ for the opening $o_{i+1}=f_{i+1}(z^2)$.\\ +If the checks pass, V is convinced that $f_1(x)$ was committed honestly. + +Now, sets $i := i+1$ and starts a new iteration. + +For the last iteration, V checks that the obtained $f_i^L(z^{2^i}),~f_i^R(z^{2^i})$ are equal to the constant values $\{f_k^L,~f_k^R\}$ received from P. -The number of queries needed is $2 \cdot log(d)$. +\vspace{10px} +It needs $log(d)$ iterations, and the number of queries (commitments + openings sent and verified) needed is $2 \cdot log(d)$. \section{FRI as polynomial commitment} \emph{[WIP. 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