math/notes_fri_stir.tex
notes_fri_stir.texRaw
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27
28\title{Notes on FRI and STIR}
29\author{arnaucube}
30\date{February 2023}
31
32\begin{document}
33
34\maketitle
35
36\begin{abstract}
37 Notes taken from \href{https://sites.google.com/site/vincenzoiovinoit/}{Vincenzo Iovino} \cite{vincenzoiovino} explanations about FRI \cite{fri}, \cite{cryptoeprint:2022/1216}, \cite{cryptoeprint:2019/1020}.
38
39 These notes are for self-consumption, are not complete, don't include all the steps neither all the proofs.
40
41 An implementation of FRI can be found at\\ \href{https://github.com/arnaucube/fri-commitment}{https://github.com/arnaucube/fri-commitment} \cite{fri-impl}.
42
43 Update(2024-03-22): notes on STIR \cite{cryptoeprint:2024/390} from explanations by \href{https://hecmas.github.io/}{Héctor Masip Ardevol} \cite{hectormasipardevol}.
44\end{abstract}
45
46\tableofcontents
47
48\section{Preliminaries}
49\subsection{General degree d test}
50
51Query at points $\{ x_i \}_0^{d+1},~z$ (with rand $z \overset{R}{\in} \mathbb{F}$).
52Interpolate $p(x)$ at $\{f(x_i)\}_0^{d+1}$ to reconstruct the unique polynomial $p$ of degree $d$ such that $p(x_i)=f(x_i)~\forall i=1, \ldots, d+1$.
53
54V checks $p(z)=f(z)$, if the check passes, then V is convinced with high probability.
55
56This needs $d+2$ queries, is linear, $\mathcal{O}(n)$. With FRI we will have the test in $\mathcal{O}(\log{}d)$.
57
58\section{FRI protocol}
59Allows to test if a function $f$ is a poly of degree $\leq d$ in $\mathcal{O}(\log{}d)$.
60
61Note: "P \emph{sends} $f(x)$ to V", "\emph{sends}", in the ideal IOP model means that all the table of $f(x)$ is sent, in practice is sent a commitment to $f(x)$.
62
63\subsection{Intuition}
64V wants to check that two functions $g,~h$ are both polynomials of degree $\leq d$.
65
66Consider the following protocol:
67
68\begin{enumerate}
69 \item V sends $\alpha \in \mathbb{F}$ to P. P sends $f(x) = g(x) + \alpha h(x)$ to V.
70 \item P sends $f(x)=g(x) + \alpha h(x)$ to V.
71 \item V queries $f(r), ~g(r), ~h(r)$ for rand $r \in \mathbb{F}$.
72 \item V checks $f(r)=g(r) + \alpha h(r)$. (Schwartz-Zippel lema).
73 If holds, V can be certain that $f(x)=g(x)+ \alpha h(x)$.
74 \item P proves that $deg(f) \leq d$.
75 \item If V is convinced that $deg(f) \leq d$, V believes that both $g, h$ have $deg \leq d$.
76\end{enumerate}
77
78%/// TODO tabulate this next lines
79With high probablility, $\alpha$ will not cancel the coeffs with $deg \geq d+1$. % TODO check which is the name of this theorem or why this is true
80
81Let $g(x)=a \cdot x^{d+1}, ~~ h(x)=b \cdot x^{d+1}$, and set $f(x) = g(x) + \alpha h(x)$.
82Imagine that P can chose $\alpha$ such that $a x^{d+1} + \alpha \cdot b x^{d+1} = 0$, then, in $f(x)$ the coefficients of degree $d+1$ would cancel.
83%///
84
85\quad
86
87Here, P proves $g,~h$ both have $deg \leq d$, but instead of doing $2 \cdot (d+2)$ queries ($d+2$ for $g$, and $d+2$ for $h$), it is done in $d+2$ queries (for $f$).
88So we halved the number of queries.
89
90
91\subsection{FRI-LDT}\label{sec:fri-ldt}
92FRI low degree testing.\\
93Both P and V have oracle access to function $f$.
94
95V wants to test if $f$ is polynomial with $deg(f) \leq d$.
96
97Let $f_0(x)=f(x)$.
98
99Each polynomial $f(x)$ of degree that is a power of $2$, can be written as
100$$f(x) = f^L(x^2) + x f^R(x^2)$$
101for some polynomials $f^L,~f^R$ of degree $\frac{deg(f)}{2}$, each one containing the even and odd degree coefficients as follows:
102
103% $f^L(x)$ is built from the even degree coefficients divided by $x$, and $f^R(x)$ from the odd degree coefficients divided by $x$.
104
105$$f^L(x)= \sum_0^{\frac{d+1}{2}-1} c_{2i} x^i ,~~ f^R(x)= \sum_0^{\frac{d+1}{2}-1} c_{2i+1} x^i$$
106
107eg. for $f(x)=x^4+x^3+x^2+x+1$,
108\begin{align*}
109 \begin{rcases}
110 f^L(x)=x^2+x+1\\
111 f^R(x)=x+1
112 \end{rcases}
113 ~f(x) = f^L(x^2) &+ x \cdot f^R(x^2)\\
114 = (x^2)^2 + (x^2) + 1 &+ x \cdot ((x^2) + 1)\\
115 = x^4 + x^2 + 1 &+ x^3 + x
116\end{align*}
117
118% \begin{enumerate}
119% \item V sends to P some $\alpha_0 \in \mathbb{F}$.
120% Let
121% \begin{equation}\tag{$A_0$}
122% f_0(x) = f_0^L(x^2) + x f_0^R(x^2)
123% \end{equation}
124% \item P sends
125% \begin{equation}\tag{$B_0$}
126% f_1(x) = f_0^L(x) + \alpha_0 f_0^R(x)
127% \end{equation}
128% to V.
129%
130% (remember that "sends" in IOP model is that P commits to it)
131% \item V sends to P some $\alpha_1 \in \mathbb{F}$.
132% Let
133% \begin{equation}\tag{$A_1$}
134% f_1(x) = f_1^L(x^2) + x f_1^R(x^2)
135% \end{equation}
136% \item P sends
137% \begin{equation}\tag{$B_1$}
138% f_2(x) = f_1^L(x) + \alpha_1 f_1^R(x)
139% \end{equation}
140% to V.
141% \item Keep repeating the process, eg. let
142% \begin{equation}\tag{$A_2$}
143% f_2(x) = f_2^L(x^2) + x f_2^R(x^2)
144% \end{equation}
145% until $f_i^L,~ f_i^R$ are constant (degree 0 polynomials).
146% \item Once $f_i^L,~ f_i^R$ are constant, P sends them to V.
147% \end{enumerate}
148%
149% Notice that at each step, $deg(f_i)$ halves.
150
151
152\vspace{30px}
153
154\paragraph{Proof generation}
155
156\emph{(Commitment phase)}
157P starts from $f(x)$, and for $i=0$ sets $f_0(x)=f(x)$.
158\begin{enumerate}
159 \item $\forall~i \in \{0, log(d)\}$, with $d = deg~f(x)$,\\
160 P computes $f_i^L(x),~ f_i^R(x)$ for which
161 \begin{equation}\tag{eq. $A_i$}
162 f_i(x) = f_i^L(x^2) + x f_i^R(x^2)
163 \end{equation}
164 holds.
165 \item V sends challenge $\alpha_i \in \mathbb{F}$
166 \item P commits to the random linear combination $f_{i+1}$, for
167 \begin{equation}\tag{eq. $B_i$}
168 f_{i+1}(x) = f_i^L(x) + \alpha_i f_i^R(x)
169 \end{equation}
170 \item P sets $f_i(x) := f_{i+1}(x)$ and starts again the iteration.
171\end{enumerate}
172Notice that at each step, $deg(f_i)$ halves.
173
174This 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.
175
176\emph{(Query phase)}
177P would receive a challenge $z \in D$ set by V (where $D$ is the evaluation domain, $D \in \mathbb{F}$), and P would open the commitments at $\{z^{2^i}, -z^{2^i}\}$ for each step $i$.
178(Recall, "opening" means that would provide a proof (MerkleProof) of it).
179
180\paragraph{Data sent from P to V}
181\begin{itemize}
182 \item[] Commitments: $\{Comm(f_i)\}_0^{log(d)}$\\
183 {\scriptsize eg. $\{Comm(f_0),~ Comm(f_1),~ Comm(f_2),~ ...,~ Comm(f_{log(d)})\}$ }
184 \item[] Openings: $\{ f_i(z^{2^i}),~f_i(-(z^{2^i})) \}_0^{log(d)}$\\
185 for a challenge $z \in D$ set by V\\
186 {\scriptsize eg. $f_0(z),~ f_0(-z),~ f_1(z^2),~ f_1(-z^2),~ f_2(z^4),~ f_2(-z^4),~ f_3(z^8),~ f_3(-z^8),~ \ldots$}
187 \item[] Constant values of last iteration: $\{f_k^L,~f_k^R\}$, for $k=log(d)$
188\end{itemize}
189
190% \begin{figure}[htp]
191% \centering
192% \begin{footnotesize}
193% \begin{sequencediagram}
194% \newinst[0]{p}{Prover}
195% \newinst[5]{v}{Verifier}
196%
197% \mess{p}{$\{Comm(f_i)\}_0^{log(d)},~ \{f_i(z^{2^i}),~f_i(-(z^{2^i})) \}_0^{log(d)},~ \{f_k^L,~ f_k^R\}$}{v}
198%
199% \end{sequencediagram}
200% \end{footnotesize}
201% \caption[FRI-LDT]{sketch of the FRI-LDT flow}
202% \label{fig:fri-ldt}
203% \end{figure}
204
205
206\paragraph{Verification}
207
208V receives:
209\begin{align*}
210 \text{Commitments:}~ &Comm(f_i),~ \forall i \in \{0, log(d)\}\\
211 \text{Openings:}~ &\{o_i, o_i'\}=\{ f_i(z^{2^i}),~f_i(-(z^{2^i})) \},~ \forall i \in \{0, log(d)\}\\
212 \text{Constant vals:}~ &\{f_k^L,~f_k^R\}
213\end{align*}
214
215\vspace{20px}
216
217For all $i \in \{0, log(d)\}$, V knows the openings at $z^{2^i}$ and $-(z^{2^i})$ for\\
218$Comm(f_i(x))$, which are $o_i=f_i(z^{2^i})$ and $o_i'=f_i(-(z^{2^i}))$ respectively.
219
220V, from (eq. $A_i$), knows that
221$$f_i(x)=f_i^L(x^2) + x f_i^R(x^2)$$
222should hold, thus
223$$f_i(z)=f_i^L(z^2) + z f_i^R(z^2)$$
224where $f_i(z)$ is known, but $f_i^L(z^2),~f_i^R(z^2)$ are unknown.
225But, V also knows the value for $f_i(-z)$, which can be represented as
226$$f_i(-z)=f_i^L(z^2) - z f_i^R(z^2)$$
227(note that when replacing $x$ by $-z$, it loses the negative in the power, not in the linear combination).
228
229Thus, we have the system of independent linear equations
230\begin{align*} % TODO add braces on left
231 f_i(z)&=f_i^L(z^2) + z f_i^R(z^2)\\
232 f_i(-z)&=f_i^L(z^2) - z f_i^R(z^2)
233\end{align*}
234for which V will find the value of $f_i^L(z^{2^i}),~f_i^R(z^{2^i})$.
235Equivalently it can be represented by
236$$
237\begin{pmatrix}
238 1 & z\\
239 1 & -z
240\end{pmatrix}
241\begin{pmatrix}
242 f_i^L(z^2)\\
243 f_i^R(z^2)
244\end{pmatrix}
245=
246\begin{pmatrix}
247 f_i(z)\\
248 f_i(-z)
249\end{pmatrix}
250$$
251where V will find the values of $f_i^L(z^{2^i}),~f_i^R(z^{2^i})$ being
252\begin{align*}
253 f_i^L(z^{2^i})=\frac{f_i(z) + f_i(-z)}{2}\\
254 f_i^R(z^{2^i})=\frac{f_i(z) - f_i(-z)}{2z}\\
255\end{align*}
256
257Once, V has computed $f_i^L(z^{2^i}),~f_i^R(z^{2^i})$, can use them to compute the linear combination of
258$$
259f_{i+1}(z^{2^i}) = f_i^L(z^{2^i}) + \alpha_i f_i^R(z^{2^i})
260$$
261obtaining then $f_{i+1}(z^{2^i})$. This comes from (eq. $B_i$).
262
263Now, V checks that the obtained $f_{i+1}(z^{2^i})$ is equal to the received opening $o_{i+1}=f_{i+1}(z^{2^i})$ from the commitment done by P.
264V checks also the commitment of $Comm(f_{i+1}(x))$ for the opening $o_{i+1}=f_{i+1}(z^{2^i})$.\\
265If the checks pass, V is convinced that $f_1(x)$ was committed honestly.
266
267Now, sets $i := i+1$ and starts a new iteration.
268
269For 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.
270
271\vspace{10px}
272It needs $log(d)$ iterations, and the number of queries (commitments + openings sent and verified) needed is $2 \cdot log(d)$.
273
274\subsection{Parameters}
275
276P commits to $f_i$ restricted to a subfield $F_0 \subset \mathbb{F}$.
277Let $0<\rho<1$ be the \emph{rate} of the code, such that
278$$|F_0| = \rho^{-1} \cdot d$$
279
280\begin{theorem}
281 For $\delta \in (0, 1-\sqrt{\rho})$, we have that if V accepts, then w.v.h.p. (with very high probability) $\Delta(f_0,~ p^d) \leq \delta$.
282\end{theorem}
283
284\section{FRI as polynomial commitment scheme}
285This section overviews the trick from \cite{cryptoeprint:2019/1020} to convert FRI into a polynomial commitment.
286
287Want to check that the evaluation of $f(x)$ at $r$ is $f(r)$, which is equivalent to proving that $\exists ~Q \in \mathbb{F}[x]$ with $deg(Q)=d-1$, such that
288
289$$
290f(x)-f(r) = Q(x) \cdot (x-r)
291$$
292
293note that $f(x)-f(r)$ evaluated at $r$ is $0$, so $(x-r) | (f(x)-f(r))$, in other words
294$(f(x)-f(r))$ is a multiple of $(x-r)$ for a polynomial $Q(x)$.
295
296Let us define $g(x) = \frac{f(x)-f(r)}{x-r}$.
297
298Prover uses FRI-LDT \ref{sec:fri-ldt} to commit to $g(x)$, and then prove w.v.h.p that $deg(g) \leq d-1$ ($\Longleftrightarrow \Delta(g,~ p^{d-1} \leq \delta$).
299
300Prover was already proving that $deg(f) \leq d$.
301
302Now, the missing thing to prove is that $g(x)$ has the right shape. We can relate $g$ to $f$ as follows:
303V does the normal FRI-LDT, but in addition, at the first iteration:
304V has $f(z)$ and $g(z)$ openings, so can verify
305$$g(z) = (f(z)-f(r))\cdot (z-r)^{-1}$$
306
307
308\section{STIR (main idea)}
309\emph{Update from 2024-03-22, notes from Héctor Masip Ardevol (\href{https://hecmas.github.io/}{https://hecmas.github.io}) explanations.}
310
311\vspace{0.3cm}
312Let $p \in \mathbb{F}[x]^{<n}$.
313
314In FRI we decompose $p(x)$ as
315$$p(x) = p_e(x^2) + x \cdot p_o(x^2)$$
316with $p_e, p_o \in \mahtbb{F}[x]^{<n}$ containing the even and odd powers respectively.
317
318The next FRI polynomial is
319$$p_1(x) = p_e(x) + \alpha p_o(x)$$
320for $\alpha \in^R \mathbb{F}$.
321
322In STIR, this would be $q(x)=x^2$,
323$$Q(z,y) = p_e(y) + z \cdot p_o(y)$$
324and then, $p(x) = Q(x, q(x))$. And $Q$ fullfills the degree from Fact 4.6 from the STIR paper.
325
326We can generalize to a $q$ with bigger degree, or with another shape, and adapting $Q$ on the choice of $q$.
327
328eg. for $q(x)=x^3$, we can take
329$$Q(z,y) = p_1(y) + z \cdot p_2(y) + z^2 \cdot p_3(y)$$
330with $p_1, p_2, p_3 \in \mathbb{F}[x]^{<n/3}$ with coefficients taken every 3 powers alternating.
331
332\bibliography{paper-notes.bib}
333\bibliographystyle{unsrt}
334
335\end{document}