Continue Stern part

macros.tex

@@ -73,6 +73,8 @@
 \newcommand{\QQ}{\xspace\ensuremath{\mathbb{Q}}\xspace}
 \newcommand{\Zq}{\xspace\ensuremath{\mathbb{Z}_q}\xspace}
 \newcommand{\bit}{\xspace\ensuremath{\{0,1\}}\xspace}
+\newcommand{\nbit}{\xspace\ensuremath{\{-1,0,1\}}\xspace}
+\newcommand{\permutations}{\ensuremath{\mathfrak S}\xspace}
 %% Pairings
 \newcommand{\Zp}{\xspace\ensuremath{\mathbb{Z}_p}\xspace}
 \newcommand{\GG}{\xspace\ensuremath{\mathbb{G}}\xspace}

sec-stern.tex

@@ -7,44 +7,27 @@ On the other hand, Stern's protocol has been originally introduced in the contex
 \index{Syndrome Decoding Problem}
 Initially, it was introduced for Syndrome Decoding Problem (\SDP): given a matrix $\mathbf P \in \FF_2^{n \times m}$ and a syndrome $\mathbf v \in \FF_2^n$, the goal is to find a binary vector $\mathbf x \in \FF_2^m$ with fixed hamming weight $w$ such that $\mathbf P \cdot \mathbf x = \mathbf v \bmod 2$.
 
-The proof is mainly combinatorial and relies on the fact that applying a permutation on a binary vector leaves its Hamming weight invariant.
-% TODO
-
 This problem shows similarities with the $\ISIS$ problem defined in \cref{de:sis} where the constraints on the norm of $\mathbf x$ is a constraint on Hamming weight, and operations are in $\FF_2$ instead of $\Zq$.
 
 After the first works of Kawachi, Tanaka and Xagawa~\cite{KTX08} that extended Stern's proofs to statements $\bmod q$, the work of Ling, Nguyen, Stehlé and Wang~\cite{LNSW13} enables the use of Stern's protocol to prove general $\SIS$ or $\LWE$ statements (meaning the knowledge of a solution to these problems).
 These advances in the expressivity of Stern-like protocols has been used to further improve it and therefore enable privacy-based primitives for which no constructions existed in the post-quantum world, such as dynamic group signatures~\cite{LLM+16}, group encryption~\cite{LLM+16a}, electronic cash~\cite{LLNW17}, etc.
 
+Unlike the Schnorr-like proof we saw in the previous section, Stern's proof is mainly combinatorial and relies on the fact that every permutation on a binary vector $\mathbf x \in \bit^m_{}$ leaves its Hamming weight $w$ invariant. As a consequence, for $\pi \in \permutations_m$, $\mathbf x$ satisfies these conditions if and only if $\pi(\mathbf x)$ also does.
+Therefore, the randomness of $\pi$ is used to verify these two constraints (being binary and the hamming weight) in a zero-knowledge fashion.
+We can notice that this can be extended to vectors $\mathbf x \in \nbit^m$ of fixed numbers of $-1$ and $1$ that allowed~\cite{LNSW13} to propose the generalization of this protocol to any $\ISIS_{n,m,q,\beta}$ statements.
+In \cref{sse:stern-abstraction}, we describes these permutations while abstracting the set of ZK-provable statements as the set $\mathsf{VALID}$ that satisfies conditions~\eqref{eq:zk-equivalence}.
+
+It is worth noticing that this argument on knowledge does not form a $\Sigma$-protocol, as the challenge space is ternary as described in \cref{sse:stern-abstraction}.
+Thus standard theorems on $\Sigma$-protocols has to be adapted in this setting.
+
 In this Section, we describe in a high-level view how Stern's protocol works, and then we detail it.
 
-\subsection{Abstraction of Stern's Protocol}
+
+\subsection{Abstraction of Stern's Protocol} \label{sse:stern-abstraction}
 \addcontentsline{tof}{subsection}{\protect\numberline{\thesubsection} Abstraction du protocole de Stern}
 
 %%%% TODO
-
+\begin{figure}[t]
-
-Let $K$, $D$, $q$ be positive integers with $D \geq K$ and $q \geq 2$, and let $\mathsf{VALID}$ be a subset of $\mathbb{Z}^D$. Suppose that $\mathcal{S}$ is a finite set such that   every $\phi \in \mathcal{S}$ can be associated with a permutation $\Gamma_\phi$ of $D$ elements  satisfying the following conditions:
-\begin{eqnarray}\label{eq:zk-equivalence}
-\begin{cases}
-\mathbf{w} \in \mathsf{VALID} \hspace*{2.5pt} \Longleftrightarrow \hspace*{2.5pt} \Gamma_\phi(\mathbf{w}) \in \mathsf{VALID}, \\
-\text{If } \mathbf{w} \in \mathsf{VALID} \text{ and } \phi \text{ is uniform in } \mathcal{S}, \text{ then }  \Gamma_\phi(\mathbf{w}) \text{ is uniform in } \mathsf{VALID}.~~~~~
-\end{cases}
-\end{eqnarray}
-We aim to construct a statistical \textsf{ZKAoK} for the following abstract relation:
-\begin{eqnarray*}
-\mathrm{R_{abstract}} = \big\{ \big((\mathbf{M}, \mathbf{v}), \mathbf{w} \big) \in \mathbb{Z}_q^{K \times D} \times \mathbb{Z}_q^D \times \mathsf{VALID}: \mathbf{M}\cdot \mathbf{w} = \mathbf{v} \bmod q.\big\}
-\end{eqnarray*}
-
-Note that, Stern's original protocol corresponds to the special case when $\mathsf{VALID} = \{
-\mathbf{w} \in \{0,1\}^D: \mathsf{wt}(\mathbf{w}) = k\}$, where $\mathsf{wt}(\cdot)$ denotes the Hamming weight and $k < D$ is a given integer, $\mathcal{S} = \mathcal{S}_D$ is  the set of all permutations of~$D$ elements and $\Gamma_{\phi}(\mathbf{w}) = \phi(\mathbf{w})$.
-
-The conditions in \eqref{eq:zk-equivalence} play a crucial role in proving in \textsf{ZK} that $\mathbf{w} \in \mathsf{VALID}$. To this end, the prover samples a random $\phi \hookleftarrow U(\mathcal{S})$ and lets the verifier check that $\Gamma_\phi(\mathbf{w}) \in \mathsf{VALID}$ without learning any additional information about $\mathbf{w}$ due to the randomness of $\phi$. Furthermore, to prove in a zero-knowledge manner that the linear equation is satisfied, the prover samples a masking vector $\mathbf{r}_w \hookleftarrow U(\mathbb{Z}_q^D)$, and convinces the verifier instead that $\mathbf{M}\cdot (\mathbf{w} + \mathbf{r}_w) = \mathbf{M}\cdot \mathbf{r}_w + \mathbf{v} \bmod q.$
-
-
-The interaction between prover $\mathcal{P}$ and verifier $\mathcal{V}$ is described in Figure~\ref{Figure:Interactive-Protocol}. The protocol uses a statistically hiding and computationally binding string commitment scheme \textsf{COM} (e.g., the \textsf{SIS}-based scheme from~\cite{KTX08}).
-
-
-\begin{figure}[!htbp]
 
   \small
   \begin{enumerate}
@@ -70,8 +53,7 @@ The interaction between prover $\mathcal{P}$ and verifier $\mathcal{V}$ is descr
   \end{enumerate}
   \textbf{Verification:}  Receiving $\mathrm{RSP}$, the verifier proceeds as follows:
   \smallskip
-%\vspace{-0.25cm}
+  \begin{itemize}
-  \begin{itemize}%[leftmargin=0.2cm,itemindent=.2cm,labelwidth=\itemindent,labelsep=0.2cm,align=left]
     \item $Ch = 1$: Check that
       \begin{gather*}
         \mathbf{t}_w \in \mathsf{VALID},\\
@@ -93,11 +75,31 @@ The interaction between prover $\mathcal{P}$ and verifier $\mathcal{V}$ is descr
 
   \end{itemize}
   In each case, the verifier outputs $1$ if and only if all the conditions hold.
-%\rule{0pt}{3ex}
   \caption{Stern-like \textsf{ZKAoK} for the relation $\mathrm{R_{abstract}}$.}
   \label{Figure:Interactive-Protocol}
 \end{figure}
-%The properties of the given protocol are summarized in Theorem~\ref{Theorem:zk-protocol}.
+
+
+Let $K$, $D$, $q$ be positive integers with $D \geq K$ and $q \geq 2$, and let $\mathsf{VALID}$ be a subset of $\mathbb{Z}^D$. Suppose that $\mathcal{S}$ is a finite set such that   every $\phi \in \mathcal{S}$ can be associated with a permutation $\Gamma_\phi \in \permutations_D$ satisfying the following conditions:
+\begin{eqnarray}\label{eq:zk-equivalence}
+\begin{cases}
+\mathbf{w} \in \mathsf{VALID} ~ \iff ~ \Gamma_\phi(\mathbf{w}) \in \mathsf{VALID}, \\
+\text{If } \mathbf{w} \in \mathsf{VALID} \text{ and } \phi \text{ is uniform in } \mathcal{S}, \text{ then }  \Gamma_\phi(\mathbf{w}) \text{ is uniform in } \mathsf{VALID}. \quad
+\end{cases}
+\end{eqnarray}
+We aim to construct a statistical Zero-Knowledge Argument of Knowledge (\textsf{ZKAoK}) for the following abstract relation:
+\begin{eqnarray*}
+\mathrm{R_{abstract}} = \big\{ \big((\mathbf{M}, \mathbf{v}), \mathbf{w} \big) \in \mathbb{Z}_q^{K \times D} \times \mathbb{Z}_q^D \times \mathsf{VALID}: \mathbf{M}\cdot \mathbf{w} = \mathbf{v} \bmod q.\big\}
+\end{eqnarray*}
+
+Note that, Stern's original protocol corresponds to the special case when the set
+$\mathsf{VALID} = \{
+\mathbf{w} \in \{0,1\}^D: \mathsf{wt}(\mathbf{w}) = k\}$, where $\mathsf{wt}(\cdot)$ denotes the Hamming weight and $k < D$ is a given integer, $\mathcal{S} = \mathcal{S}_D$ is  the set of all permutations of~$D$ elements and $\Gamma_{\phi}(\mathbf{w}) = \phi(\mathbf{w})$.
+
+The conditions in \eqref{eq:zk-equivalence} play a crucial role in proving in \textsf{ZK} that $\mathbf{w} \in \mathsf{VALID}$. To this end, the prover samples a random $\phi \hookleftarrow U(\mathcal{S})$ and lets the verifier check that $\Gamma_\phi(\mathbf{w}) \in \mathsf{VALID}$ without learning any additional information about $\mathbf{w}$ due to the randomness of $\phi$. Furthermore, to prove in a zero-knowledge manner that the linear equation is satisfied, the prover samples a masking vector $\mathbf{r}_w \hookleftarrow U(\mathbb{Z}_q^D)$, and convinces the verifier instead that $\mathbf{M}\cdot (\mathbf{w} + \mathbf{r}_w) = \mathbf{M}\cdot \mathbf{r}_w + \mathbf{v} \bmod q.$
+
+
+The interaction between prover $\mathcal{P}$ and verifier $\mathcal{V}$ is described in Figure~\ref{Figure:Interactive-Protocol}. The protocol uses a statistically hiding and computationally binding string commitment scheme \textsf{COM} (e.g., the \textsf{SIS}-based scheme from~\cite{KTX08}).
 
 \begin{theorem}\label{Theorem:zk-protocol}
 The protocol in Figure~\ref{Figure:Interactive-Protocol} is a statistical \emph{\textsf{ZKAoK}} with perfect completeness, soundness error~$2/3$, and communication cost~$\mathcal{O}(D\log q)$. Namely:
@@ -111,3 +113,21 @@ The proof of the theorem relies on standard simulation and extraction techniques
 
 
 %%%% END TODO
+
+%%%%%%%%%%%%%%%%%%%%%
+%%%% Recap Table %%%%
+%%%%%%%%%%%%%%%%%%%%%
+\begin{figure}
+  \begin{itemize}
+    \item $\mathsf B^{2}_{\mathfrak m}$: the set of vectors in $\bit^{2\mathfrak m}$ with Hamming weight $\mathfrak m$.
+    \item $\mathsf B^{3}_{\mathfrak m}$: the set of vectors in $\nbit^{3\mathfrak m}$ which has exactly $\mathfrak m$ coordinates equal to $j$ for each $j \in \nbit$.
+  \end{itemize}
+  \caption{Notations for Stern-like protocols.}
+  \label{fig:stern-notations}
+\end{figure}
+
+\subsection{The Decomposition-Extension Framework}
+\addcontentsline{tof}{subsection}{\protect\numberline{\thesubsection} Méthode de décomposition-extension}
+
+A method used in~\cite{LNSW13} to prove knowledge of an \ISIS preimage consists in first \textit{decomposing} the secret $\mathbf{x} = (x_1, \ldots, x_m)  \in [-B,B]^m$ into a vector $\tilde{\mathbf x}$ of $\nbit^{m \delta_B}$ such that $\tilde{\mathbf x} = [ \tilde{\mathbf u}_1^{T} \mid \cdots \mid \tilde{\mathbf u}_{\delta_B}^T]^T$ and for all $j \in \{1, \ldots, m\}$, $(1, 2, \ldots, 2^{\delta_B - 1})^T \cdot \tilde{\mathbf u}_j^{} = x_j$.
+Once that is done, we fix the hamming weight of the resulting vector by \textit{extending} its components $\tilde{\mathbf u}_j^{}$ into $\mathbf u_j \in \mathsf B^3_{m}$.

symbols.tex

@@ -8,8 +8,9 @@
   $\ppt$     & Probabilistic Polynomial Time            \\
   $\epsilon$ & empty word                               \\
   $\mathbf A$ & bold uppercase letters represent matrices\\
-  $\mathbf b$ & bold lowercase letters represent vectors\\
+  $\mathbf b$ & bold lowercase letters represent column vectors\\
   $\widetilde{\mathbf A}$ & Gram-Schmidt orthogonalization of matrix $\mathbf A$\\
+  $\mathbf{A}^T_{}, \mathbf{u}^T_{}$ & the transpose of a matrix or a vector respectively\\
   [1ex] \multicolumn{2}{l}{\scbf{Usual sets}}           \\
   $\QQ$      & the set of rational numbers              \\
   $\RR$      & the set of real numbers                  \\