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+\section{A Lattice-Based Signature  with Efficient Protocols} \label{se:gs-lwe-sigep}
+
+%We first specify the parameters used in our scheme. Let $\lambda$ be the security parameter, and let $n = \bigO(\lambda)$, $q = \mathsf{poly}(n)$, and $m \geq 2n \log q$.
+
+
+
+%We assume that    messages are  vectors of $N$ blocks $\mathsf{Msg}=(\mathfrak{m}_1,\ldots,\mathfrak{m}_N)$, where  each
+%block is an $L$-bit string $\mathfrak{m}_k = \mathfrak{m}_k[1] \ldots \mathfrak{m}_k[L] \in \{0,1\}^L$ for $k \in \{1,\ldots, N\}$.
+
+Our scheme can be seen as a variant of the B\"ohl \textit{et al.} signature \cite{BHJ+15}, where
+each signature is a triple $(\tau,\mathbf{v},\mathbf{s})$, made of a tag $\tau \in \{0,1\}^\ell$ and   integer vectors $(\mathbf{v},\mathbf{s})$ satisfying
+ $[\mathbf{A} \mid \mathbf{A}_0 + \sum_{j=1}^\ell  \tau[j]  \cdot \mathbf{A}_j ] \cdot \mathbf{v} = \mathbf{u} + \mathbf{D} \cdot \mathbf{h} \bmod q$,
+ where matrices $\mathbf{A}, \mathbf{A}_0, \ldots, \mathbf{A}_\ell, \mathbf{D} \in \Zq^{n \times m}$
+are public random matrices and $\mathbf{h} \in \{0,1\}^m$ is a chameleon hash of the message which is computed using randomness $\mathbf{s}$.
+A difference is that, while \cite{BHJ+15} uses a short single-use tag $\tau \in \Zq$,
+we need the tag to be an $\ell$-bit string $\tau \in \{0,1\}^{\ell}$ which will assume the same role as the prime exponent of Camenisch-Lysyanskaya signatures
+\cite{CL02a} in the security proof.
+
+We show that a suitable chameleon hash function makes the scheme compatible with Stern-like zero-knowledge arguments \cite{LNSW13,LNW15} for arguing possession of a valid message-signature pair. \cref{sse:stern} shows how to translate such a statement into asserting that a short  witness vector $\mathbf{x}$ with a particular structure satisfies
+ a relation of the form
+$\mathbf{P} \cdot \mathbf{x} = \mathbf{v} \bmod q$, for some public matrix $\mathbf{P}$ and vector~$\mathbf{v}$.
+The underlying chameleon hash can be seen as a composition of the chameleon hash of \cite[Se. 4.1]{CHKP10} with
+a technique used in \cite{PSTY13,LLNW16}: on input of a message $(\mathfrak{m}_1,\ldots,\mathfrak{m}_N)$, it outputs the binary decomposition of
+$\mathbf{D}_0 \cdot \mathbf{s} + \sum_{k=1}^N \mathbf{D}_k \cdot \mathfrak{m}_k$,  for some discrete Gaussian vector $\mathbf{s}$.
+
+\subsection{Description} \label{desc-sig-protoc}
+
+ We assume that   messages are  vectors of $N$ blocks $\mathsf{Msg}=(\mathfrak{m}_1,\ldots,\mathfrak{m}_N)$, where  each
+block is a $2m$-bit string $\mathfrak{m}_k = \mathfrak{m}_k[1] \ldots \mathfrak{m}_k[2m] \in \{0,1\}^{2m}$ for $k \in \{1,\ldots, N\}$.  \\
+\indent For each vector $\mathbf{v} \in \Zq^L$, we denote by $\bit(\mathbf{v}) \in \{0,1\}^{L \lceil \log q \rceil}$ the vector obtained by replacing each
+coordinate of $\mathbf{v}$ by its binary representation.
+
+
+
+\begin{description}
+  \item[\textsf{Keygen}$(1^\lambda,1^N)$:] Given a security parameter $\lambda>0$  and the   number of  blocks $N = \mathsf{poly}(\lambda)$, choose the following parameters: $n = \bigO(\lambda)$; a prime modulus $q = \widetilde{\bigO}(N\cdot n^{4})$; dimension  $m =2n \lceil \log q \rceil $; an integer $\ell = \Theta(\lambda)$; and  Gaussian parameters $\sigma = \Omega(\sqrt{n\log q}\log n)$,  $\sigma_0 = 2\sqrt{2}(N+1) \sigma m^{3/2}$, and $\sigma_1 = \sqrt{\sigma_0^2 + \sigma^2}$. Define the message space as $(\{0,1\}^{2m})^N$.
+    \smallskip   
+    \begin{itemize}
+      \item[1.] Run $\TrapGen(1^n,1^m,q)$ to get~$\mathbf{A} \in
+        \Zq^{n \times m}$ and a short basis $\mathbf{T}_{\mathbf{A}}$ of
+        $\Lambda_q^{\perp}(\mathbf{A}).$  This basis allows computing short vectors in $\Lambda_q^{\perp}(\mathbf{A})$ with a Gaussian parameter $\sigma$.
+%	$\sigma \geq \| \widetilde{\mathbf{T}_{\mathbf{A}}} \| \cdot \omega (\sqrt{\log m})$.
+        Next, choose $\ell+1$ random   $\mathbf{A}_0,\mathbf{A}_1,\ldots,\mathbf{A}_{\ell} \sample U(\Zq^{n \times m})$. %, where $\ell = \Theta(\lambda)$.
+      \item[2.]  Choose  random matrices $\mathbf{D} \sample U(\Zq^{n \times m})$, $\mathbf{D}_0,\mathbf{D}_1,\ldots,\mathbf{D}_{N} \sample U(\Zq^{2n \times 2m})$ as well as a random     vector
+        $\mathbf{u} \sample U(\Zq^n)$. \smallskip
+    \end{itemize}
+    The  private key consists of $SK:= \mathbf{T}_{\mathbf{A}} \in \ZZ^{m \times m}$ and the public key is
+    $${PK}:=\big( \mathbf{A}, ~ \{\mathbf{A}_j \}_{j=0}^{\ell},  ~ \{\mathbf{D}_k\}_{k=0}^{N},~\mathbf{D}, ~\mathbf{u} \big).$$
+  % \smallskip
+  \item[\textsf{Sign}$\big(SK, \mathsf{Msg} \big)$:] To sign an $N$-block message
+    $\mathsf{Msg}=\left(\mathfrak{m}_1,\ldots,\mathfrak{m}_N \right) \in \left(\{0,1\}^{2m} \right)^N$,    
+    \begin{enumerate}[1.]
+      \item Choose a   random  string $\tau \sample U(\{0,1\}^\ell )$. Then, using  $SK:=
+        \mathbf{T}_{\mathbf{A}}$, compute with $\ExtBasis$ a short delegated basis $\mathbf{T}_\tau  \in \ZZ^{2m \times 2m}$
+        for the matrix
+        \begin{eqnarray} \label{tau-matrix}
+          \mathbf{A}_{\tau}=
+          [ \mathbf{A} \mid  \mathbf{A}_0 +
+            \sum_{j=1}^\ell \tau[j] \mathbf{A}_j
+          ] \in  \Zq^{ n  \times 2m}.
+        \end{eqnarray}
+      \item Sample a     vector $\mathbf{s} \sample D_{\ZZ^{2m},\sigma_1 }$. Compute   $\mathbf{c}_M \in \Zq^{2n}$ as a chameleon hash of $\left(\mathfrak{m}_1,\ldots,\mathfrak{m}_N \right)$: i.e., compute
+         $\mathbf{c}_M =       \mathbf{D}_{0} \cdot \mathbf{s}   + \sum_{k=1}^N   \mathbf{D}_k \cdot \mathfrak{m}_k \in \mathbb{Z}_q^{2n} ,$ 
+        which is used to define  $\mathbf{u}_M=\mathbf{u} + \mathbf{D} \cdot \bit( \mathbf{c}_M) \in \Zq^n .$
+        Then,
+        using the delegated basis $\mathbf{T}_\tau \in \ZZ^{2m \times 2m}$, sample  a short vector $\mathbf{v} \in \ZZ^{2m}$ in $D_{\Lambda_q^{\mathbf{u}_M}(\mathbf{A}_\tau), \sigma}$.
+    \end{enumerate}
+    Output the signature $sig=(\tau,\mathbf{v},\mathbf{s}) \in \{0,1\}^\ell \times  \ZZ^{2m} \times \ZZ^{2m}$. \smallskip
+  \item[\textsf{Verify}$\big(PK,\mathsf{Msg},sig\big)$:] Given  $PK$, a message $\mathsf{Msg}=(\mathfrak{m}_1,\ldots,\mathfrak{m}_N) \in (\{0,1\}^{2m})^N$ and a purported
+    signature $sig=(\tau,\mathbf{v},\mathbf{s}) \in \{0,1\}^\ell \times  \ZZ^{2m} \times \ZZ^{2m}$,
+    return $1$ if
+    \begin{eqnarray} \label{ver-eq-block}
+      \mathbf{A}_{\tau} \cdot \mathbf{v} &=& \mathbf{u} +  \mathbf{D} \cdot \bit(  \mathbf{D}_0 \cdot \mathbf{s} + \sum_{k=1}^N     \mathbf{D}_k \cdot \mathfrak{m}_k ) \bmod q.
+    \end{eqnarray}
+    and $\| \mathbf{v} \| < \sigma \sqrt{2m}$, $\| \mathbf{s} \| <  \sigma_1 \sqrt{2m}$.
+\end{description}
+When the scheme is used for obliviously signing committed messages,
+the security proof follows Bai \textit{et al.} \cite{BLL+15} in that it applies an argument based on the R\'enyi divergence  in   one signing query. This argument requires
+ to sample $\mathbf{s}$ from a Gaussian distribution whose  standard deviation $\sigma_1$ is polynomially larger than $\sigma$. \\
+\indent
+We note that, instead of being included in the public key, the matrices $ \{\mathbf{D}_k\}_{k=0}^{N}$ can be part of common public parameters shared by many signers. Indeed,
+ only the matrices $(\mathbf{A},\{\mathbf{A}_i\}_{i=0}^\ell)$ should be specific to the user who holds the secret key $SK=\mathbf{T}_{\mathbf{A}}$. In Section \ref{commit-sig}, we use a variant where  $ \{\mathbf{D}_k\}_{k=0}^{N}$
+belong to public parameters.
+
+
+\subsection{Security Analysis}
+The security analysis in Theorem  \ref{th:gs-lwe-security-cma-sig} requires that   $q>\ell$.
+
+
+\begin{theorem} \label{th:gs-lwe-security-cma-sig}
+The signature scheme is secure under chosen-message attacks under the $\mathsf{SIS}$ assumption.
+\end{theorem}
+
+\begin{proof}
+To prove the result, we will distinguish three kinds of attacks:
+\begin{description}
+\item[Type I attacks]  are attacks where, in the adversary's forgery $sig^\star=(\tau^\star,\mathbf{v}^\star,\mathbf{s}^\star)$, $\tau^\star$ did not appear in any output
+of the signing oracle.
+\item[Type II attacks]  are such that, in the adversary's forgery $sig^\star=(\tau^\star,\mathbf{v}^\star,\mathbf{s}^\star)$, $\tau^\star$ is recycled from an output
+ $sig^{(i^\star)}=(\tau^{(i^\star)},\mathbf{v}^{(i^\star)},\mathbf{s}^{(i^\star)})$ of the signing oracle, for some index $i^\star \in \{1,\ldots,Q\}$. However,
+if $\mathsf{Msg}^\star=(\mathfrak{m}_1^\star,\ldots,\mathfrak{m}_N^\star)$ and $\mathsf{Msg}^{(i^\star)}=(\mathfrak{m}_1^{(i^\star)},\ldots,\mathfrak{m}_N^{(i^\star)})$ denote the forgery
+message and the $i^\star$-th signing query, respectively, we have
+$\mathbf{D}_0 \cdot \mathbf{s}^\star + \sum_{k=1}^N \mathbf{D}_k   \cdot \mathfrak{m}_k^\star \neq  \mathbf{D}_0 \cdot \mathbf{s}^{(i^\star)} + \sum_{k=1}^N \mathbf{D}_k  \cdot \mathfrak{m}_k^{(i^\star)}. $
+\item[Type III attacks]  are those where the adversary's forgery $sig^\star=(\tau^\star,\mathbf{v}^\star,\mathbf{s}^\star)$ recycles $\tau^\star $ from an output
+ $sig^{(i^\star)}=(\tau^{(i^\star)},\mathbf{v}^{(i^\star)},\mathbf{s}^{(i^\star)})$ of the signing oracle (i.e.,
+$\tau^{(i^\star)}= \tau^\star$  for some index $i^\star \in \{1,\ldots,Q\}$) and  we have the collision
+\begin{eqnarray} \label{collision}
+\mathbf{D}_0 \cdot \mathbf{s}^\star + \sum_{k=1}^N \mathbf{D}_k   \cdot \mathfrak{m}_k^\star =  \mathbf{D}_0 \cdot \mathbf{s}^{(i^\star)} + \sum_{k=1}^N \mathbf{D}_k  \cdot \mathfrak{m}_k^{(i^\star)}.
+\end{eqnarray}
+\end{description}
+Type III attacks   imply a collision for the  chameleon hash function of Kawachi \textit{et al.} \cite{KTX08}: if (\ref{collision}) holds,
+  a short vector
+of $\Lambda_q^{\perp}([ \mathbf{D}_0 \mid \mathbf{D}_1 \mid \ldots \mid \mathbf{D}_N])$ is obtained as
+  $$\big({\mathbf{s}^\star}^T- {\mathbf{s}^{(i^\star)}}^T \mid {\mathfrak{m}_1^\star }^T - {\mathfrak{m}_1^{(i^\star)} }^T \mid \ldots \mid {\mathfrak{m}_N^\star }^T - {\mathfrak{m}_N^{(i^\star)} }^T \big)^T,$$   so that a collision breaks  the $\mathsf{SIS}$ assumption. \\ \indent
+The security against Type I attacks is proved by \cref{le:lwe-gs-type-I-attacks} which applies the same technique  as in \cite{Boy10,MP12}. In particular, the prefix guessing technique
+of \cite{HW09} allows keeping the modulus smaller than the number $Q$ of adversarial queries as in \cite{MP12}.
+ In order to deal with Type II attacks, we can leverage the technique of~\cite{BHJ+15}. In \cref{le:lwe-gs-type-II-attacks}, we prove that Type II attack would also contradict   $\mathsf{SIS}$.
+\end{proof}
+
+\begin{lemma} \label{le:lwe-gs-type-I-attacks}
+The scheme is secure against Type I attacks if the $\mathsf{SIS}_{n,m,q,\beta'}$ assumption holds for $\beta' = m^{3/2} \sigma^2 ( \ell+3) + m^{1/2} \sigma_1 $
+\end{lemma}
+
+\begin{proof}
+ Let $\adv$ be a $\ppt$ adversary that can mount a Type I attack with non-negligible success probability $\varepsilon$. We construct a $\ppt$
+   algorithm $\bdv$ that uses $\adv$ to break the~$\SIS_{n,m,q,\beta'}$ assumption.  It takes as input~$\bar{\mathbf{A}} \in
+   \Zq^{n \times m}$ and computes  $\mathbf{v} \in
+   \Lambda_q^{\perp}(\bar{\mathbf{A}})$ with~$0 < \|\mathbf{v}\| \leq \beta'$. \\
+	\indent
+		Algorithm~$\bdv$ first chooses the $\ell$-bit strings $\tau^{(1)},\ldots,\tau^{(Q)} \sample U(\{0,1\}^\ell)$ to be used in signing queries. As in \cite{HW09}, it
+		guesses the shortest prefix such that the string $\tau^\star$ contained in $\adv$'s forgery differs from all prefixes of  $\tau^{(1)},\ldots,\tau^{(Q)}$. To this
+		end, $\bdv$ chooses $i^\dagger \sample U(\{1,\ldots, Q\})$ and $t^\dagger \sample U(\{1,\ldots,\ell\})$ so that, with probability $1/(Q \cdot \ell)$, the longest
+		common prefix between $\tau^\star$ and one of the $\left\{\tau^{(i)} \right\}_{i=1}^Q$ is the string
+		$\tau^\star[1] \ldots \tau^\star[t^\dagger-1] =\tau^{(i^\dagger)}[1] \ldots \tau^{(i^\dagger)}[t^\dagger-1]  \in \{0,1\}^{t^\dagger -1}$ comprised of the
+		first $(t^\dagger-1)$-th bits of $\tau^\star \in \{0,1\}^\ell$. We define $ \tau^\dagger \in \{0,1\}^{t^\dagger}$ as the $t^\dagger$-bit string  $\tau^\dagger=\tau^\star[1] \ldots \tau^\star[t^\dagger] $. By construction, with probability $1/(Q \cdot \ell)$, we have $\tau^\dagger \not \in \left\{\tau^{(1)}_{|{t^\dagger}}, \ldots ,\tau^{(Q)}_{|{t^\dagger}} \right\}$, where $\tau^{(i)}_{|{t^\dagger}}$ denotes
+		the $t^\dagger$-th prefix of $\tau^{(i)}$ for each $i\in \{1,\ldots,Q\}$. \\
+		\indent
+				Then, $\bdv$  runs
+   $\TrapGen(1^n,1^m,q)$ to obtain $\mathbf{C} \in \Zq^{n \times m}$ and a
+   basis $\mathbf{T}_{\mathbf{C}}$ of~$\Lambda_q^{\perp}(\mathbf{C})$ with
+   $\|\widetilde{\mathbf{T}_{\mathbf{C}}}\| \leq \bigO(\sqrt{n \log q})$. Then,
+   it picks~$\ell+1$ matrices~$\mathbf{Q}_0,\ldots, \mathbf{Q}_{\ell} \in \ZZ^{m \times m}$, where
+   each matrix $\mathbf{Q}_i$ has its columns  sampled independently from~$D_{\ZZ^m, \sigma}$.   The
+   reduction $\bdv$ defines the matrices $\{ \mathbf{A}_j\}_{j=0}^{\ell}$ as
+ \begin{eqnarray*}
+ \left\{
+ \begin{array}{ll}
+  \mathbf{A}_0 =  \bar{\mathbf{A}} \cdot \mathbf{Q}_0 +  (\sum_{j=1}^{t^\dagger} {\tau^\star[j]}) \cdot
+   \mathbf{C} \\
+ \mathbf{A}_j =  \bar{\mathbf{A}} \cdot \mathbf{Q}_j  + (-1)^{\tau^\star[j]} \cdot
+   \mathbf{C}, \qquad \quad \text{ for  }  j \in
+   [1,t^\dagger] \\
+	\mathbf{A}_j =  \bar{\mathbf{A}} \cdot \mathbf{Q}_j , \qquad \quad \qquad \quad~~ \qquad \quad \text{ for  }  j \in
+   [t^\dagger+1,\ell]
+ \end{array}
+  \right.
+ \end{eqnarray*}
+ It also sets $\mathbf{A}=\bar{\mathbf{A}}$.
+  We note that  we have
+% \vspace*{-.1cm}
+ \begin{eqnarray*}
+   \mathbf{A}_{\tau^{(i)}} &=& \left[ \begin{array}{c|c} \bar{\mathbf{A}}  &   \mathbf{A}_0 +
+ \sum_{j=1}^\ell   \tau^{(i)}[j] \mathbf{A}_j
+ \end{array} \right] \\
+ & = & \left[
+\begin{array}{c|c}
+	 \bar{\mathbf{A}} ~ & ~  \bar{\mathbf{A}} \cdot (\mathbf{Q}_0 +
+   \sum_{j=1}^{\ell}  \tau^{(i)}[j] \mathbf{Q}_j) + (
+   \sum_{j=1}^{t^\dagger}  \tau^\star[j]  +(-1)^{\tau^\star[j]}  \tau^{(i)}[j])\cdot  \mathbf{C}
+\end{array} \right]
+\\
+ &=&
+\left[
+\begin{array}{c|c}
+ \bar{\mathbf{A}} ~ & ~ \bar{\mathbf{A}} \cdot  (\mathbf{Q}_0 +
+   \sum_{j=1}^{\ell} \tau^{(i)}[j] \mathbf{Q}_j) +  h_{\tau^{(i)}} \cdot \mathbf{C}
+ \end{array} \right]
+ \end{eqnarray*}
+ where $  h_{\tau^{(i)}} \in [1,t^\dagger] \subset [1,\ell]$ stands for the Hamming distance between
+ $\tau^{(i)}_{|t^\dagger}$ and $\tau^\star_{|t^\dagger}$. Note that, with probability $1/(Q \cdot \ell)$ and since $q>\ell$, we have
+ $  h_{\tau^{(i)}} \neq 0 \bmod q$ whenever $\tau^{(i)}_{|t^\dagger} \neq \tau^\star_{|t^\dagger}$. \\
+\indent  Next, $\bdv$ chooses the matrices $\mathbf{D}_k \sample U(\Zq^{2n  \times 2m})$ uniformly at random for each $k \in [0,N]$. Then, it picks a random short matrix $\mathbf{R}  \in \ZZ^{m \times m}$ which  has its columns independently sampled from $D_{\ZZ^m,\sigma}$
+ and computes
+\begin{eqnarray*}
+\mathbf{D} &=&   \bar{\mathbf{A}} \cdot \mathbf{R}. % \qquad \qquad \qquad \quad~ \forall k \in \{0,\ldots,N\}.
+\end{eqnarray*}
+Finally, $\bdv$ samples a short vector $\mathbf{e}_u \sample D_{\ZZ^m,\sigma_1}$  and computes the vector   $\mathbf{u} \in \Zq^n$
+as    $\mathbf{u} =      \bar{\mathbf{A}} \cdot \mathbf{e}_u  \in \Zq^n$. The public key  $${PK}:=\big( \mathbf{A}, ~
+  \{\mathbf{A}_j \}_{j=0}^{\ell},  ~ \{\mathbf{D}_k\}_{k=0}^{N},~\mathbf{D}, ~\mathbf{u} \big)$$
+is given to $\adv$.
+
+%Hence,
+%   $\bdv$ is able to compute a trapdoor $\mathbf{T}_{\tau^{(i)}} \in \ZZ^{2m \times 2m}$  for each matrix $\{\mathbf{A}_{\tau^{(i)}} \}_{i=1}^Q $  (see~\cite[Se.~4.2]{ABB1},
+% using the basis~$\mathbf{T}_{\mathbf{C}}$ of~$\Lambda_q^{\perp}(\mathbf{C})$.
+
+
+At the $i$-th signing query $\mathsf{Msg}^{(i)}=(\mathfrak{m}_1^{(i)},\ldots,\mathfrak{m}_N^{(i)}) \in (\{0,1\}^{2m})^N$,  $\bdv$ can use the trapdoor $\mathbf{T}_{\mathbf{C}} \in \ZZ^{m \times m}$ to generate a signature.
+To do this, $\bdv$ first samples    $\mathbf{s}^{(i)} \sample D_{\ZZ^{2m},\sigma_1}$ and computes a vector $\mathbf{u}_M \in \Zq^m$ as
+ $$\mathbf{u}_M  = \mathbf{u}  +  \mathbf{D} \cdot \bit \bigl( \sum_{k=1}^N    \mathbf{D}_k  \cdot {\mathfrak{m}_k^{(i)} }  +   \mathbf{D}_{0} \cdot {\mathbf{s}^{(i)} } \bigr)    ~~  \bmod q.$$
+ Using $\mathbf{T}_{\mathbf{C}} \in \ZZ^{m \times m}$, $\bdv$ can then sample  a short vector $\mathbf{v}^{(i)} \in \ZZ^{2m}$ in $D^{\mathbf{u}_M}_{\Lambda^{\perp}(\mathbf{A}_{\tau^{(i)}}), \sigma}$ such
+that $\big(\tau^{(i)},\mathbf{v}^{(i)},\mathbf{s}^{(i)} \big)$ satisfies the verification equation (\ref{ver-eq-block}).  \\
+\indent  When $\adv$ halts, it outputs a valid signature $sig^\star=\big(\tau^{(i^\dagger)},\mathbf{v}^\star,\mathbf{s}^\star \big)$ on a
+message $\mathsf{Msg}^\star=(\mathfrak{m}_1^\star,\ldots,\mathfrak{m}_N^\star)$ with $\| \mathbf{v}^\star \| \leq \sigma \sqrt{2m}$ and $\| \mathbf{s}^\star \| \leq \sigma_1 \sqrt{2m}$.
+At this point, $\bdv$ aborts and declares failure if it was unfortunate in its choice of $i^\dagger \in \{1,\ldots,Q\}$ and $t^\dagger \in \{1,\ldots,\ell\}$. Otherwise,
+with probability $1/(Q \cdot \ell)$, $\bdv$ correctly guessed $i^\dagger \in \{1,\ldots,Q\}$ and $t^\dagger \in \{1,\ldots,\ell\}$, in which case it can solve the given $\mathsf{SIS}$ instance as follows.
+
+If we parse $\mathbf{v}^\star \in \ZZ^{2m}$ as $({\mathbf{v}_1^\star }^T \mid {\mathbf{v}_2^\star }^T )^T$ with $\mathbf{v}_1^\star,\mathbf{v}_2^\star \in \ZZ^m$, we have the equality
+ \begin{align*}
+   &\left[ \begin{array}{c|c} \bar{\mathbf{A}} ~&~  \bar{\mathbf{A}}    \cdot  (\mathbf{Q}_0 +
+       \sum_{j=1}^{\ell}  \tau^\star[j] \mathbf{Q}_j)
+   \end{array} \right] \cdot 
+   \left[\begin{array}{c} {\mathbf{v}_1^\star } \\ \hline {\mathbf{v}_2^\star } \end{array} \right] \\
+   & \hspace{3cm}=   \mathbf{u} +  \mathbf{D} \cdot \bit \bigl( \mathbf{D}_0 \cdot {\mathbf{s}^{\star} } +
+   \sum_{k=1}^N \mathbf{D}_k \cdot  {\mathfrak{m}_k^{\star} } \bigr) \bmod q \\
+   & \hspace{3cm}=  \bar{\mathbf{A}} \cdot \Bigl( \mathbf{e}_u +    \mathbf{R}  \cdot \bit \bigl( \mathbf{D}_0 \cdot {\mathbf{s}^{\star} } +
+   \sum_{k=1}^N \mathbf{D}_k \cdot  {\mathfrak{m}_k^{\star} } \bigr)  \Bigr)   \bmod q ,
+ \end{align*}
+which implies that the vector
+\begin{eqnarray*}
+\mathbf{w}  &=& {\mathbf{v}_1^\star }  +        (\mathbf{Q}_0 +
+   \sum_{j=1}^{\ell}  \tau^\star[j] \mathbf{Q}_j) \cdot {\mathbf{v}_2^\star } - \mathbf{e}_u -     \mathbf{R}  \cdot \bit \bigl( \mathbf{D}_0 \cdot {\mathbf{s}^{\star} } +
+  \sum_{k=1}^N \mathbf{D}_k \cdot  {\mathfrak{m}_k^{\star} } \bigr)  \in \ZZ^m
+\end{eqnarray*}
+is in $\Lambda_q^{\perp}(\bar{\mathbf{A}})$. Moreover, with overwhelming probability, this vector is non-zero since, in $\adv$'s view, the distribution of
+ $\mathbf{e}_u \in \ZZ^m$ is $D_{\Lambda_q^{\mathbf{u}}(\bar{\mathbf{A}}),\sigma_1}$, which ensures that $\mathbf{e}_u$ is  statistically hidden by
+the syndrome $\mathbf{u}  =   \bar{\mathbf{A}}  \cdot  \mathbf{e}_u $. Finally, the norm of $\mathbf{w}$ is smaller than
+% modified by Khoa: $\| \mathbf{w} \| \leq m^{3/2} \sigma ( \sigma_1 + N / \sqrt{2}) + m^{1/2} ( \sigma + \sigma_1) + (\ell+1) \sigma m$,
+$\beta' = m^{3/2} \sigma^2 ( \ell+3) + m^{1/2} \sigma_1 $
+ which yields a valid solution of the given $\mathsf{SIS}_{n,m,q,\beta'}$ instance
+ with overwhelming probability.
+\end{proof}
+
+\begin{lemma} \label{le:lwe-gs-type-II-attacks}
+The scheme is secure against Type II attacks if the $\mathsf{SIS}_{n,m,q,\beta''}$ assumption holds for $\beta'' = \sqrt{2}   (\ell+2)   \sigma^2  m^{3/2} + m^{1/2} $.
+\end{lemma}
+
+\begin{proof}
+We prove the result using a sequence of games. For each $i$, we denote by $W_i$ the event that the adversary wins by outputting a Type II forgery in \textsf{Game}   $i$.
+\medskip
+
+\begin{description}
+  \item[\textsf{Game} 0:] This is the real game where, at the $i$-th signing query $\mathsf{Msg}^{(i)}=(\mathfrak{m}_1^{(i)},\ldots,\mathfrak{m}_N^{(i)})$,
+    the adversary obtains a signature $sig^{(i)}=(\tau^{(i)},\mathbf{v}^{(i)},\mathbf{s}^{(i)})$ for each $i \in \{1,\ldots,Q\}$ from the signing oracle. At the end of the game, the adversary
+    outputs a forgery  $sig^\star=(\tau^\star,\mathbf{v}^\star,\mathbf{s}^\star)$ on a message $\mathsf{Msg}^{\star}=(\mathfrak{m}_1^{\star},\ldots,\mathfrak{m}_N^{\star})$.
+    By hypothesis, the adversary's advantage is $\varepsilon = \Pr[W_0]$. We assume without loss of generality that the random $\ell$-bit strings $\tau^{(1)}, \ldots, \tau^{(Q)}$ are chosen
+    at the very beginning of the game.
+    Since $(\mathsf{Msg}^\star,sig^\star)$ is a Type II forgery, there exists an index $i^\star \in \{1,\ldots,Q\}$ such that $\tau^\star =\tau^{(i^\star)} $.
+
+  \item[\textsf{Game} 1:] This game is identical to \textsf{Game} $0$ with the difference that the reduction aborts the experiment in the unlikely event that, in the adversary's forgery
+    $sig^\star=(\tau^\star,\mathbf{v}^\star,\mathbf{s}^\star)$, $\tau^\star$ coincides with more than one of the random $\ell$-bit strings $\tau^{(1)}, \ldots, \tau^{(Q)}$
+    used by the challenger. If we call $F_1$ the latter event, we have $\Pr[F_1]  < Q^2/2^\ell$ since we are guaranteed to have $\neg F_1$ as long as no two $\tau^{(i)}$, $\tau^{(i')}$ collide.
+    Given that  \textsf{Game} $1$ is identical to \textsf{Game} $0$ until $F_1$ occurs, we have $|\Pr[W_1]-\Pr[W_0]| \leq \Pr[F_1] < Q^2/2^\ell$.
+
+  \item[\textsf{Game} 2:] This game is like  \textsf{Game} $1$ with the following difference. At the outset of the game, the challenger $\bdv$ chooses a random index
+    $i^\dagger \sample U(\{1,\ldots,Q\})$ as a guess that $\adv$'s forgery will  recycle the $\ell$-bit string $\tau^{(i^\dagger)} \in \{0,1\}^\ell$ of the $i^\dagger$-th signing query.
+    When $\adv$ outputs its Type II forgery  $sig^\star=(\tau^\star,\mathbf{v}^\star,\mathbf{s}^\star)$,  the challenger aborts
+    in the event that   $\tau^{(i^\dagger)} \neq \tau^\star$ (i.e., $i^\dagger \neq i^\star$). Since the choice of $i^\dagger $ in $\{1,\ldots,Q\}$  is  independent of $\adv$'s view, we
+    have $\Pr[W_2]=\Pr[W_1]/Q$.
+
+  \item[\textsf{Game} 3:] In this game, we modify the key generation phase and the way to answer signing queries.
+    First, the challenger $\bdv$ randomly picks $h_0,h_1,\ldots,h_\ell \in \Zq$ subject to the constraints
+    \begin{eqnarray*}
+      h_0 + \sum_{j=1}^\ell \tau^{(i^\dagger)}[j] \cdot h_j &=& 0 \bmod q  \\
+      h_0 + \sum_{j=1}^\ell \tau^{(i)}[j] \cdot h_j & \neq & 0 \bmod q   \qquad \qquad i \in \{1,\ldots,Q\} \setminus \{i^\dagger\}
+    \end{eqnarray*}
+    It   runs $(\mathbf{C},\mathbf{T}_{\mathbf{C}}) \leftarrow \mathsf{TrapGen}(1^n,1^m,q)$,
+    $(\mathbf{D}_0,\mathbf{T}_{\mathbf{D}_0}) \leftarrow \mathsf{TrapGen}(1^{2n},1^{2m},q)$ so as to obtain  statistically random matrices $\mathbf{C} \in \Zq^{n \times m} $, $\mathbf{D}_0 \in \Zq^{2n \times 2m}$ with
+    trapdoors $\mathbf{T}_{\mathbf{C}} \in \ZZ^{m \times m}$,  $\mathbf{T}_{\mathbf{D}_0}  \in \ZZ^{2m \times 2m}$ consisting of  short bases of $\Lambda_q^{\perp}(\mathbf{C})$ and $\Lambda_q^{\perp}(\mathbf{D}_0)$, respectively. Then,
+    $\bdv$
+    chooses
+    a uniformly random  $\mathbf{D} \sample U(\Zq^{n \times m})$ and re-randomizes it using short matrices
+    $\mathbf{S},\mathbf{S}_0,\mathbf{S}_1,\ldots,\mathbf{S}_{\ell} \sample \ZZ^{m \times m}$, which are obtained
+    by sampling their columns from the distribution $D_{\ZZ^m,\sigma}$. Namely, from  $\mathbf{D}  \in \Zq^{n \times m}$, $\bdv$
+    defines
+    \begin{eqnarray} \nonumber
+      \mathbf{A} &=&  \mathbf{D} \cdot \mathbf{S} \\ \label{setup-sig3}
+      \mathbf{A}_0 &=&  \mathbf{D} \cdot \mathbf{S}_0 + h_0 \cdot \mathbf{C} \\ \nonumber
+      \mathbf{A}_j &=&  \mathbf{D} \cdot \mathbf{S}_j + h_j \cdot \mathbf{C} \qquad \qquad \forall j \in \{1,\ldots,\ell\} %\\ \nonumber
+%\mathbf{D}_k &=&  \mathbf{D} \cdot \mathbf{R}_k  \qquad \qquad \qquad \quad~ \forall k \in \{1,\ldots,N\}.
+    \end{eqnarray}
+    In addition, $\bdv$ picks random matrices $\mathbf{D}_1,\ldots,\mathbf{D}_N \sample U(\Zq^{2n \times 2m})$ and a random vector $\mathbf{c}_M \sample U(\Zq^{2n})$. It samples
+    short vectors $\mathbf{v}_1 ,\mathbf{v}_2 \sample D_{\ZZ^m,\sigma}$ and computes $\mathbf{u} \in \Zq^n$
+    as    $\mathbf{u}  =     \mathbf{A}_{\tau^{(i^\dagger)}} \cdot
+    \left[
+      \begin{array}{c}
+        \mathbf{v}_1 \\ \hline \mathbf{v}_2
+    \end{array} \right]
+    -  \mathbf{D}  \cdot \bit( \mathbf{c}_M )  \bmod q$, where
+    \begin{eqnarray*}
+      \mathbf{A}_{\tau^{(i^\dagger)}} &=&  \left[
+        \begin{array}{c|c}   \mathbf{A} ~ & ~ \mathbf{A}_0 +
+          \sum_{j=1}^\ell \tau^{(i^\dagger)}[j] \cdot \mathbf{A}_j
+      \end{array} \right]    \\ &=&     \left[
+        \begin{array}{c|c}    \mathbf{D} \cdot \mathbf{S} ~  & ~  \mathbf{D}\cdot (\mathbf{S}_0 +
+          \sum_{j=1}^\ell \tau^{(i^\dagger)}[j] \cdot \mathbf{S}_j)
+      \end{array} \right]  .
+    \end{eqnarray*}
+    The adversary's signing queries are then answered as follows.
+    \begin{itemize}
+      \item At the $i$-th signing query $ (\mathfrak{m}_1^{(i)},\ldots,\mathfrak{m}_N^{(i)})$, whenever $i \neq i^\dagger$, we have
+        \begin{eqnarray*}
+          \mathbf{A}_{\tau^{(i)}} &=&  \left[
+            \begin{array}{c|c} \mathbf{A} ~& ~    \mathbf{A}_0 +
+              \sum_{j=1}^\ell \tau^{(i)}[j] \cdot \mathbf{A}_j
+          \end{array} \right]  \\
+          &=&   \left[
+            \begin{array}{c|c}  \mathbf{A} ~ &    ~   \mathbf{D} \cdot (\mathbf{S}_0 +
+              \sum_{j=1}^\ell \tau^{(i)}[j] \cdot \mathbf{S}_j) + h_{\tau^{(i)}} \cdot \mathbf{C}
+          \end{array} \right]
+          \in  \Zq^{ n  \times 2m},
+        \end{eqnarray*}
+        with $h_{\tau^{(i)}} = h_0 + \sum_{j=1}^\ell \tau^{(i)}[j] \cdot h_j \neq 0$. This implies that $\bdv$ can use the trapdoor $\mathbf{T}_{\mathbf{C}} \in \ZZ^{m \times m}$ to generate a signature.
+        To this end, $\bdv$ first samples  a discrete Gaussian vector $\vec{s}^{(i)} \sample D_{\ZZ^{2m},\sigma_1}$ and computes  $\mathbf{u}_M \in \Zq^n$ as
+        $$\mathbf{u}_M  = \mathbf{u}  + \mathbf{D} \cdot \bit( \sum_{k=1}^N   \mathbf{D}_k \cdot {\mathfrak{m}_k^{(i)} }  +   \mathbf{D}_{0} \cdot {\mathbf{s}^{(i)} } ) ~~  \bmod q.$$ Then,
+        using $\mathbf{T}_{\mathbf{C}} \in \ZZ^{m \times m}$, it samples  a short vector $\mathbf{v}^{(i)} \in \ZZ^{2m}$ in $D^{\mathbf{u}_M}_{\Lambda^{\perp}(\mathbf{A}_{\tau^{(i)}}), \sigma}$ such
+        that $(\tau^{(i)},\mathbf{v}^{(i)},\mathbf{s}^{(i)})$ satisfies (\ref{ver-eq-block}).
+      \item At the $i^\dagger$-th signing query $ (\mathfrak{m}_1^{(i^\dagger)},\ldots,\mathfrak{m}_N^{(i^\dagger)})$, we have
+        \begin{eqnarray} \nonumber
+          \mathbf{A}_{\tau^{(i^\dagger)}} &=&   \left[
+            \begin{array}{c|c}   \mathbf{A} ~& ~    \mathbf{A}_0 +
+              \sum_{j=1}^\ell \tau^{(i^\dagger)}[j] \cdot \mathbf{A}_j
+          \end{array} \right] \\
+          \label{i-mat} &=&   \left[
+            \begin{array}{c|c}  \mathbf{D} \cdot  \mathbf{S}  ~&~     \mathbf{D}   \cdot (\mathbf{S}_0 +
+              \sum_{j=1}^\ell \tau^{(i^\dagger)}[j] \cdot \mathbf{S}_j)
+          \end{array} \right]
+          \in  \Zq^{ n \times 2m} \quad
+        \end{eqnarray}
+        due to the constraint $h_0 + \sum_{j=1}^\ell \tau^{(i^\dagger)}[j] \cdot h_j = 0 \bmod q $.
+        To answer the query, $\bdv$ uses the trapdoor $\mathbf{T}_{\mathbf{D}_0} \in \ZZ^{2m \times 2m}$ of $\Lambda_q^{\perp}(\mathbf{D}_0)$ to sample a short vector
+        $\mathbf{s}^{(i^\dagger)} \in D_{\Lambda_q^{\mathbf{c}'_M} (\mathbf{D}_0), \sigma_1}$, where $\mathbf{c}'_M = \mathbf{c}_M - \sum_{k=1}^N \mathbf{D}_k  \cdot  {\mathfrak{m}_k^{(i^\dagger)} } \in \Zq^{2n}$.
+        The obtained vector $\mathbf{s}^{(i^\dagger)} \in \ZZ^{2m}$ thus verifies
+
+        \begin{eqnarray} \label{sim-s}
+          \mathbf{D}_0 \cdot {\mathbf{s}^{(i^\dagger)} }  &=&
+          \mathbf{c}_M - \sum_{k=1}^N \mathbf{D}_k  \cdot  {\mathfrak{m}_k^{(i^\dagger)} } ~\bmod q,
+          \quad
+        \end{eqnarray}
+        and $\adv$ receives $sig^{(i^\dagger)}=(\tau^{(i^\dagger)},\mathbf{v}^{(i^\dagger)},\mathbf{s}^{(i^\dagger)})$, where $ \mathbf{v}^{(i^\dagger)} = (\mathbf{v}_1^T \mid \mathbf{v}_2^T)^T $.
+        By construction, the returned signature $sig^{(i^\dagger)}$ satisfies
+        \begin{eqnarray*}
+          \mathbf{A}_{\tau^{(i^\dagger)}}
+          \cdot \left[ \begin{array}{c}
+          \mathbf{v}_1 \\ \hline  \mathbf{v}_2 \end{array} \right]
+          &=& \mathbf{u} +  \mathbf{D} \cdot \bit \bigl( \mathbf{D}_0 \cdot  {\mathbf{s}^{(i^\dagger)} }  + \sum_{k=1}^N  \mathbf{D}_k \cdot {\mathfrak{m}_k^{(i^\dagger)} } \bigr) \quad \bmod q,
+        \end{eqnarray*}
+        and the distribution of $(\tau^{(i^\dagger)},\mathbf{v}^{(i^\dagger)},\mathbf{s}^{(i^\dagger)})$ is statistically the same as in \textsf{Game} $2$.
+    \end{itemize}
+\end{description}
+We   conclude that  $\Pr[W_2]$ is negligibly far apart from  $\Pr[W_3]$ since, by the  Leftover Hash Lemma (see \cite[Le. 13]{ABB10}), the public key $PK$ in \textsf{Game} $3$ is statistically close to its distribution in \textsf{Game} $2$.
+\medskip
+
+\noindent
+In \textsf{Game} $3$, we claim that the challenger $\bdv$ can use $\adv$ to solve the $\mathsf{SIS}$ problem by finding a short vector of $\Lambda_q^\perp(\mathbf{D})$ with probability $\Pr[W_3]$. Indeed,
+  with proba\-bility $\Pr[W_3]$,  the adversary outputs a valid signature $sig^\star=(\tau^{(i^\dagger)},\mathbf{v}^\star,\mathbf{s}^\star)$ on a message $\mathsf{Msg}^\star=(\mathfrak{m}_1^\star,\ldots,\mathfrak{m}_N^\star)$ with $\| \mathbf{v}^\star \| \leq \sigma \sqrt{2m}$ and $\| \mathbf{s}^\star \| \leq \sigma_1 \sqrt{2m}$.
+If we parse $\mathbf{v}^\star \in \ZZ^{2m}$ as $({\mathbf{v}_1^\star }^T \mid {\mathbf{v}_2^\star }^T )^T$ with $\mathbf{v}_1^\star,\mathbf{v}_2^\star \in \ZZ^m$, we have
+the equality
+\begin{eqnarray} \label{first-sol}
+  \mathbf{A}_{\tau^{(i^\dagger)}} \cdot \left[ \begin{array}{c}
+ \mathbf{v}_1^\star  \\ \hline    \mathbf{v}_2^\star
+\end{array} \right]
+ &=& \mathbf{u} + \mathbf{D} \cdot \bit (  \mathbf{D}_0 \cdot {\mathbf{s}^{\star} }
++ \sum_{k=1}^N   \mathbf{D}_k  \cdot {\mathfrak{m}_k^{\star} } ) \quad \bmod q.
+\end{eqnarray}
+
+Due to the way $\mathbf{u} \in \Zq^n$ was defined at the outset of the game,  $\bdv$ also knows short vectors $\mathbf{v}^{(i^\dagger)}=(\mathbf{v}_1^T \mid \mathbf{v}_2^T)^T \in \ZZ^{2m}$
+such that
+\begin{eqnarray} \label{second-sol}    \mathbf{A}_{\tau^{(i^\dagger)}} \cdot
+  \left[\begin{array}{c}
+ \mathbf{v}_1 \\ \hline \mathbf{v}_2
+\end{array} \right] = \mathbf{u} +  \mathbf{D}  \cdot \bit( \mathbf{c}_M )  \bmod q. \end{eqnarray}
+ Relation (\ref{sim-s}) implies that  $ \mathbf{c}_M \neq \mathbf{D}_0 \cdot {\mathbf{s}^{\star} }
++ \sum_{k=1}^N   \mathbf{D}_k  \cdot {\mathfrak{m}_k^{\star} }  \bmod q$ by hypothesis. It follows  that  $\bit(\mathbf{c}_M) - \bit (   \mathbf{D}_0 \cdot {\mathbf{s}^{\star} }
++ \sum_{k=1}^N   \mathbf{D}_k  \cdot {\mathfrak{m}_k^{\star} } )  $ is a non-zero vector in $\{-1,0,1\}^m$. Subtracting (\ref{second-sol}) from (\ref{first-sol}), we get
+\begin{eqnarray*}
+  \mathbf{A}_{\tau^{(i^\dagger)}} \cdot \left[\begin{array}{c}
+ \mathbf{v}_1^\star - \mathbf{v}_1  \\ \hline    \mathbf{v}_2^\star - \mathbf{v}_1
+\end{array} \right]
+ &=&  \mathbf{D} \cdot \bigl(  \bit(\mathbf{c}_M) - \bit (   \mathbf{D}_0 \cdot {\mathbf{s}^{\star} }
++ \sum_{k=1}^N   \mathbf{D}_k  \cdot {\mathfrak{m}_k^{\star} } ) \bigr)  \mod q,
+\end{eqnarray*}
+which implies
+\begin{multline} \label{eq-un}
+    \left[
+\begin{array}{c|c}  \mathbf{D} \cdot  \mathbf{S}  ~ &~   \mathbf{D} \cdot (\mathbf{S}_0 +
+\sum_{j=1}^\ell \tau^{(i^\dagger)}[j] \cdot \mathbf{S}_j)
+\end{array} \right]  \cdot  \left[ \begin{array}{c}  {\mathbf{v}_1^\star -\mathbf{v}_1 } \\ \hline   {\mathbf{v}_2^\star - \mathbf{v}_2 } \end{array} \right] \\ =  \mathbf{D} \cdot \bigl(  \bit(\mathbf{c}_M) - \bit (   \mathbf{D}_0 \cdot {\mathbf{s}^{\star} }
++ \sum_{k=1}^N   \mathbf{D}_k  \cdot {\mathfrak{m}_k^{\star} } ) \bigr)  \mod q .
+\end{multline}
+  The above implies that the vector
+\begin{eqnarray} \nonumber
+  \mathbf{w} &=&
+  \mathbf{S}  \cdot (\mathbf{v}_1^\star - \mathbf{v}_1) + (\mathbf{S}_0  + \sum_{j=1}^\ell \tau^{(i^\dagger)}[j] \cdot \mathbf{S}_j ) \cdot (\mathbf{v}_2^\star - \mathbf{v}_2)  \\
+  \nonumber && \hspace{2.75cm} ~+ \bit \big(   \mathbf{D}_0 \cdot {\mathbf{s}^{\star} } + \sum_{k=1}^N   \mathbf{D}_k  \cdot {\mathfrak{m}_k^{\star} }  \big) -  \bit(\mathbf{c}_M)
+\end{eqnarray}
+is a short integer vector of $\Lambda_q^{\perp}(\mathbf{D})$. Indeed,   its  norm can be bounded as  $\| \mathbf{w} \| \leq \beta'' = \sqrt{2}   (\ell+2)   \sigma^2  m^{3/2} + m^{1/2} $. We argue that it is non-zero with overwhelming probability. We already observed that
+$ \bit (   \mathbf{D}_0 \cdot {\mathbf{s}^{\star} }
++ \sum_{k=1}^N   \mathbf{D}_k  \cdot {\mathfrak{m}_k^{\star} } ) - \bit(\mathbf{c}_M)$ is a non-zero vector of $\{-1,0,1\}^m$, which rules out the event that
+ $({\mathbf{v}_1^\star}, {\mathbf{v}_2^\star} ) =({\mathbf{v}_1} , {\mathbf{v}_2}) $. Hence, we can only have $\mathbf{w}=\mathbf{0}^m$ when the equality
+\begin{multline} \label{final-eq}
+ \mathbf{S}  \cdot (\mathbf{v}_1^\star - \mathbf{v}_1) + (\mathbf{S}_0  +
+\sum_{j=1}^\ell \tau^{(i^\dagger)}[j] \cdot \mathbf{S}_j ) \cdot (\mathbf{v}_2^\star - \mathbf{v}_2) \\ = \bit(\mathbf{c}_M)  -  \bit \big(   \mathbf{D}_0 \cdot {\mathbf{s}^{\star} }
++ \sum_{k=1}^N   \mathbf{D}_k  \cdot {\mathfrak{m}_k^{\star} }  \big) \qquad
+\end{multline}
+holds over $\ZZ$. However, as long as either $\mathbf{v}_1^\star \neq \mathbf{v}_1$ or $\mathbf{v}_2^\star \ne \mathbf{v}_2$, the left-hand-side member of (\ref{final-eq})
+ 	is information theoretically unpredictable since the columns of matrices $\mathbf{S}$ and $\{\mathbf{S}_j\}_{j=0}^\ell$ are statistically hidden in the view of $\adv$.
+ Indeed, conditionally on  the public key, each column of  $\mathbf{S}$ and $\{\mathbf{S}_j\}_{j=0}^\ell$ has at least $n$ bits
+ of min-entropy, as shown by, e.g., \cite[Le. 2.7]{MP12}.
+\end{proof}
+
+
+\subsection{Protocols for Signing a Committed Value and Proving Possession of a Signature} \label{commit-sig}
+
+
+
+We first show  a two-party protocol  whereby a user can interact with the signer in order to obtain a signature on a committed message. \\
+\indent In order to prove that the scheme still guarantees unforgeability for obliviously signed messages,
+ we will assume that each message block $\mathfrak{m}_k \in \{0,1\}^{2m}$ is obtained by encoding
+the actual message $M_k =M_k[1] \ldots M_k[m] \in \{0,1\}^m$ as $\mathfrak{m}_k= \mathsf{Encode}(M_k)=( \bar{M}_k[1] , M_k[1],\ldots, \bar{M}_k[m] , M_k[m] ) $. Namely,
+each $0$ (respectively each $1$) is encoded as a pair $(1,0)$ (resp. $(0,1)$). The reason for this encoding is that the proof of   Theorem \ref{commit-thm} requires that at least one block
+$\mathfrak{m}_k^\star $ of the forgery message is $1$ while the same bit is $0$ at some specific signing query. We will show (see \cref{se:gs-lwe-stern}) that the correctness of this encoding can
+be efficiently proved using Stern-like~\cite{Ste96} protocols.
+
+To sign committed messages, a first idea is exploit the fact that our signature of Section  \ref{desc-sig-protoc} blends well with the $\mathsf{SIS}$-based commitment scheme suggested by Kawachi \textit{et al.}~\cite{KTX08}.
+  In the latter scheme, the commitment key consists of matrices $(\mathbf{D}_0,\mathbf{D}_1) \in \Zq^{2n \times 2m} \times \Zq^{2n \times 2m}$, so that    message
+	$\mathfrak{m} \in \{0,1\}^{2m}$ can be committed to by sampling a   Gaussian vector $\mathbf{s} \sample D_{\ZZ^{2m},\sigma}$ and computing
+	$\mathbf{C}=  \mathbf{D}_0 \cdot \mathbf{s} + \mathbf{D}_1 \cdot \mathfrak{m}  \in \Zq^{2n}$. This scheme  extends to commit to multiple messages $(\mathfrak{m}_1,\ldots,\mathfrak{m}_N)$ at once by computing
+$\mathbf{C}=\mathbf{D}_0 \cdot \mathbf{s} + \sum_{k=1}^N \mathbf{D}_k \cdot \mathfrak{m}_k  \in \Zq^{2n}$ using a longer
+ commitment key $(\mathbf{D}_0,\mathbf{D}_1,\ldots,\mathbf{D}_N) \in (\Zq^{2n \times 2m})^{N+1} $. It is easy to see that the resulting commitment remains statistically hiding and computationally
+ binding under the $\mathsf{SIS}$ assumption.
+
+%If we assume that the signer only sees  perfectly hiding commitments $ \mathbf{c}_{\mathfrak{m}} = \mathbf{D}_0 \cdot \mathbf{s}' + \sum_{k=1}^N \mathbf{D}_k \cdot \mathfrak{m}_k$ and $\mathbf{C}= \mathbf{B}_0 \cdot %\mathbf{r} + \sum_{k=1}^N \mathbf{B}_k \cdot \mathfrak{m}_k$ to the message    $(\mathfrak{m}_1,\ldots,\mathfrak{m}_N) \in (\{0,1\}^m)^N$ on which the
+%user wants to obtain a signature, a simple way for the
+%user to prove that $\mathbf{C}$ and $ \mathbf{c}_{\mathfrak{m}}$ are commitments to the same message is to
+%  generate a  witness indistinguishable proof  of knowledge of a short vector
+%  $$\mathbf{v}=[ \mathfrak{m}_1^T \mid \ldots \mid \mathfrak{m}_N^T  \mid  \mathbf{r}^T \mid {\mathbf{s}'}^T  ]^T \in   (\{0,1\}^m)^N \times (\ZZ^m)^2    $$ satisfying
+%	\begin{eqnarray*}
+%   \left[ \begin{array}{c|c|c|c|c|c}
+%\mathbf{B}_1  ~ &  ~  \mathbf{B}_2 ~ & ~  \ldots ~ &~ \mathbf{B}_{N}  ~& ~   \mathbf{B}_0 ~ &      \\ \hline
+% \mathbf{D}_1  ~ & ~ \mathbf{D}_2~ & ~ \ldots  ~ & ~\mathbf{D}_N~ & &   ~ \mathbf{D}_0~
+% \end{array} \right] \cdot \mathbf{v}
+%= \begin{bmatrix}
+%\mathbf{C} \\ \hline   \mathbf{c}_{\mathfrak{m}}
+%\end{bmatrix}.
+%\end{eqnarray*}
+
+In order to make our construction usable in  the definitional framework of Camenisch \textit{et al.} \cite{CKL+15}, we   assume common public parameters
+ (i.e., a common reference string) and encrypt all witnesses of which knowledge is being proved under a public key included in the common reference string. The resulting ciphertexts thus serve as statistically binding commitments
+ to the witnesses.
+ To enable this,  the common public parameters   comprise public keys $\mathbf{G}_0 \in \Zq^{n \times \ell}$, $\mathbf{G}_1 \in \Zq^{n \times 2m}$
+for multi-bit variants of the dual Regev cryptosystem \cite{GPV08} and  all parties are denied access to the underlying private keys. The flexibility of Stern-like protocols allows us to prove that  the content of a perfectly hiding commitment $ \mathbf{c}_{\mathfrak{m}}$ is consistent with
+encrypted values.%, the protocols of Ling \textit{et al.} \cite{LNW15} come in handy.
+
+
+\begin{description}
+\item[\textsf{Global}\textrm{-}\textsf{Setup}:] Let $B = \sqrt{n} \omega(\log n)$ and let $\chi$ be a $B$-bounded distribution.
+Let $p = \sigma \cdot \omega(\sqrt{m})$ upper-bound  entries of vectors sampled from the distribution~$D_{\ZZ^{2m},\sigma}$.
+    Generate two public  keys  for the dual Regev encryption scheme
+    in its multi-bit variant. These keys consists of a public random matrix
+    $\mathbf{B} \sample U(\Zq^{n \times m})$ and  random matrices $\mathbf{G}_0 = \mathbf{B} \cdot \mathbf{E}_0 \in \Zq^{n \times \ell }$, $\mathbf{G}_1  = \mathbf{B} \cdot \mathbf{E}_1  \in \Zq^{n \times 2m}$,
+		 where $\mathbf{E}_0 \in \ZZ^{ m \times \ell}$ and $\mathbf{E}_1 \in \ZZ^{m \times 2m}$ are short Gaussian matrices with columns sampled from $D_{\ZZ^{m},\sigma}$. These matrices  will be
+		used to encrypt integer vectors of dimension $\ell$ and $2m$, respectively.  Finally, generate public parameters $CK:=\{ \mathbf{D}_k \}_{k=0}^N$ consisting of uniformly
+		random matrices $\mathbf{D}_k \sample U(\Zq^{2n \times 2m})$ for a statistically hiding commitment
+			to vectors in $(\{0,1\}^{2m})^N$.
+		Return public parameters consisting of
+		$$ \mathsf{par}:= \{ ~\mathbf{B} \in \Zq^{n \times m} ,~\mathbf{G}_0 \in \Zq^{n \times \ell},~\mathbf{G}_1 \in \Zq^{n \times 2m},~CK  \}.   $$
+%where $p > \sigma_1 \sqrt{m}$ upper-bounds  entries of vectors sampled from the distribution $D_{\ZZ^m,\sigma_1}$,
+
+\item[\textsf{Issue} $\leftrightarrow$ \textsf{Obtain} :]  The signer $S$, who holds a key pair $PK:=\{ \mathbf{A} , ~\{\mathbf{A}_j\}_{j=0}^\ell,~\mathbf{D},~\mathbf{u} \}$, $SK:=\mathbf{T}_{\mathbf{A}}$, interacts with the user $U$
+ who has a message $(\mathfrak{m}_1,\ldots,\mathfrak{m}_N)$, in the following interactive protocol. \smallskip
+\begin{itemize}
+\item[1.] $U$ samples  $\mathbf{s}' \sample D_{\ZZ^{2m},\sigma} $ and computes $ \mathbf{c}_{\mathfrak{m}} = \mathbf{D}_0 \cdot \mathbf{s}' + \sum_{k=1}^N \mathbf{D}_k \cdot \mathfrak{m}_k \in \mathbb{Z}_q^{2n}$
+ which is sent to $S$ as a commitment to $(\mathfrak{m}_1,\ldots,\mathfrak{m}_N)$. In addition, $U$ encrypts  $\{\mathfrak{m}_k\}_{k=1}^N$ and $\mathbf{s}'$ under the dual-Regev public key $(\mathbf{B},\mathbf{G}_1)$
+   by computing for all $k \in \{1,\ldots,N\}$:
+\begin{eqnarray} \label{enc-Mk} \nonumber
+\mathbf{c}_{k} &=& (\mathbf{c}_{k,1},\mathbf{c}_{k,2}) \\ &=&  \big( \mathbf{B}^T \cdot \mathbf{s}_{k} + \mathbf{e}_{k,1} ,~  \mathbf{G}_1^T \cdot \mathbf{s}_{k} + \mathbf{e}_{k,2} + \mathfrak{m}_k \cdot \lfloor q/2 \rfloor  \big) \in \Zq^m \times \Zq^{2m} \qquad %\forall k\in \{1,\ldots,N\}
+%\qquad
+\end{eqnarray}
+for  randomly chosen $\mathbf{s}_{k} \sample \chi^n$, $\mathbf{e}_{k,1} \sample \chi^m$, $\mathbf{e}_{k,2} \sample  \chi^{2m}$,
+and \begin{eqnarray} \label{enc-s} \nonumber
+\mathbf{c}_{s'} &=& (\mathbf{c}_{s',1},\mathbf{c}_{s',2}) \\ &=&  \big( \mathbf{B}^T \cdot \mathbf{s}_{0} + \mathbf{e}_{0,1} ,~  \mathbf{G}_1^T \cdot \mathbf{s}_{0} + \mathbf{e}_{0,2} + \mathbf{s}' \cdot \lfloor q/p \rfloor  \big)  \in \Zq^m \times \Zq^{2m}
+\end{eqnarray}
+ where $\mathbf{s}_{0} \sample \chi^n$, $\mathbf{e}_{0,1} \sample \chi^m$, $\mathbf{e}_{0,2} \sample  \chi^{2m}$.  The ciphertexts $\{\mathbf{c}_k\}_{k=1}^N$ and $\mathbf{c}_{s'}$ are
+sent to $S$ along with $\mathbf{c}_{\mathfrak{m}}$.
+
+Then, $U$ generates an interactive zero-knowledge argument to convince~$S$ that
+$ \mathbf{c}_{\mathfrak{m}}$ is a commitment to $(\mathfrak{m}_1, \ldots, \mathfrak{m}_N)$ with the randomness   $\mathbf{s}'$  such that $\{\mathfrak{m}_k\}_{k=1}^N$ and
+  $\mathbf{s}'$   were honestly encrypted to $\{ \mathbf{c}_{k} \}_{i=1}^N$ and $\mathbf{c}_{s'}$, as in~(\ref{enc-Mk}) and~(\ref{enc-s}).
+%is consistent with the messages encrypted in $\{ \mathbf{c}_{k} \}_{i=1}^N$ and $\mathbf{c}_{s'}$.
+For convenience, this argument system will be described in Section~\ref{subsection:zk-for-commitments}, where we demonstrate that, together with other zero-knowledge protocols used in this work, it can be derived from a Stern-like~\cite{Ste96} protocol constructed in \cref{se:gs-lwe-stern}.
+
+\item[2.]  If the argument of step 1 properly verifies, $S$ samples $\mathbf{s}'' \sample D_{\ZZ^{2m},\sigma_0}$ and computes
+ a vector $\mathbf{u}_{\mathfrak{m}}= \mathbf{u} + \mathbf{D} \cdot \bit \bigl( \mathbf{c}_{\mathfrak{m}} + \mathbf{D}_0 \cdot \mathbf{s}'' \bigr) \in \Zq^n$.
+Next, $S$ randomly picks  $\tau \sample \{0,1\}^\ell$ and
+ uses $\mathbf{T}_{\mathbf{A}}$ to compute a delegated basis $\mathbf{T}_{\tau} \in \ZZ^{2m \times 2m}$ for the matrix $\mathbf{A}_{\tau} \in \Zq^{n \times 2m}$ of (\ref{tau-matrix}).
+Using  $\mathbf{T}_\tau \in \ZZ^{2m \times 2m}$, $S$ samples  a short vector $\mathbf{v} \in \ZZ^{2m}$ in $D^{\mathbf{u}_M}_{\Lambda^{\perp}(\mathbf{A}_\tau), \sigma}$.  It returns
+the vector $( \tau,\mathbf{v},\mathbf{s}'') \in \{0,1\}^\ell \times  \ZZ^{2m} \times  \ZZ^{2m} $  to $U$.
+\item[3.] $U$ computes $\mathbf{s} = \mathbf{s}'+\mathbf{s}''$ over $\ZZ$ and verifies that $$\mathbf{A}_{\tau} \cdot \mathbf{v} = \mathbf{u} + \mathbf{D} \cdot \bit
+\bigl( \mathbf{D}_0 \cdot \mathbf{s} + \sum_{k=1}^N \mathbf{D}_k \cdot \mathfrak{m}_k \bigr) \bmod q.$$ If so, it outputs $(\tau,\mathbf{v},\mathbf{s})$. Otherwise, it outputs $\perp$.
+ \end{itemize}
+\end{description}
+Note that, if both parties faithfully run the protocol, the user obtains a valid signature $(\tau,\mathbf{v},\mathbf{s})$ for which the distribution of $\mathbf{s}$ is $D_{\ZZ^{2m},\sigma_1}$,
+where $\sigma_1=\sqrt{\sigma^2 + \sigma_0^2}$. \\
+\indent The following protocol allows proving possession of a message-signature pair.
+
+\begin{description}
+\item[\textsf{Prove}:] On input of a signature $(\tau,\mathbf{v}=(\mathbf{v}_1^T \mid \mathbf{v}_2^T)^T,\mathbf{s}) \in \{0,1\}^\ell \times \ZZ^{2m} \times \ZZ^{2m}$ on the message $(\mathfrak{m}_1,\ldots,\mathfrak{m}_N)$, the user
+does the following. \smallskip \smallskip
+\begin{itemize}
+\item[1.] Using  $(\mathbf{B},\mathbf{G}_0)$ and $(\mathbf{B},\mathbf{G}_1)$   generate perfectly binding commitments to $\tau \in \{0,1\}^\ell$, $\{\mathfrak{m}_k \}_{k=1}^N$,
+ $\mathbf{v}_1,\mathbf{v}_2 \in \ZZ^m$ and $\mathbf{s} \in \ZZ^{2m}$.  Namely, compute
+\begin{eqnarray*}  \nonumber
+\mathbf{c}_{\tau} &=& (\mathbf{c}_{\tau,1},\mathbf{c}_{\tau,2}) \\ &=&  \big( \mathbf{B}^T \cdot \mathbf{s}_{\tau} + \mathbf{e}_{\tau,1} ,~  \mathbf{G}_0^T \cdot \mathbf{s}_{\tau} + \mathbf{e}_{\tau,2} + \tau
+  \cdot \lfloor q/2 \rfloor  \big)  \in \Zq^m \times \Zq^\ell, \\
+  \mathbf{c}_{k} &=& (\mathbf{c}_{k,1},\mathbf{c}_{k,2}) \\ &=&  \big( \mathbf{B}^T \cdot \mathbf{s}_{k} + \mathbf{e}_{k,1} ,~  \mathbf{G}_1^T \cdot \mathbf{s}_{k} + \mathbf{e}_{k,2} + \mathfrak{m}_k \cdot \lfloor q/2 \rfloor  \big) \in \Zq^m \times \Zq^{2m}
+  \\ && \hspace{7.6cm} \forall k\in \{1,\ldots,N\} \qquad
+\end{eqnarray*}
+where $\mathbf{s}_{\tau}, \mathbf{s}_{k}  \sample \chi^n$, $\mathbf{e}_{\tau,1} , \mathbf{e}_{k,1} \sample \chi^m$, $\mathbf{e}_{\tau,2} \sample  \chi^\ell$,        $\mathbf{e}_{k,2} \sample  \chi^{2m}$,
+as well as \begin{eqnarray*} \nonumber
+\mathbf{c}_{\mathbf{v}} &=& (\mathbf{c}_{\mathbf{v},1},\mathbf{c}_{\mathbf{v},2}) \\ &=&  \big( \mathbf{B}^T \cdot \mathbf{s}_{\mathbf{v}} + \mathbf{e}_{\mathbf{v},1} ,~  \mathbf{G}_1^T \cdot \mathbf{s}_{\mathbf{v}} + \mathbf{e}_{\mathbf{v},2} + \mathbf{v}   \cdot \lfloor q/p \rfloor  \big)  \in \Zq^m \times \Zq^{2m}
+ \\
+%\mathbf{c}_{\mathbf{v}_2} &=& (\mathbf{c}_{\mathbf{v}_2,1},\mathbf{c}_{\mathbf{v}_2,2}) \\ &=&  \big( \mathbf{B}^T \cdot \mathbf{s}_{\mathbf{v}_2} + \mathbf{e}_{\mathbf{v}_2,1} ,~  \mathbf{G}_1^T %\cdot \mathbf{s}_{\mathbf{v}_2} + \mathbf{e}_{\mathbf{v}_2,2} + \mathbf{v}_2  \cdot \lfloor q/p \rfloor  \big)  \in \Zq^m \times \Zq^m \\
+\mathbf{c}_{s} &=& (\mathbf{c}_{s,1},\mathbf{c}_{s,2}) \\ &=&  \big( \mathbf{B}^T \cdot \mathbf{s}_{0} + \mathbf{e}_{0,1} ,~  \mathbf{G}_1^T \cdot \mathbf{s}_{0} + \mathbf{e}_{0,2} + \mathbf{s}  \cdot \lfloor q/p \rfloor  \big)  \in \Zq^m \times \Zq^{2m} ,
+\end{eqnarray*}
+ where $\mathbf{s}_{\mathbf{v}}, \mathbf{s}_{0}  \sample \chi^n$, $\mathbf{e}_{\mathbf{v},1},\mathbf{e}_{0,1} \sample \chi^m$, 
+    $\mathbf{e}_{\mathbf{v},2},\mathbf{e}_{0,2}\sample \chi^{2m}$.
+\item[2.] Prove in zero-knowledge that $\mathbf{c}_{\tau}$, $\mathbf{c}_{s}$, $\mathbf{c}_{\mathbf{v} }$,  $\{\mathbf{c}_k\}_{k=1}^N$ encrypt a valid message-signature pair. In Section~\ref{subsection:zk-for-signature}, we show that this involved zero-knowledge protocol can be derived from the statistical zero-knowledge argument of knowledge for a simpler, but more general  relation  that we explicitly present in \cref{se:gs-lwe-stern}. The proof system can be made statistically ZK for a malicious verifier using standard techniques (assuming a common reference string, we can use  \cite{Dam00}).  In the random oracle model, it can
+  be made non-interactive using the Fiat-Shamir heuristic \cite{FS86}.
+
+\end{itemize}
+\end{description}
+
+%To establish the security of the protocol,
+\noindent We   require that the adversary be unable to prove possession of a signature of a message $(\mathfrak{m}_1,\ldots,\mathfrak{m}_N)$ for which it did not legally
+obtain a credential by interacting with the issuer.  Note that the messages that are blindly signed by the issuer are uniquely defined since, at each signing
+query, the adversary is required to supply perfectly binding commitments $\{\mathbf{c}_k\}_{k=1}^N$ to $(\mathfrak{m}_1,\ldots,\mathfrak{m}_N)$. 
+
+In instantiations using non-interactive proofs, we assume that  these can be bound to a verifier-chosen nonce    to prevent replay attacks, as suggested    in   \cite{CKL+15}.
+
+The security proof (in Theorem \ref{commit-thm}) makes crucial use of the R\'enyi divergence using arguments in the spirit of Bai \textit{et al.} \cite{BLL+15}. The
+reduction has to guess upfront the index $i^\star \in \{1,\ldots,Q\}$ of the specific signing query for which the adversary will re-use $\tau^{(i^\star)}$. For
+this query, the reduction will have to make sure that the simulation trapdoor of Agrawal \textit{et al.} \cite{ABB10} (used by the $\mathsf{SampleRight}$ algorithm
+of Lemma \ref{lem:sampler}) vanishes: otherwise, the adversary's forgery would not be usable for solving $\mathsf{SIS}$. This means that, as in  the proof of
+\cite{BHJ+15}, the reduction must answer exactly one signing query in a different way, without using the trapdoor.  While B\"ohl \textit{et al.} solve this
+problem by exploiting the fact that they only need to prove security against non-adaptive forgers, we  directly use a built-in chameleon hash function mechanism
+which is implicitly realized by the matrix $\mathbf{D}_0$ and the vector $\mathbf{s}$. Namely, in the signing query for which the Agrawal \textit{et al.}
+trapdoor~\cite{ABB10} cancels, we assign a special value to the vector $\mathbf{s} \in \ZZ^{2m}$, which depends on the adaptively-chosen signed message
+$(\mathsf{Msg}_1^{(i^\star)},\ldots,\mathsf{Msg}_N^{(i^\star)})$ and some Gaussian matrices $\{\mathbf{R}_k\}_{k=1}^N$  hidden behind $\{\mathbf{D}_k\}_{k=1}^N$.
+
+One issue is that this results in a different distribution  for the vector   $\mathbf{s} \in \ZZ^m$. However, we can still view $\mathbf{s}$ as a vector sampled from a
+Gaussian distribution centered away from $\mathbf{0}^{2m}$. Since this specific situation occurs only once during the simulation, we can apply a result proved in
+\cite{LSS14} which upper-bounds the R\'enyi divergence between two Gaussian distributions with identical standard  deviations but different centers.  By
+choosing the standard deviation $\sigma_1$ of $\mathbf{s} \in \ZZ^{2m}$ to be polynomially larger than that of the columns of matrices $\{\mathbf{R}_k\}_{k=1}^N$, we can
+keep the R\'enyi divergence between the two distributions of $\mathbf{s}$ (i.e., the one of the simulation and the one of the real game) sufficiently small to apply
+the probability preservation property (which still gives a polynomial reduction since the argument must only be applied on one signing query).  Namely, the
+latter implies that, if the R\'enyi divergence $R_2(\mathbf{s}^{\mathsf{real}}||\mathbf{s}^{\mathsf{sim}})$ is polynomial, the probability that  the simulated vector
+$\mathbf{s}^{\mathsf{sim}} \in \ZZ^{2m}$  passes the verification test will  only be polynomially smaller than in the real game and so will be the adversary's
+probability of success.
+
+Another option would have been to keep the statistical distance between $\mathbf{s}^{\mathsf{real}}$ and $\mathbf{s}^{\mathsf{sim}}$ negligible using the smudging
+technique of \cite{AJL+12}. However, this would have implied to use an exponentially large modulus $q$ since $\sigma_1$ should have been exponentially larger
+than the standard deviations of the columns of $\{\mathbf{R}_k\}_{k=1}^N$.
+
+\begin{theorem} \label{commit-thm}
+  Under the $\mathsf{SIS}_{n,2m, q, \hat{\beta}}$ assumption, where $\hat{\beta} = N \sigma (2m)^{3/2} + 4 \sigma_1 m^{3/2}$\hspace*{-1.5pt}, the above
+  protocols are  secure protocols for obtaining a signature on a committed message and proving possession of a valid message-signature pair.
+\end{theorem}
+
+\begin{theorem} \label{anon-cred}
+The scheme provides anonymity under the $\mathsf{LWE}_{n,q,\chi}$ assumption.
+\end{theorem}
+
+\section{Subprotocols for Stern-like Argument}
+\addcontentsline{tof}{section}{\protect\numberline{\thesection} Protocoles pour les preuves à la Stern}
+\label{se:gs-lwe-stern}
+
+\subsection{Proving the Consistency of Commitments}\label{subsection:zk-for-commitments}
+The argument system used in our protocol for signing a committed value in Section~\ref{commit-sig} can be summarized as follows.
+\begin{description}
+ \item[Common Input:] Matrices $\{\mathbf{D}_k\in \ZZ_q^{2n \times 2m}\}_{k=0}^N$; $\mathbf{B}\in \ZZ_q^{n \times m}$; $\mathbf{G}_1 \in \mathbb{Z}_q^{n \times 2m}$;
+
+     \smallskip
+     \hspace*{-7.5pt}vectors $\mathbf{c}_{\mathfrak{m}} \in \mathbb{Z}_q^{2n}$; $\{\mathbf{c}_{k,1} \in \ZZ_q^{m}\}_{k=1}^N$; $\{\mathbf{c}_{k,2} \in \ZZ_q^{2m}\}_{k=1}^N$; $\mathbf{c}_{\mathbf{s}', 1} \in \ZZ_q^{m}$; $\mathbf{c}_{\mathbf{s}',2} \in \mathbb{Z}_q^{2m}$. \medskip
+  \item[Prover's Input:] $\mathfrak{m} = (\mathfrak{m}_1^T \| \ldots \| \mathfrak{m}_N^T)^T \in \mathsf{CorEnc}(mN)$;
+
+ $\{\mathbf{s}_{k} \in [-B,B]^n, \hspace*{2.5pt} \mathbf{e}_{k,1}\in [-B,B]^m; \hspace*{2.5pt} \mathbf{e}_{k,2}\in [-B,B]^{2m}\}_{k=1}^N$; \hspace*{5pt} $\mathbf{s}_0\in [-B,B]^n$;
+
+ $\mathbf{e}_{0,1}\in [-B,B]^m; \hspace*{5pt} \mathbf{e}_{0,2}\in [-B,B]^{2m}$; \hspace*{5pt} $\mathbf{s}' \in [-(p-1), (p-1)]^{2m}$ \smallskip
+  \item[Prover's Goal:] Convince the verifier in \textsf{ZK} that:
+\end{description}
+\vspace*{-10pt}
+\begin{eqnarray}\label{equation:R-commit-statement}
+\hspace*{-5pt}
+\begin{cases}
+\mathbf{c}_{\mathfrak{m}}= \mathbf{D}_0 \cdot \mathbf{s}' + \sum_{k=1}^N \mathbf{D}_k \cdot \mathfrak{m}_k \bmod q; \\[2.5pt]
+
+\mathbf{c}_{\mathbf{s}', 1}= \mathbf{B}^T\cdot \mathbf{s}_0 + \mathbf{e}_{0,1} \bmod q; \hspace*{5pt}\mathbf{c}_{\mathbf{s}',2}= \mathbf{G}_1^T\cdot \mathbf{s}_0 + \mathbf{e}_{0,2} + \lfloor q/p \rfloor\cdot \mathbf{s}'\bmod q; \\[2.5pt]
+\forall k\in [N]: \mathbf{c}_{k,1}= \mathbf{B}^T\cdot\mathbf{s}_{k} + \mathbf{e}_{k,1}; \hspace*{5pt}\mathbf{c}_{k,2}= \mathbf{G}_1^T\cdot \mathbf{s}_{k} + \mathbf{e}_{k,2} + \lfloor q/2 \rfloor\cdot \mathfrak{m}_k.
+\end{cases}
+\end{eqnarray}
+We will show that the above argument system can be obtained from the one in \cref{sse:stern-abstraction}. We proceed in two steps.
+
+\smallskip \smallskip
+
+\noindent \textbf{Step 1:} \emph{Transforming the equations in~(\ref{equation:R-commit-statement}) into a unified one of the form $\mathbf{P}\cdot \mathbf{x} = \mathbf{v} \bmod q$, where $\|\mathbf{x}\|_\infty =1$ and $\mathbf{x} \in \mathsf{VALID}$ - a ``specially-designed'' set.}
+
+To do so, we first form the following vectors and matrices:
+\[
+  \scriptsize
+\begin{cases}
+\mathbf{x}_1 \hspace*{-1pt}= \hspace*{-1pt}\big(\mathbf{s}_0^T \| \mathbf{e}_{0,1}^T \| \mathbf{e}_{0,2}^T \| \mathbf{s}_{1}^T \| \mathbf{e}_{1,1}^T \| \mathbf{e}_{1,2}^T \| \ldots \| \mathbf{s}_{N}^T \| \mathbf{e}_{N,1}^T \| \mathbf{e}_{N,2}^T \big)^T\hspace*{-3.5pt} \in \hspace*{-1.5pt}[-B,B]^{(n+3m)(N+1)}; \\[2.5pt]
+%\mathbf{x}_2 = \big(\mathfrak{m}_1^T \| \ldots\| \mathfrak{m}_N^T\big)^T \in \mathsf{CorEnc}(mN); \hspace*{10pt} \mathbf{x}_3 = \mathbf{s}' \in [-(p-1), (p-1)]^{2m};\\[2.5pt]
+\mathbf{v} = \big(\mathbf{c}_{\mathfrak{m}}^T \| \mathbf{c}_{\mathbf{s}',1}^T\| \mathbf{c}_{\mathbf{s}',2}^T\| \mathbf{c}_{1,1}^T \|\mathbf{c}_{1,2}^T \| \ldots \|\mathbf{c}_{N,1}^T \|\mathbf{c}_{N,2}^T \big)^T \in \mathbb{Z}_q^{2n + 3m(N+1)};\\[5pt]
+%\mathbf{D} = [\mathbf{D}_1 | \ldots | \mathbf{D}_N]; \hspace*{5pt}
+\mathbf{P}_1 = \left(
+                         \begin{array}{ccc}
+                         \begin{array}{c}
+                         \mathbf{B}^T \\
+                         \hline
+                         \rule{0pt}{3ex}\mathbf{G}_1^T
+                             \end{array}
+                           & \vline
+                            & \mathbf{I}_{3m}
+                         \end{array}
+                       \right); \hspace*{10pt}
+\mathbf{Q}_2 = \left(
+                                                                                           \begin{array}{c}
+                         \mathbf{0} \\
+                         \hline
+                         \rule{0pt}{3ex}\lfloor\frac{q}{2}\rfloor\mathbf{I}_{2m}
+                             \end{array}
+                                                                                         \right); \hspace*{10pt}
+\mathbf{Q}_p = \left(
+                                                                                           \begin{array}{c}
+                         \mathbf{0} \\
+                         \hline
+                         \rule{0pt}{3ex}\lfloor\frac{q}{p}\rfloor\mathbf{I}_{2m}
+                             \end{array}
+                                                                                         \right)\\[5pt]
+
+\mathbf{M}_1 = \left(
+                 \begin{array}{c}
+                    \mathbf{0} \\
+                         \hline
+                         \rule{0pt}{3ex}
+
+                     \begin{array}{cccc}
+                       \mathbf{P}_1 &  &  &  \\
+                        & \mathbf{P}_1 &  &  \\
+                        &  & \xddots &  \\
+                        &  &  & \mathbf{P}_1 \\
+                     \end{array}
+
+                    \\
+                 \end{array}
+               \right); \hspace*{15pt}
+\mathbf{M}_2 = \left(
+                 \begin{array}{c}
+                    \mathbf{D}_1 | \ldots | \mathbf{D}_N \\
+                         \hline
+                         \rule{0pt}{3ex}
+\mathbf{0} \\
+                         \hline
+                         \rule{0pt}{3ex}
+                     \begin{array}{ccc}
+                        \mathbf{Q}_2 &  &  \\
+                          & \xddots &  \\
+                          &  & \mathbf{Q}_2 \\
+                     \end{array}
+
+                    \\
+                 \end{array}
+               \right); \hspace*{15pt}
+\mathbf{M}_3 = \left(
+  \begin{array}{c}
+    \mathbf{D}_0 \\
+    \hline
+    \rule{0pt}{3ex}
+    \mathbf{Q}_p \\
+    \hline
+                         \rule{0pt}{3ex}
+     \\
+    \mathbf{0} \\
+     \\
+  \end{array}
+\right).
+\end{cases}
+\]
+
+We then observe that (\ref{equation:R-commit-statement}) can be rewritten as:
+\begin{eqnarray}\label{equation:R-commit-unified}
+\vspace*{-5pt}
+\mathbf{M}_1 \cdot \mathbf{x}_1 + \mathbf{M}_2 \cdot \mathfrak{m} + \mathbf{M}_3 \cdot \mathbf{s}' = \mathbf{v} \in \mathbb{Z}_q^D,
+\end{eqnarray}
+where $D = 2n + 3m(N+1)$.
+Now we employ the techniques from \cref{sse:stern-abstraction} to convert~\eqref{equation:R-commit-unified} into the form $\mathbf{P}\cdot \mathbf{x} = \mathbf{v} \bmod q$. Specifically, if we let:
+\[
+\vspace*{-5pt}
+\begin{cases}
+\mathsf{DecExt}_{(n+3m)(N+1),B}(\mathbf{x}_1) \rightarrow \hat{\mathbf{x}}_1 \in  \mathsf{B}^3_{(n+3m)(N+1)\delta_B}; \\[2.5pt]
+{\mathbf{M}}'_1 = \mathbf{M}_1 \cdot \widehat{\mathbf{K}}_{(n+3m)(N+1),B} \in \ZZ_q^{D \times 3(n+3m)(N+1)\delta_B}; \\[2.5pt]
+%\mathsf{Ext}_{2mN}(\mathbf{x}_2) \rightarrow \hat{\mathbf{x}}_2 \in \mathsf{B}_{2(2mN)}; \hspace*{5pt}
+%{\mathbf{M}}'_2 = \big[\mathbf{M}_2 | \mathbf{0}^{D \times 2mN}] \in \mathbb{Z}_q^{D \times 4mN}; \\[5pt]
+\mathsf{DecExt}_{2m, p-1}(\mathbf{s}') \rightarrow \hat{\mathbf{s}} \in \mathsf{B}^3_{2m\delta_{p-1}}; \hspace*{5pt}
+{\mathbf M}'_3 = \mathbf{M}_3 \cdot \widehat{\mathbf K}_{2m,p-1} \in \mathbb{Z}_q^{D \times 6m\delta_{p-1}},
+\end{cases}
+\]
+$L = 3(n+3m)(N+1)\delta_B + 2mN + 6m\delta_{p-1}$, and $\mathbf{P} \hspace*{-1pt}= \hspace*{-1pt}\big[\mathbf{M}'_1 | \mathbf{M}_2 | \mathbf{M}'_3\big] \hspace*{-2pt}\in \hspace*{-1pt}\mathbb{Z}_q^{D \times L}$, and $\mathbf{x} = \big(\hat{\mathbf{x}}_1^T \| \mathfrak{m}^T \| \hat{\mathbf{s}}^T\big)^T$, then we will obtain the desired equation:
+\[
+\mathbf{P}\cdot \mathbf{x} = \mathbf{v} \bmod q.
+\]
+Having performed the above unification, we now define $\mathsf{VALID}$ as the set of all vectors $\mathbf{t} \hspace*{-1pt}\in\hspace*{-1pt} \{-1,0,1\}^L$ of the form $\mathbf{t}\hspace*{-1pt} =\hspace*{-1pt} \big(\mathbf{t}_1^T \| \mathbf{t}_2^T \| \mathbf{t}_3^T\big)^T$\hspace*{-2.5pt}, where $\mathbf{t}_1 \in \mathsf{B}^3_{(n+3m)(N+1)\delta_B}$, $\mathbf{t}_2 \in \mathsf{CorEnc}(mN)$, and $\mathbf{t}_3 \in  \mathsf{B}^3_{2m\delta_{p-1}}$. Note that $\mathbf{x} \in \mathsf{VALID}$. \\
+
+\smallskip
+
+\noindent
+\textbf{Step 2:} \emph{Specifying the set $\mathcal{S}$ and permutations of $L$ elements $\{T_\pi: \pi \in \mathcal{S}\}$ for which the conditions in~(\ref{eq:zk-equivalence}) hold.}
+
+\begin{itemize}
+\item Define $\mathcal{S}: = \mathcal{S}_{3(n+3m)(N+1)\delta_B} \times \{0,1\}^{mN} \times \mathcal{S}_{6m\delta_{p-1}}$. \smallskip
+\item For $\pi = (\pi_1, \mathbf{b}, \pi_3) \in \mathcal{S}$, and for vector $\mathbf{w} = \big(\mathbf{w}_1^T \| \mathbf{w}_2^T \| \mathbf{w}_3^T\big)^T \in \mathbb{Z}_q^L$, where $\mathbf{w}_1 \in \ZZ_q^{3(n+3m)(N+1)\delta_B}$, $\mathbf{w}_2 \in \ZZ_q^{2mN}$, $\mathbf{w}_3 \in \ZZ_q^{6m\delta_{p-1}}$, we define: \vspace*{-5pt}
+    \[
+    T_{\pi} = \big(\pi_1(\mathbf{w}_1)^T \| E_{\mathbf{b}}(\mathbf{w}_2)^T \| \pi_3(\mathbf{w}_3)^T\big)^T.
+    \]
+\end{itemize}
+\vspace*{-2.5pt}
+By inspection, it can be seen that the properties in~(\ref{eq:zk-equivalence}) are satisfied, as desired. As a result, we can obtain the required argument system by running the protocol in \cref{sse:stern-abstraction} with common input $(\mathbf{P}, \mathbf{v})$ and prover's input $\mathbf{x}$.
+
+%--------------------------------------------------
+\subsection{Proving the Possession of a Signature on a Committed Value}\label{subsection:zk-for-signature}
+We now describe how to derive the protocol for proving the possession of a signature on a committed value, that is used in Section~\ref{commit-sig}.
+\begin{description}
+  \item[Common Input:] Matrices $\mathbf{A}, \{\mathbf{A}_j\}_{j=0}^\ell, \mathbf{D} \in \ZZ_q^{n \times m}$; $\{\mathbf{D}_k\in \ZZ_q^{2n \times 2m}\}_{k=0}^N$; $\mathbf{B}\in \ZZ_q^{n \times m}$; $\mathbf{G}_1 \in \mathbb{Z}_q^{n \times 2m}$;
+       $\mathbf{G}_0 \in \mathbb{Z}_q^{n \times \ell}$; vectors
+       $ \{\mathbf{c}_{k,1}\}_{k=1}^N, \mathbf{c}_{\tau,1}, \mathbf{c}_{\mathbf{v}, 1},   \mathbf{c}_{s, 1} \in \ZZ_q^m$;  $\{\mathbf{c}_{k,2}\}_{k=1}^N,\mathbf{c}_{\mathbf{v}, 2}, \mathbf{c}_{s,2} \in \ZZ_q^{2m}$;  $\mathbf{c}_{\tau,2} \in \ZZ_q^\ell$; $\mathbf{u} \in \mathbb{Z}_q^n$.
+
+       \smallskip
+
+  \item[Prover's Input:] $\mathbf{v} = \left(
+                                                                  \begin{array}{c}
+                                                                    \mathbf{v}_1 \\
+                                                                    \mathbf{v}_2 \\
+                                                                  \end{array}
+                                                                \right)
+  $, where $\mathbf{v}_1, \mathbf{v}_2\in [-\beta, \beta]^m$ and $\beta = \sigma\cdot \omega(\log m)$ - the infinity norm bound of signatures; $\tau \in \{0,1\}^\ell$; $\mathbf{s} \in [-(p-1), (p-1)]^{2m}$;
+
+      \smallskip
+      $\mathfrak{m} = (\mathfrak{m}_1^T \| \ldots \| \mathfrak{m}_N^T)^T \in \mathsf{CorEnc}(mN)$; $\{\mathbf{s}_{k}\}_{k=1}^N$, $\mathbf{s}_{\mathbf{v}}$,  $\mathbf{s}_0$, $\mathbf{s}_\tau \in [-B,B]^n$;
+
+      \smallskip
+      $\{\mathbf{e}_{k,1}\}_{k=1}^N$, $\mathbf{e}_{\mathbf{v}, 1}$,  $\mathbf{e}_{0,1}$, $\mathbf{e}_{\tau,1} \in [-B,B]^m$;
+      $\{\mathbf{e}_{k,2}\}_{k=1}^N, \mathbf{e}_{0,2},\mathbf{e}_{\mathbf{v},2} \in [-B,B]^{2m}$;
+
+    \smallskip
+      $\mathbf{e}_{\tau,2} \in [-B,B]^\ell$.
+\end{description}
+
+\noindent
+\textbf{Prover's Goal:} Convince the verifier in \textsf{ZK} that: \vspace*{-7.5pt}
+\begin{eqnarray}\label{equation:R-sign-signature}
+\hspace*{-5pt}
+\mathbf{A}\hspace*{-1.5pt}\cdot\hspace*{-1.5pt}\mathbf{v}_1 + \mathbf{A}_0\hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \mathbf{v}_2 + \sum_{i=1}^\ell \mathbf{A}_i\hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \tau[i] \mathbf{v}_2 - \mathbf{D}\hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \mathsf{bin}(\mathbf{D}_0\hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \mathbf{s} + \sum_{k=1}^N \mathbf{D}_i\hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \mathfrak{m}_k) = \mathbf{u} \bmod q,\vspace*{-10pt}
+\end{eqnarray}
+and that (modulo $q$)
+\begin{eqnarray}\label{equation:R-sign-ciphertext}
+\hspace*{-12.5pt}
+\begin{cases}
+\forall k\in [N]: \mathbf{c}_{k,1}= \mathbf{B}^T\cdot\mathbf{s}_{k} + \mathbf{e}_{k,1} ; \hspace*{5pt}\mathbf{c}_{k,2}= \mathbf{G}_1^T\cdot \mathbf{s}_{k} + \mathbf{e}_{k,2} + \lfloor q/2 \rfloor\cdot \mathfrak{m}_k  ;  \\
+
+\mathbf{c}_{\mathbf{v}, 1}= \mathbf{B}^T\cdot \mathbf{s}_{\mathbf{v}} + \mathbf{e}_{\mathbf{v},1}  ; \\
+\mathbf{c}_{\mathbf{v},2}= \mathbf{G}_1^T \hspace*{-2pt}\cdot\hspace*{-2pt} \mathbf{s}_{\mathbf{v}} \hspace*{-2pt}+\hspace*{-2pt} \mathbf{e}_{\mathbf{v},2}\hspace*{-2pt}+\hspace*{-2pt} \lfloor\frac{q}{p}\rfloor \hspace*{-2pt}\cdot\hspace*{-2pt} \mathbf{v} \hspace*{-2pt}=\hspace*{-2pt} \mathbf{G}_1^T \hspace*{-2pt}\cdot\hspace*{-2pt} \mathbf{s}_{\mathbf{v}} \hspace*{-2pt}+\hspace*{-2pt} \mathbf{e}_{\mathbf{v},2}\hspace*{-2pt}+\hspace*{-2pt} \left(\hspace*{-2pt}
+                                                                                                                         \begin{array}{c}
+                                                                                                                           \lfloor\frac{q}{p}\rfloor \mathbf{I}_m \\
+                                                                                                                           \mathbf{0}\\
+                                                                                                                         \end{array}
+                                                                                                                       \hspace*{-2pt}\right)\cdot \mathbf{v}_1
+                                                                                                                       \hspace*{-2pt}+ \hspace*{-2pt} \left(\hspace*{-2pt}
+                                                                                                                         \begin{array}{c}
+                                                                                                                           \mathbf{0}\\
+                                                                                                                           \lfloor\frac{q}{p}\rfloor \mathbf{I}_m \\
+                                                                                                                         \end{array}
+                                                                                                                       \hspace*{-2pt}\right)\hspace*{-2pt}\cdot\hspace*{-2pt} \mathbf{v}_2
+ ; \\
+
+%\mathbf{c}_{\mathbf{v}_2, 1}= \mathbf{B}^T\cdot \mathbf{s}_{\mathbf{v}_2} + \mathbf{e}_{\mathbf{v}_2,1}  ; \hspace*{2.5pt}
+%\mathbf{c}_{\mathbf{v}_2,2}= \mathbf{G}_1^T \cdot \mathbf{s}_{\mathbf{v}_2} + \mathbf{e}_{\mathbf{v}_2,2}+ \lfloor\frac{q}{p}\rfloor \cdot %\mathbf{v}_2  ; \\
+
+\mathbf{c}_{\mathbf{s}, 1}= \mathbf{B}^T\cdot \mathbf{s}_0 + \mathbf{e}_{0,1}  ; \hspace*{5pt}\mathbf{c}_{\mathbf{s},2}= \mathbf{G}_1^T\cdot \mathbf{s}_0 + \mathbf{e}_{0,2} + \lfloor q/p \rfloor\cdot \mathbf{s} ; \\
+
+\mathbf{c}_{\tau,1} = \mathbf{B}^T\cdot \mathbf{s}_\tau + \mathbf{e}_{\tau,1}  ; \hspace*{2.5pt} \mathbf{c}_{\tau,2}= \mathbf{G}_0^T\cdot \mathbf{s}_\tau + \mathbf{e}_{\tau,2} + \lfloor q/2 \rfloor\cdot \tau  .      
+\end{cases}
+\end{eqnarray} 
+$~$ \\
+We proceed in two steps.
+\medskip  \smallskip 
+
+\noindent \textbf{Step 1:} \emph{Transforming the equations in~(\ref{equation:R-sign-signature}) and~(\ref{equation:R-sign-ciphertext}) into a unified one of the form $\mathbf{P}\cdot \mathbf{x} = \mathbf{c} \bmod q$, where $\|\mathbf{x}\|_\infty =1$ and $\mathbf{x} \in \mathsf{VALID}$ - a ``specially-designed'' set.}
+
+Note that, if we let $\mathbf{y} = \mathsf{bin}(\mathbf{D}_0\hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \mathbf{s} + \sum_{k=1}^N \mathbf{D}_i\hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \mathfrak{m}_k) \in \{0,1\}^{m}$, then we have $\mathbf{H}_{2n \times m}\cdot \mathbf{y} = \mathbf{D}_0\hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \mathbf{s} + \sum_{k=1}^N \mathbf{D}_i\hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \mathfrak{m}_k \bmod q$, and~(\ref{equation:R-sign-signature}) can be equivalently written as:
+\begin{eqnarray*}\label{equation:R-sign-signature-2}
+\hspace*{-10pt}
+\left(
+  \begin{array}{c}
+    \mathbf{A} \\
+    \mathbf{0} \\
+  \end{array}
+\right)\hspace*{-1.5pt}\cdot\hspace*{-1.5pt}
+\mathbf{v}_1 +
+\left(
+  \begin{array}{c}
+    \mathbf{A}_0 \\
+    \mathbf{0} \\
+  \end{array}
+\right)\hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \mathbf{v}_2 &+&
+\sum_{i=1}^\ell \left(
+  \begin{array}{c}
+    \mathbf{A}_i \\
+    \mathbf{0} \\
+  \end{array}
+\right)\hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \tau[i] \mathbf{v}_2 +
+\left(
+  \begin{array}{c}
+    \mathbf{0} \\
+    \mathbf{D}_0 \\
+  \end{array}
+\right)\cdot \mathbf{s} + \left(
+  \begin{array}{c}
+    -\mathbf{D} \\
+    -\mathbf{H}_{2n \times m} \\
+  \end{array}
+\right)\cdot \mathbf{y} \\
+&+&\left(
+  \begin{array}{c}
+    \mathbf{0} \\
+    \mathbf{D}_1 | \ldots | \mathbf{D}_N \\
+  \end{array}
+\right)\cdot \mathfrak{m} = \left(
+                      \begin{array}{c}
+                        \mathbf{u} \\
+                        \mathbf{0}^{2n} \\
+                      \end{array}
+                    \right) ~\bmod q.
+\end{eqnarray*}
+Next, we use linear algebra to combine this equation and~(\ref{equation:R-sign-ciphertext}) into (modulo $q$):
+\begin{align}\label{equation:R-sign-almost}
+\hspace*{-10pt}
+\mathbf{F}\hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \mathbf{v}_1 \hspace*{-1.5pt}+\hspace*{-1.5pt} \mathbf{F}_0\hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \mathbf{v}_2 \hspace*{-1.5pt}+\hspace*{-1.5pt} \sum_{i=1}^\ell \mathbf{F}_i \hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \tau[i]\mathbf{v}_2 + \mathbf{M}_1 \hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \tau \hspace*{-1.5pt}+\hspace*{-1.5pt} \mathbf{M}_2\hspace*{-1.5pt}\cdot\hspace*{-1.5pt}\mathbf{y} + \mathbf{M}_3\hspace*{-1.5pt}\cdot\hspace*{-1.5pt}\mathfrak{m}
+ \hspace*{-1.5pt}+ \hspace*{-1.5pt}\mathbf{M}_4 \hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \mathbf{s} \hspace*{-2pt}+\hspace*{-2pt} \mathbf{M}_5\hspace*{-1.5pt}\cdot\hspace*{-1.5pt} \mathbf{e} \hspace*{-2pt}=\hspace*{-2pt} \mathbf{c},
+\end{align}
+where, for dimensions  $D = \ell + 3n + 7m + 3mN$ and $L_0 = D + nN$,
+\begin{itemize}
+\item Matrices $\mathbf{F}, \mathbf{F}_0, \mathbf{F}_1, \ldots, \mathbf{F}_\ell \in \mathbb{Z}_q^{D \times m}$, $\mathbf{M}_1 \in \mathbb{Z}_q^{D \times \ell}$, $\mathbf{M}_2 \in \mathbb{Z}_q^{D \times m}$, $\mathbf{M}_3 \in \mathbb{Z}_q^{D \times 2mN}$, $\mathbf{M}_4 \in \mathbb{Z}_q^{D \times 2m}$, $\mathbf{M}_5 \in \mathbb{Z}_q^{D \times L_0}$ and vector $\mathbf{c} \in \mathbb{Z}_q^D$ are built from the public input.
+\item Vector $\mathbf{e} = \big(\hspace*{1pt}\mathbf{s}_1^T \hspace*{1pt}\|\hspace*{1pt} \ldots \hspace*{1pt}\|\hspace*{1pt} \mathbf{s}_N^T \hspace*{1pt}\|\hspace*{1pt} \mathbf{s}_{\mathbf{v}}^T \hspace*{1pt}\|\hspace*{1pt} \mathbf{s}_0^T \hspace*{1pt}\|\hspace*{1pt} \mathbf{s}_\tau^T \hspace*{1pt}\|\hspace*{1pt} \mathbf{e}_{1,1}^T \hspace*{1pt}\|\hspace*{1pt} \ldots \hspace*{1pt}\|\hspace*{1pt} \mathbf{e}_{N,1}^T \hspace*{1pt}\|\hspace*{1pt} \mathbf{e}_{\mathbf{v},1}^T \hspace*{1pt}\|\hspace*{1pt} \mathbf{e}_{0,1}^T \hspace*{1pt}\|\hspace*{1pt} \mathbf{e}_{\tau, 1}^T \hspace*{1pt}\| \\
+    ~~~~~~~~~~~~~~~~~\|\hspace*{1pt}\mathbf{e}_{1,2}^T \hspace*{1pt}\|\hspace*{1pt} \ldots \hspace*{1pt}\|\hspace*{1pt} \mathbf{e}_{N,2}^T \hspace*{1pt}\|\hspace*{1pt} \mathbf{e}_{0,2}^T \hspace*{1pt}\|\hspace*{1pt} \mathbf{e}_{\mathbf{v},2}^T \hspace*{1pt}\|\hspace*{1pt} \mathbf{e}_{\tau,2}^T\hspace*{1pt}\big)^T \in [-B,B]^{L_0}$.
+\end{itemize}
+
+Now we further transform~\eqref{equation:R-sign-almost} using the techniques from \cref{sse:stern-abstraction}. Specifically, we form the following:
+\[
+\begin{cases}
+\mathsf{DecExt}_{m, \beta}(\mathbf{v}_1) \rightarrow \hat{\mathbf{v}}_1 \in \mathsf{B}^3_{m\delta_\beta}; \hspace*{5pt}\mathsf{DecExt}_{m, \beta}(\mathbf{v}_2)\rightarrow \hat{\mathbf{v}}_2 \in \mathsf{B}^3_{m\delta_\beta}; \\[2.5pt]
+{\mathbf{F}}' = \big[\mathbf{F} \cdot \widehat{\mathbf{K}}_{m, \beta} | \mathbf{F}_0 \cdot \widehat{\mathbf{K}}_{m, \beta} | \mathbf{F}_1 \cdot \widehat{\mathbf{K}}_{m, \beta} | \ldots | \mathbf{F}_\ell \cdot \widehat{\mathbf{K}}_{m, \beta} | \mathbf{0}^{D \times 3m\delta_\beta \ell}\big] \in \mathbb{Z}_q^{D \times 3m\delta_\beta(2\ell+2)}; \\[2.5pt]
+\mathsf{Ext}_{2\ell}(\tau) \rightarrow \hat{\tau} = (\tau[1], \ldots, \tau[\ell], \ldots, \tau[2\ell])^T \in \mathsf{B}^2_{\ell}; \hspace*{2.5pt}\mathbf{M}'_1 = [\mathbf{M}_1 | \mathbf{0}^{D \times \ell}] \in \mathbb{Z}_q^{D \times 2\ell};\\[2.5pt]
+\mathsf{Ext}_{2m}(\mathbf{y})\rightarrow \hat{\mathbf{y}} \in \mathsf{B}^2_{m}; \hspace*{2.5pt}\mathbf{M}'_2 = [\mathbf{M}_2 | \mathbf{0}^{D \times m}] \in \ZZ_q^{D \times 2m }; \\[2.5pt]
+\mathsf{DecExt}_{2m, p-1}(\mathbf{s}) \rightarrow \hat{\mathbf{s}} \in \mathsf{B}^3_{2m\delta_{p-1}}; \hspace*{2.5pt} \mathbf{M}'_4 = \mathbf{M}_4 \cdot \widehat{\mathbf{K}}_{2m, p-1} \in \mathbb{Z}_q^{D \times 6m\delta_{p-1}}; \\[2.5pt]
+\mathsf{DecExt}_{L_0, B}(\mathbf{e}) \rightarrow \hat{\mathbf{e}} \in \mathsf{B}^3_{L_0\delta_{B}}; \hspace*{2.5pt}
+\mathbf{M}'_5 = \mathbf{M}_5 \cdot \widehat{\mathbf{K}}_{L_0, B} \in \mathbb{Z}_q^{D \times 3L_0\delta_B}.
+\end{cases}
+\]
+Now, let $L = 3m\delta_\beta(2\ell+2) + 2\ell + 2m + 2mN + 6m\delta_{p-1} + 3L_0\delta_B$, and construct matrix
+$\mathbf{P} = \big[\hspace*{1pt}\mathbf{F}' \hspace*{1pt}|\hspace*{1pt} \mathbf{M}'_1 \hspace*{1pt}| \hspace*{1pt}\mathbf{M}'_2 \hspace*{1pt}|\hspace*{1pt} \mathbf{M}_3\hspace*{1pt}|\hspace*{1pt} \mathbf{M}'_4\hspace*{1pt}| \hspace*{1pt} \mathbf{M}'_5 \hspace*{1pt}\big] \in \ZZ_q^{D \times L}$ and vector
+\[
+\mathbf{x} = \big(\hspace*{1.5pt}\hat{\mathbf{v}}_1^T\hspace*{1.5pt} \|\hspace*{1.5pt} \hat{\mathbf{v}}_2^T \hspace*{1.5pt}\| \hspace*{1.5pt}\tau[1]\hat{\mathbf{v}}_2^T\hspace*{1.5pt} \| \ldots \|\hspace*{1.5pt} \tau[\ell]\hat{\mathbf{v}}_2^T\hspace*{1.5pt}\| \ldots \| \hspace*{1.5pt}\tau[2\ell]\hat{\mathbf{v}}_2^T\hspace*{1.5pt}\| \hspace*{1.5pt} \hat{\tau}^T \hspace*{1.5pt}\| \hspace*{1.5pt}\hat{\mathbf{y}}^T\hspace*{1.5pt} \|\hspace*{1.5pt} \mathfrak{m}^T \hspace*{1.5pt}\|\hspace*{1.5pt} \hat{\mathbf{s}}^T\hspace*{1.5pt}\| \hspace*{1.5pt} \hat{\mathbf{e}}^T\hspace*{1.5pt}\big)^T,
+\]
+then we will obtain the equation $\mathbf{P}\cdot \mathbf{x} = \mathbf{c} \bmod q$.
+
+Before going on, we define $\mathsf{VALID}$ as the set of
+    $\mathbf{w} \in \{-1,0,1\}^L$ of the form:
+\vspace*{-5pt}
+\[
+\mathbf{w} = \big(\mathbf{w}_{1}^T \| \mathbf{w}_2^T \| g_1 \mathbf{w}_2^T\| \ldots \| g_{2\ell}\mathbf{w}_2^T \| \mathbf{g}^T\| \mathbf{w}_3^T\| \mathbf{w}_4^T \| \mathbf{w}_5^T \| \mathbf{w}_6^T\big)^T
+\vspace*{-5pt}
+\]
+for some $\mathbf{w}_1, \mathbf{w}_2 \in \mathsf{B}^3_{m\delta_\beta}$, $\mathbf{g} = (g_1, \ldots, g_{2\ell}) \in \mathsf{B}_{2\ell}$, $\mathbf{w}_3 \in \mathsf{B}^2_{m}$, $\mathbf{w}_4 \in \mathsf{CorEnc}(mN)$, $\mathbf{w}_5 \in \mathsf{B}^3_{2m\delta_{p-1}}$, and $\mathbf{w}_6 \in \mathsf{B}^3_{L_0\delta_B}$.
+It can be checked that the constructed vector $\mathbf{x}$ belongs to this tailored set $\mathsf{VALID}$.\\
+
+
+
+ 
+
+\noindent
+\textbf{Step 2:} \emph{Specifying the set $\mathcal{S}$ and permutations of $L$ elements $\{T_\pi: \pi \in \mathcal{S}\}$ for which the conditions in~(\ref{eq:zk-equivalence}) hold.}
+
+\begin{itemize}
+\item Define $\mathcal{S} = \mathcal{S}_{3m\delta_\beta} \times \mathcal{S}_{3m\delta_\beta} \times \mathcal{S}_{2\ell} \times\mathcal{S}_{2m} \times \{0,1\}^{mN}\times \mathcal{S}_{6m\delta_{p-1}} \times \mathcal{S}_{3L_0\delta_B}$. \medskip
+\item For $\pi = (\phi, \psi, \gamma, \rho, \mathbf{b}, \eta, \xi) \in \mathcal{S}$ and $\mathbf{z} =  \big(\mathbf{z}_0^1 \| \mathbf{z}_0^2 \| \mathbf{z}_1 \| \ldots \| \mathbf{z}_{2\ell} \| \mathbf{g} \| \mathbf{t}_1 \| \mathbf{t}_2 \|\mathbf{t}_3 \| \mathbf{t}_4\big) \in \mathbb{Z}_q^L$,
+    where ${\mathbf{z}_0^1}, {\mathbf{z}_0^2}, \mathbf{z}_1, \ldots, \mathbf{z}_{2\ell}  \in \mathbb{Z}_q^{3m\delta_\beta}$, $\mathbf{g} \in \mathbb{Z}_q^{2\ell}$, $\mathbf{t}_1\in \mathbb{Z}_q^{2m}$, $\mathbf{t}_2 \in \mathbb{Z}_q^{2mN}$, $\mathbf{t}_3 \in \mathbb{Z}_q^{6m\delta_{p-1}}$, and $\mathbf{t}_4 \in \mathbb{Z}_q^{3L_0\delta_B}$, we define:
+\begin{eqnarray*}
+\hspace*{-15pt}
+T_{\pi}(\mathbf{z}) = \big(\phi(\mathbf{z}_0^1)^T\hspace*{1pt} \| \psi(\mathbf{z}_0^2)^T \hspace*{1pt}\| \psi(\mathbf{z}_{\gamma(1)})^T \hspace*{1pt}\| \ldots \| \psi(\mathbf{z}_{\gamma(2\ell)})^T \hspace*{1pt}\| \gamma(\mathbf{g})^T\hspace*{1pt} \| \\
+~~~~~~~~\|\rho(\mathbf{t}_1)^T \| E_{\mathbf{b}}(\mathbf{t}_2)^T \hspace*{1pt}\| \eta(\mathbf{t}_3)^T \| \xi(\mathbf{t}_4)^T\hspace*{1pt}\big)^T
+\end{eqnarray*}
+as the permutation that transforms $\mathbf{z}$ as follows:
+\begin{enumerate}
+\item It rearranges the order of the $2\ell$ blocks $\mathbf{z}_1, \ldots, \mathbf{z}_{2\ell}$ according to $\gamma$.
+\item It then {permutes} block $\mathbf{z}_0^1$ according to $\phi$, blocks $\mathbf{z}_0^2$, $\{\mathbf{z}_i\}_{i=1}^{2\ell}$ according to~$\psi$, block $\mathbf{g}$ according to $\gamma$, block $\mathbf{t}_1$ according to $\rho$, block $\mathbf{t}_2$ according to $E_{\mathbf{b}}$, block $\mathbf{t}_3$ according to~$\eta$, and block $\mathbf{t}_4$ according to $\xi$.
+\end{enumerate}
+\end{itemize}
+It can be check that~(\ref{eq:zk-equivalence}) holds. Therefore, we can obtain a statistical \textsf{ZKAoK} for the given relation by running the protocol in \cref{sse:stern-abstraction}.