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\section{A Lattice-Based Signature with Efficient Protocols} \label{se:gs-lwe-sigep}
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%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 $m \geq 2n \log q$.
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%We assume that messages are vectors of $N$ blocks $\mathsf{Msg}=(\mathfrak{m}_1,\ldots,\mathfrak{m}_N)$, where each
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%block is an $L$-bit string $\mathfrak{m}_k = \mathfrak{m}_k[1] \ldots \mathfrak{m}_k[L] \in \{0,1\}^L$ for $k \in \{1,\ldots, N\}$.
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Our scheme can be seen as a variant of the B\"ohl \textit{et al.} signature \cite{BHJ+15}, where
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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
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$[\mathbf{A} \mid \mathbf{A}_0 + \sum_{j=1}^\ell \tau[j] \cdot \mathbf{A}_j ] \cdot \mathbf{v} = \mathbf{u} + \mathbf{D} \cdot \mathbf{h} \bmod q$,
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where matrices $\mathbf{A}, \mathbf{A}_0, \ldots, \mathbf{A}_\ell, \mathbf{D} \in \Zq^{n \times m}$
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are public random matrices and $\mathbf{h} \in \{0,1\}^m$ is a chameleon hash of the message which is computed using randomness $\mathbf{s}$.
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A difference is that, while \cite{BHJ+15} uses a short single-use tag $\tau \in \Zq$,
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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
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\cite{CL02a} in the security proof.
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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
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a relation of the form
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$\mathbf{P} \cdot \mathbf{x} = \mathbf{v} \bmod q$, for some public matrix $\mathbf{P}$ and vector~$\mathbf{v}$.
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The underlying chameleon hash can be seen as a composition of the chameleon hash of \cite[Se. 4.1]{CHKP10} with
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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
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$\mathbf{D}_0 \cdot \mathbf{s} + \sum_{k=1}^N \mathbf{D}_k \cdot \mathfrak{m}_k$, for some discrete Gaussian vector $\mathbf{s}$.
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\subsection{Description} \label{desc-sig-protoc}
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We assume that messages are vectors of $N$ blocks $\mathsf{Msg}=(\mathfrak{m}_1,\ldots,\mathfrak{m}_N)$, where each
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block is a $2m$-bit string $\mathfrak{m}_k = \mathfrak{m}_k[1] \ldots \mathfrak{m}_k[2m] \in \{0,1\}^{2m}$ for $k \in \{1,\ldots, N\}$. \\
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\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
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coordinate of $\mathbf{v}$ by its binary representation.
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\begin{description}
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\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 $(\{0,1\}^{2m})^N$.
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\smallskip
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\begin{itemize}
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\item[1.] Run $\TrapGen(1^n,1^m,q)$ to get~$\mathbf{A} \in
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\Zq^{n \times m}$ and a short basis $\mathbf{T}_{\mathbf{A}}$ of
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$\Lambda_q^{\perp}(\mathbf{A}).$ This basis allows computing short vectors in $\Lambda_q^{\perp}(\mathbf{A})$ with a Gaussian parameter $\sigma$.
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% $\sigma \geq \| \widetilde{\mathbf{T}_{\mathbf{A}}} \| \cdot \omega (\sqrt{\log m})$.
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Next, choose $\ell+1$ random $\mathbf{A}_0,\mathbf{A}_1,\ldots,\mathbf{A}_{\ell} \sample U(\Zq^{n \times m})$. %, where $\ell = \Theta(\lambda)$.
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\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
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$\mathbf{u} \sample U(\Zq^n)$. \smallskip
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\end{itemize}
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The private key consists of $SK:= \mathbf{T}_{\mathbf{A}} \in \ZZ^{m \times m}$ and the public key is
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$${PK}:=\big( \mathbf{A}, ~ \{\mathbf{A}_j \}_{j=0}^{\ell}, ~ \{\mathbf{D}_k\}_{k=0}^{N},~\mathbf{D}, ~\mathbf{u} \big).$$
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% \smallskip
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\item[\textsf{Sign}$\big(SK, \mathsf{Msg} \big)$:] To sign an $N$-block message
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$\mathsf{Msg}=\left(\mathfrak{m}_1,\ldots,\mathfrak{m}_N \right) \in \left(\{0,1\}^{2m} \right)^N$,
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\begin{enumerate}[1.]
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\item Choose a random string $\tau \sample U(\{0,1\}^\ell )$. Then, using $SK:=
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\mathbf{T}_{\mathbf{A}}$, compute with $\ExtBasis$ a short delegated basis $\mathbf{T}_\tau \in \ZZ^{2m \times 2m}$
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for the matrix
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\begin{eqnarray} \label{tau-matrix}
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\mathbf{A}_{\tau}=
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[ \mathbf{A} \mid \mathbf{A}_0 +
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\sum_{j=1}^\ell \tau[j] \mathbf{A}_j
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] \in \Zq^{ n \times 2m}.
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\end{eqnarray}
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\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
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$\mathbf{c}_M = \mathbf{D}_{0} \cdot \mathbf{s} + \sum_{k=1}^N \mathbf{D}_k \cdot \mathfrak{m}_k \in \mathbb{Z}_q^{2n} ,$
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which is used to define $\mathbf{u}_M=\mathbf{u} + \mathbf{D} \cdot \bit( \mathbf{c}_M) \in \Zq^n .$
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Then,
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using the delegated basis $\mathbf{T}_\tau \in \ZZ^{2m \times 2m}$, sample a short vector $\mathbf{v} \in \ZZ^{2m}$ in $D_{\Lambda_q^{\mathbf{u}_M}(\mathbf{A}_\tau), \sigma}$.
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\end{enumerate}
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Output the signature $sig=(\tau,\mathbf{v},\mathbf{s}) \in \{0,1\}^\ell \times \ZZ^{2m} \times \ZZ^{2m}$. \smallskip
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\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
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signature $sig=(\tau,\mathbf{v},\mathbf{s}) \in \{0,1\}^\ell \times \ZZ^{2m} \times \ZZ^{2m}$,
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return $1$ if
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\begin{eqnarray} \label{ver-eq-block}
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\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.
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\end{eqnarray}
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and $\| \mathbf{v} \| < \sigma \sqrt{2m}$, $\| \mathbf{s} \| < \sigma_1 \sqrt{2m}$.
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\end{description}
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When the scheme is used for obliviously signing committed messages,
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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
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to sample $\mathbf{s}$ from a Gaussian distribution whose standard deviation $\sigma_1$ is polynomially larger than $\sigma$. \\
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\indent
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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,
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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}$
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belong to public parameters.
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\subsection{Security Analysis}
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The security analysis in Theorem \ref{th:gs-lwe-security-cma-sig} requires that $q>\ell$.
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\begin{theorem} \label{th:gs-lwe-security-cma-sig}
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The signature scheme is secure under chosen-message attacks under the $\mathsf{SIS}$ assumption.
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\end{theorem}
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\begin{proof}
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To prove the result, we will distinguish three kinds of attacks:
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\begin{description}
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\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
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of the signing oracle.
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\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
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$sig^{(i^\star)}=(\tau^{(i^\star)},\mathbf{v}^{(i^\star)},\mathbf{s}^{(i^\star)})$ of the signing oracle, for some index $i^\star \in \{1,\ldots,Q\}$. However,
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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
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message and the $i^\star$-th signing query, respectively, we have
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$\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)}. $
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\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
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$sig^{(i^\star)}=(\tau^{(i^\star)},\mathbf{v}^{(i^\star)},\mathbf{s}^{(i^\star)})$ of the signing oracle (i.e.,
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$\tau^{(i^\star)}= \tau^\star$ for some index $i^\star \in \{1,\ldots,Q\}$) and we have the collision
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\begin{eqnarray} \label{collision}
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\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)}.
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\end{eqnarray}
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\end{description}
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Type III attacks imply a collision for the chameleon hash function of Kawachi \textit{et al.} \cite{KTX08}: if (\ref{collision}) holds,
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a short vector
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of $\Lambda_q^{\perp}([ \mathbf{D}_0 \mid \mathbf{D}_1 \mid \ldots \mid \mathbf{D}_N])$ is obtained as
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$$\big({\mathbf{s}^\star}^T- {\mathbf{s}^{(i^\star)}}^T \mid {\mathfrak{m}_1^\star }^T - {\mathfrak{m}_1^{(i^\star)} }^T \mid \ldots \mid {\mathfrak{m}_N^\star }^T - {\mathfrak{m}_N^{(i^\star)} }^T \big)^T,$$ so that a collision breaks the $\mathsf{SIS}$ assumption. \\ \indent
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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
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of \cite{HW09} allows keeping the modulus smaller than the number $Q$ of adversarial queries as in \cite{MP12}.
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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 $\mathsf{SIS}$.
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\end{proof}
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\begin{lemma} \label{le:lwe-gs-type-I-attacks}
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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 $
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\end{lemma}
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\begin{proof}
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Let $\adv$ be a $\ppt$ adversary that can mount a Type I attack with non-negligible success probability $\varepsilon$. We construct a $\ppt$
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algorithm $\bdv$ that uses $\adv$ to break the~$\SIS_{n,m,q,\beta'}$ assumption. It takes as input~$\bar{\mathbf{A}} \in
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\Zq^{n \times m}$ and computes $\mathbf{v} \in
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\Lambda_q^{\perp}(\bar{\mathbf{A}})$ with~$0 < \|\mathbf{v}\| \leq \beta'$. \\
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\indent
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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
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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
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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
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common prefix between $\tau^\star$ and one of the $\left\{\tau^{(i)} \right\}_{i=1}^Q$ is the string
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$\tau^\star[1] \ldots \tau^\star[t^\dagger-1] =\tau^{(i^\dagger)}[1] \ldots \tau^{(i^\dagger)}[t^\dagger-1] \in \{0,1\}^{t^\dagger -1}$ comprised of the
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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
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the $t^\dagger$-th prefix of $\tau^{(i)}$ for each $i\in \{1,\ldots,Q\}$. \\
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\indent
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Then, $\bdv$ runs
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$\TrapGen(1^n,1^m,q)$ to obtain $\mathbf{C} \in \Zq^{n \times m}$ and a
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basis $\mathbf{T}_{\mathbf{C}}$ of~$\Lambda_q^{\perp}(\mathbf{C})$ with
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$\|\widetilde{\mathbf{T}_{\mathbf{C}}}\| \leq \bigO(\sqrt{n \log q})$. Then,
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it picks~$\ell+1$ matrices~$\mathbf{Q}_0,\ldots, \mathbf{Q}_{\ell} \in \ZZ^{m \times m}$, where
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each matrix $\mathbf{Q}_i$ has its columns sampled independently from~$D_{\ZZ^m, \sigma}$. The
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reduction $\bdv$ defines the matrices $\{ \mathbf{A}_j\}_{j=0}^{\ell}$ as
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\begin{eqnarray*}
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\left\{
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\begin{array}{ll}
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\mathbf{A}_0 = \bar{\mathbf{A}} \cdot \mathbf{Q}_0 + (\sum_{j=1}^{t^\dagger} {\tau^\star[j]}) \cdot
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\mathbf{C} \\
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\mathbf{A}_j = \bar{\mathbf{A}} \cdot \mathbf{Q}_j + (-1)^{\tau^\star[j]} \cdot
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\mathbf{C}, \qquad \quad \text{ for } j \in
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[1,t^\dagger] \\
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\mathbf{A}_j = \bar{\mathbf{A}} \cdot \mathbf{Q}_j , \qquad \quad \qquad \quad~~ \qquad \quad \text{ for } j \in
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[t^\dagger+1,\ell]
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\end{array}
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\right.
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\end{eqnarray*}
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It also sets $\mathbf{A}=\bar{\mathbf{A}}$.
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We note that we have
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% \vspace*{-.1cm}
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\begin{eqnarray*}
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\mathbf{A}_{\tau^{(i)}} &=& \left[ \begin{array}{c|c} \bar{\mathbf{A}} & \mathbf{A}_0 +
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\sum_{j=1}^\ell \tau^{(i)}[j] \mathbf{A}_j
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\end{array} \right] \\
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& = & \left[
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\begin{array}{c|c}
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\bar{\mathbf{A}} ~ & ~ \bar{\mathbf{A}} \cdot (\mathbf{Q}_0 +
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\sum_{j=1}^{\ell} \tau^{(i)}[j] \mathbf{Q}_j) + (
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\sum_{j=1}^{t^\dagger} \tau^\star[j] +(-1)^{\tau^\star[j]} \tau^{(i)}[j])\cdot \mathbf{C}
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\end{array} \right]
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\\
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&=&
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\left[
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\begin{array}{c|c}
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\bar{\mathbf{A}} ~ & ~ \bar{\mathbf{A}} \cdot (\mathbf{Q}_0 +
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\sum_{j=1}^{\ell} \tau^{(i)}[j] \mathbf{Q}_j) + h_{\tau^{(i)}} \cdot \mathbf{C}
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\end{array} \right]
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\end{eqnarray*}
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where $ h_{\tau^{(i)}} \in [1,t^\dagger] \subset [1,\ell]$ stands for the Hamming distance between
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$\tau^{(i)}_{|t^\dagger}$ and $\tau^\star_{|t^\dagger}$. Note that, with probability $1/(Q \cdot \ell)$ and since $q>\ell$, we have
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$ h_{\tau^{(i)}} \neq 0 \bmod q$ whenever $\tau^{(i)}_{|t^\dagger} \neq \tau^\star_{|t^\dagger}$. \\
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\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 $D_{\ZZ^m,\sigma}$
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and computes
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\begin{eqnarray*}
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\mathbf{D} &=& \bar{\mathbf{A}} \cdot \mathbf{R}. % \qquad \qquad \qquad \quad~ \forall k \in \{0,\ldots,N\}.
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\end{eqnarray*}
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Finally, $\bdv$ samples a short vector $\mathbf{e}_u \sample D_{\ZZ^m,\sigma_1}$ and computes the vector $\mathbf{u} \in \Zq^n$
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as $\mathbf{u} = \bar{\mathbf{A}} \cdot \mathbf{e}_u \in \Zq^n$. The public key $${PK}:=\big( \mathbf{A}, ~
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\{\mathbf{A}_j \}_{j=0}^{\ell}, ~ \{\mathbf{D}_k\}_{k=0}^{N},~\mathbf{D}, ~\mathbf{u} \big)$$
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is given to $\adv$.
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%Hence,
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% $\bdv$ 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},
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% using the basis~$\mathbf{T}_{\mathbf{C}}$ of~$\Lambda_q^{\perp}(\mathbf{C})$.
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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.
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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
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$$\mathbf{u}_M = \mathbf{u} + \mathbf{D} \cdot \bit \bigl( \sum_{k=1}^N \mathbf{D}_k \cdot {\mathfrak{m}_k^{(i)} } + \mathbf{D}_{0} \cdot {\mathbf{s}^{(i)} } \bigr) ~~ \bmod q.$$
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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
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that $\big(\tau^{(i)},\mathbf{v}^{(i)},\mathbf{s}^{(i)} \big)$ satisfies the verification equation (\ref{ver-eq-block}). \\
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\indent When $\adv$ halts, it outputs a valid signature $sig^\star=\big(\tau^{(i^\dagger)},\mathbf{v}^\star,\mathbf{s}^\star \big)$ on a
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message $\mathsf{Msg}^\star=(\mathfrak{m}_1^\star,\ldots,\mathfrak{m}_N^\star)$ with $\| \mathbf{v}^\star \| \leq \sigma \sqrt{2m}$ and $\| \mathbf{s}^\star \| \leq \sigma_1 \sqrt{2m}$.
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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,
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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.
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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
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\begin{align*}
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&\left[ \begin{array}{c|c} \bar{\mathbf{A}} ~&~ \bar{\mathbf{A}} \cdot (\mathbf{Q}_0 +
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\sum_{j=1}^{\ell} \tau^\star[j] \mathbf{Q}_j)
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\end{array} \right] \cdot
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\left[\begin{array}{c} {\mathbf{v}_1^\star } \\ \hline {\mathbf{v}_2^\star } \end{array} \right] \\
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& \hspace{3cm}= \mathbf{u} + \mathbf{D} \cdot \bit \bigl( \mathbf{D}_0 \cdot {\mathbf{s}^{\star} } +
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\sum_{k=1}^N \mathbf{D}_k \cdot {\mathfrak{m}_k^{\star} } \bigr) \bmod q \\
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& \hspace{3cm}= \bar{\mathbf{A}} \cdot \Bigl( \mathbf{e}_u + \mathbf{R} \cdot \bit \bigl( \mathbf{D}_0 \cdot {\mathbf{s}^{\star} } +
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\sum_{k=1}^N \mathbf{D}_k \cdot {\mathfrak{m}_k^{\star} } \bigr) \Bigr) \bmod q ,
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\end{align*}
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which implies that the vector
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\begin{eqnarray*}
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\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
|
||||
\[
|
||||
\mathbf{x} = \big(\hspace*{1.5pt}\hat{\mathbf{v}}_1^T\hspace*{1.5pt} \|\hspace*{1.5pt} \hat{\mathbf{v}}_2^T \hspace*{1.5pt}\| \hspace*{1.5pt}\tau[1]\hat{\mathbf{v}}_2^T\hspace*{1.5pt} \| \ldots \|\hspace*{1.5pt} \tau[\ell]\hat{\mathbf{v}}_2^T\hspace*{1.5pt}\| \ldots \| \hspace*{1.5pt}\tau[2\ell]\hat{\mathbf{v}}_2^T\hspace*{1.5pt}\| \hspace*{1.5pt} \hat{\tau}^T \hspace*{1.5pt}\| \hspace*{1.5pt}\hat{\mathbf{y}}^T\hspace*{1.5pt} \|\hspace*{1.5pt} \mathfrak{m}^T \hspace*{1.5pt}\|\hspace*{1.5pt} \hat{\mathbf{s}}^T\hspace*{1.5pt}\| \hspace*{1.5pt} \hat{\mathbf{e}}^T\hspace*{1.5pt}\big)^T,
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\]
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||||
then we will obtain the equation $\mathbf{P}\cdot \mathbf{x} = \mathbf{c} \bmod q$.
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||||
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||||
Before going on, we define $\mathsf{VALID}$ as the set of
|
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$\mathbf{w} \in \{-1,0,1\}^L$ 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}.
|
||||
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Reference in New Issue
Block a user