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Fabrice Mouhartem 2018-06-19 13:22:35 +02:00
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48
chap-GE-LWE.tex
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@ -122,7 +122,7 @@ $\mathcal{G}_r(1^\lambda)$ which samples public/secret parameters for the relati
return $1$ whenever $(x,w)\in R$. To encrypt a witness $w$ such that $(x,w) \in R$ for some public $x$, the sender fetches the pair $(\pk,\crt_{\pk})$
from $\mathsf{database}$ and runs the randomized encryption algorithm. The latter takes as input $w$, a label $L$, the receiver's pair $(\pk,\crt_{\pk})$ as
well as public keys $\pk_{\GM}$ and $\pk_{\OA}$. Its output is a ciphertext
$\Psi \leftarrow \mathsf{ENC}(\pk_{\GM},\pk_{\OA},\pk,\crt_{\pk},w,L)$.
$\Psi \gets \mathsf{ENC}(\pk_{\GM},\pk_{\OA},\pk,\crt_{\pk},w,L)$.
On input of the same elements, the certificate $\crt_{\pk}$, the ciphertext $\Psi$ and the random coins $coins_{\Psi}$ that were used to produce $\Psi$, the
non-interactive algorithm $\mathsf{PP}$ generates a proof $\pi_{\Psi}$ that there exists a certified receiver whose public key was registered in $\mathsf{database}$ and
who is able to decrypt $\Psi$ and obtain a witness $w$ such that $(x,w) \in R$. The verification algorithm $\mathcal{V}$ takes as input $\Psi$, $\pk_{\GM}$,
@ -137,7 +137,7 @@ that pk belongs to the language of valid public keys. Here, we are implicitly as
of valid public keys is dense (all matrices are valid keys, as is the case in our scheme).
In the upcoming definitions, we sometimes use the notation
\[ \langle \mathsf{output}_A |\mathsf{output}_B \rangle \allowbreak \leftarrow \langle A(\mathsf{input}_A),B(\mathsf{input}_B) \rangle (\mathsf{common\textrm{-}input}) \]
\[ \langle \mathsf{output}_A |\mathsf{output}_B \rangle \allowbreak \gets \langle A(\mathsf{input}_A),B(\mathsf{input}_B) \rangle (\mathsf{common\textrm{-}input}) \]
to denote the execution of a protocol between $A$ and $B$ obtaining their own outputs from their respective inputs.
\medskip
@ -148,7 +148,7 @@ probability.
\begin{center}
\procedure{Experiment $\Expt^{\mathrm{correctness}}(\lambda)$}{
\mathsf{param} \leftarrow
\mathsf{param} \gets
\mathsf{SETUP}_{\mathsf{init}}(1^\lambda); (\pk_{R},\sk_{R})
\gets \mathcal{G}_r (\lambda); (x,w) \leftarrow \mathsf{sample}_{R}
(\pk_{R},\sk_{R}); \\
@ -384,11 +384,11 @@ encryption of a message of its choice from a random element of the ciphertext sp
\procedure{Experiment $\Expt^{\mathrm{ROR}}_{\adv}(\lambda)$}{
\ID^\star \gets \adv(\textsf{id}, \lambda); (\mathsf{PP}, \textsf{msk}) \gets \mathsf{Setup}(1^\lambda);~\\
M \gets \adv^{\mathsf{Extract}_\mathsf{PP}(\textsf{msk}, \cdot)}_\textsf{Ch}(\mathsf{PP});\\
b \sample U(\bit);\\
b \sample \U(\bit);\\
\pcif b = 1 \pcthen\\
\pcind C^\star \gets \mathsf{Encrypt}_\mathsf{PP}(M, \ID^\star) \\
\pcelse\\
\pcind C^\star \gets U(\mathcal{C});\\
\pcind C^\star \sample \U(\mathcal{C});\\
b' \gets \adv^{\mathsf{Extract}_\mathsf{PP}(\textsf{msk}, \cdot)}(\textsf{guess},C^\star);\\
\pcif b = b' \pcthen\\
\pcind \pcreturn 1\\
@ -413,8 +413,8 @@ encryption of a message of its choice from a random element of the ciphertext sp
$q, n, \sigma, \alpha$ and define $k =\lfloor \log q \rfloor$, $\bar{m}= nk$, $m = 2 \bar{m}$ and choose a noise distribution $\chi$ for $\LWE$.
\begin{enumerate}
\item Compute $(\bar{\mathbf A}, \mathbf{T}_{\bar{\mathbf{A}}}) \gets \TrapGen(1^n, 1^m, q)$.
\item Define $\mathbf{G} = \mathbf{I}_n \otimes [1|2|\ldots |2^{k-1}] \in \ZZ_q^{n \times \bar{m}}$. Sample matrices $\mathbf B \sample U(\ZZ_q^{ n \times \bar{m}}) $,
$ \mathbf U \sample U(\Zq^{n \times m})$.
\item Define $\mathbf{G} = \mathbf{I}_n \otimes [1|2|\ldots |2^{k-1}] \in \ZZ_q^{n \times \bar{m}}$. Sample matrices $\mathbf B \sample \U(\ZZ_q^{ n \times \bar{m}}) $,
$ \mathbf U \sample \U(\Zq^{n \times m})$.
\item Let $\mathsf{FRD}: \Zq^n \to \Zq^{n \times n}$ be the full-rank difference mapping from~\cite{ABB10}.
\end{enumerate} Output
$
@ -431,7 +431,7 @@ encryption of a message of its choice from a random element of the ciphertext sp
\item[\textsf{Encrypt}$_\mathsf{PP}(\ID,\mathbf m)$:] Given an identity $\ID$ and a message $\mathbf m \in \bit^m$, \smallskip
\begin{enumerate}
\item Compute the matrix $\mathbf B_\ID = \mathbf B + \mathsf{FRD}(\ID) \cdot \mathbf G \in \Zq^{n \times \bar{m}}$.
Sample vectors $\mathbf s \sample U(\Zq^n), \mathbf x, \mathbf y \sample \chi^m$, $\mathbf R \sample D_{\ZZ,\sigma}^{m \times \bar{m}}$ and compute
Sample vectors $\mathbf s \sample \U(\Zq^n), \mathbf x, \mathbf y \sample \chi^m$, $\mathbf R \sample D_{\ZZ,\sigma}^{m \times \bar{m}}$ and compute
$\mathbf z = \mathbf R^T \cdot \mathbf y \in \ZZ^m$.
\item Compute
\begin{equation} \label{eq:ABB-c}
@ -770,7 +770,7 @@ This relation is in the same spirit as the one of Kiayias, Tsiounis and Yung \ci
\item[6.] Let $\mathsf{FRD}: \mathbb{Z}_q^{n} \rightarrow \mathbb{Z}_q^{n \times n}$ be the full-rank difference mapping from~\cite{ABB10}.
\item[7.] Pick a random matrix $\mathbf{F} \leftarrow \mathbb{Z}_q^{2n \times n \bar{m}k}$, which will be used to hash users' public keys from $\Zq^{n \times \bar{m}}$ to $\mathbb{Z}_q^n$.
% \item[7.] Pick matrices $\mathbf{A}_0,\mathbf{A}_1,\ldots,\mathbf{A}_{\ell}, \mathbf{D}_1 \xleftarrow{\$} \Zq^{n \times m}$, $\mathbf{D}, \mathbf{D}_0 \xleftarrow{\$} \mathbb{Z}_q^{n \times %nk}$ and vector $\mathbf{u} \xleftarrow{\$} \Zq^n$. These objects will be used for verifying the membership certificates issued by GM.
\item[8.] Let $\mathbf{G} \in \Zq^{n \times \bar{m}}$ be the gadget matrix $\mathbf{G}= \mathbf{I}_n \otimes \begin{bmatrix} 1 & 2 & \ldots & 2^{k-1} \end{bmatrix}$ of \cite{MP12}. Pick matrices $\bar{\mathbf{A}}, \mathbf{U} \leftarrow U(\mathbb{Z}_q^{n \times m})$ and $\mathbf{V} \leftarrow U(\mathbb{Z}_q^{n \times m})$. Looking ahead, $\mathbf{U}$ will be used to encrypt for the receiver while $\mathbf{V}$ will be used
\item[8.] Let $\mathbf{G} \in \Zq^{n \times \bar{m}}$ be the gadget matrix $\mathbf{G}= \mathbf{I}_n \otimes \begin{bmatrix} 1 & 2 & \ldots & 2^{k-1} \end{bmatrix}$ of \cite{MP12}. Pick matrices $\bar{\mathbf{A}}, \mathbf{U} \sample \U(\mathbb{Z}_q^{n \times m})$ and $\mathbf{V} \sample \U(\mathbb{Z}_q^{n \times m})$. Looking ahead, $\mathbf{U}$ will be used to encrypt for the receiver while $\mathbf{V}$ will be used
to encrypt the user's public key under the $\OA$'s public key. As for $\bar{\mathbf{A}}$, it will be used in two instances of the ABB encryption scheme \cite{ABB10}. \smallskip \smallskip
\end{itemize}
Output
@ -780,10 +780,10 @@ This relation is in the same spirit as the one of Kiayias, Tsiounis and Yung \ci
\end{eqnarray*}
\item[$\langle
\mathcal{G}_r, \mathsf{sample}_{R}
\rangle$:] Algorithm $\mathcal{G}_r(1^\lambda,1^n,1^m)$ proceeds by sampling a random matrix $\mathbf{A}_R \leftarrow U(\Zq^{n \times m})$ and outputting
\rangle$:] Algorithm $\mathcal{G}_r(1^\lambda,1^n,1^m)$ proceeds by sampling a random matrix $\mathbf{A}_R \sample \U(\Zq^{n \times m})$ and outputting
$(\pk_{R},\sk_{R})=(\mathbf{A}_R,\varepsilon)$. On input of a public key
$\pk_{R}=\mathbf{A}_R \in \Zq^{n \times m}$ for the relation $\mathrm{R}_{\ISIS}$, algorithm
$\mathsf{sample}_{R}$ picks $\mathbf{w} \leftarrow U(\{0,1\}^m)$ and outputs a pair $((\mathbf{A}_R,\mathbf{u}_R),\mathbf{w})$, where $\mathbf{u}_R =\mathbf{A}_R \cdot \mathbf{w} \in \Zq^n$.
$\mathsf{sample}_{R}$ picks $\mathbf{w} \sample \U(\{0,1\}^m)$ and outputs a pair $((\mathbf{A}_R,\mathbf{u}_R),\mathbf{w})$, where $\mathbf{u}_R =\mathbf{A}_R \cdot \mathbf{w} \in \Zq^n$.
\item[$\mathsf{SETUP_{\GM}}(\param)$:] The $\GM$ generates $(\sk_\GM,\pk_\GM) \leftarrow \mathsf{Keygen}(1^\lambda,q,n,m,\ell,\sigma)$ as a key pair for the $\SIS$-based signature scheme of \cite{LLM+16} (as recalled in \cref{se:gs-lwe-sigep}). This key pair
@ -827,7 +827,7 @@ $(\sk_{\OA},\pk_{\OA})=(\mathbf{T}_{\OA},\mathbf{B}_{\OA})$.
where $\tau= \tau[1] \ldots \tau[\ell] \in \{0,1\}^{\ell}$, as in the scheme of \cref{se:gs-lwe-sigep}. \smallskip
\end{enumerate}
$\mathsf{U}$ verifies that $\crt_{\mathsf{U}}$ is tuple of the form (\ref{eq:cert-description}) satisfying (\ref{eq:cert-verification}) and returns~$\perp$ if it is not the case.
The $\GM$ stores $(\pk_{\mathsf{U}},\crt_\mathsf{U})$ in the user database $\mathsf{database}$ and returns the certificate $\crt_\mathsf{U}$ to the new user $\U$. \medskip
The $\GM$ stores $(\pk_{\mathsf{U}},\crt_\mathsf{U})$ in the user database $\mathsf{database}$ and returns the certificate $\crt_\mathsf{U}$ to the new user $\mathsf{U}$. \medskip
% \begin{eqnarray}\label{eq:cert-pk}
%\mathsf{cert}_{\mathsf{pk}} = (\mathbf{h_M}, sig_{\mathbf{M}}).
%\end{eqnarray}
@ -842,7 +842,7 @@ $(\sk_{\OA},\pk_{\OA})=(\mathbf{T}_{\OA},\mathbf{B}_{\OA})$.
%Define $\mathbf{B}_{\vk} = \mathbf{B}_\mathsf{U} + \mathbf{H}_{\vk}\cdot \mathbf{G} \in \mathbb{Z}_q^{n \times m}$.
\item[3.] Encrypt the witness $\mathbf{w} \in \{0,1\}^m$ under $\mathsf{U}$'s public key $\mathbf{B}_\mathsf{U} \in \Zq^{n \times \bar{m}}$ using the tag $\vk$ by taking the following steps: \smallskip
\begin{enumerate}
\item[a.] Sample $\mathbf{s}_{\rec} \leftarrow U(\mathbb{Z}_q^n)$, $\mathbf{R}_{\rec} \leftarrow D_{\ZZ,\sigma}^{m \times \bar{m}} $ and
\item[a.] Sample $\mathbf{s}_{\rec} \leftarrow \U(\mathbb{Z}_q^n)$, $\mathbf{R}_{\rec} \leftarrow D_{\ZZ,\sigma}^{m \times \bar{m}} $ and
$\mathbf{x}_{\rec}, \mathbf{y}_{\rec} \leftarrow \chi^m$. Compute $\mathbf{z}_{\rec} = \mathbf{R}_{\rec}^T\cdot \mathbf{y}_{\rec} \in \mathbb{Z}^{\bar{m}}$.
\item[b.] Compute
\begin{eqnarray}\label{eq:c-recipient}
@ -862,7 +862,7 @@ $(\sk_{\OA},\pk_{\OA})=(\mathbf{T}_{\OA},\mathbf{B}_{\OA})$.
\item[4.] Encrypt the decomposition $\mathsf{vdec}_{n,q-1}(\mathbf{h_\mathsf{U}}) \in \{0,1\}^{m}$ of the hashed $\pk_\mathsf{U}$ under
the $\OA$'s public key $\mathbf{B}_{\OA} \in \Zq^{n \times \bar{m}}$ w.r.t. the tag $\vk \in \Zq^n$. Namely, conduct the following steps: \smallskip
\begin{enumerate}
\item[a.] Sample $\mathbf{s}_{\mathsf{oa}} \leftarrow U( \mathbb{Z}_q^n)$, $\mathbf{R}_{\mathsf{oa}} \leftarrow D_{\ZZ,\sigma}^{m \times \bar{m}}$,
\item[a.] Sample $\mathbf{s}_{\mathsf{oa}} \leftarrow \U( \mathbb{Z}_q^n)$, $\mathbf{R}_{\mathsf{oa}} \leftarrow D_{\ZZ,\sigma}^{m \times \bar{m}}$,
$\mathbf{x}_{\mathsf{oa}} \leftarrow \chi^{m}, \mathbf{y}_{\mathsf{oa}} \leftarrow \chi^m$. Set $\mathbf{z}_{\mathsf{oa}} = \mathbf{R}_{\mathsf{oa}}^T\cdot \mathbf{y}_{\mathsf{oa}} \in \ZZ^{\bar{m}}$.
\item[b.] Compute
\begin{eqnarray}\label{eq:c-open}
@ -1112,7 +1112,7 @@ The security results are explicited in the following theorems.
that $\mathbf{u}_R= \mathbf{A}_R \cdot \mathbf{w} \bmod q$, with $\mathbf{A}_R \in \ZZ_p^{n \times m}$, $\mathbf{u}_R \in \ZZ_q^n$ and $\mathbf{w} \in \{0,1\}^m$. In return, $\adv$ obtains, as a challenge, a
group encryption
$\Psi^\star= (\vk^\star,\mathbf{c}_{\rec}^\star, \mathbf{c}_{\oa}^\star, \Sigma^\star).$
of the witness $\mathbf{w}$ under $\pk_{\USR,b} =\mathbf{B}_{\USR,b}$, for some random bit $b \leftarrow U( \{0,1\})$ of the challenger's
of the witness $\mathbf{w}$ under $\pk_{\USR,b} =\mathbf{B}_{\USR,b}$, for some random bit $b \leftarrow \U( \{0,1\})$ of the challenger's
choice. Then, the adversary obtains proofs $\pi_{\Psi^\star}^\star$ for
$\Psi^\star$ and makes further opening and decryption queries under the
natural restrictions of Definition \ref{anonymity-def}. When the adversary $\adv$ halts, it
@ -1165,7 +1165,7 @@ The security results are explicited in the following theorems.
Next, the reduction runs the appropriate steps of the actual $\textsf{SETUP}_\textsf{init}$ algorithm to obtain
$\mathsf{COM}_{\mathsf{par}}$, $\mathbf{F} \in \ZZ_q^{2n \times n\bar{m}k}$ and $\mathbf{U} \in \ZZ_q^{n \times m}$.
Namely, $\bdv$ samples $\mathbf F \sample U(\Zq^{2m\times n \bar{m} k})$ and $\mathbf U \sample U(\Zq^{n \times m})$ like in the $\mathsf{SETUP}_\mathsf{init}$ algorithm and sends
Namely, $\bdv$ samples $\mathbf F \sample \U(\Zq^{2m\times n \bar{m} k})$ and $\mathbf U \sample \U(\Zq^{n \times m})$ like in the $\mathsf{SETUP}_\mathsf{init}$ algorithm and sends
$$ \param = \big\{\lambda, n, q, k, m, B, \chi, \sigma, \beta, \ell, \kappa, \mathcal{OTS}, \compar, \mathsf{FRD}, \bar{\mathbf{A}}, \mathbf{G}, \mathbf{F}, \mathbf{U}, \mathbf{V} \big\} $$
along with $\pk_\OA = \mathbf B \in \ZZ_q^{n \times \bar{m}}$ to the adversary $\adv$.
@ -1182,8 +1182,8 @@ The security results are explicited in the following theorems.
After a number of queries, $\adv$ decides to move to the challenge phase and sends a challenge query $\big( (\mathbf{A}_R,\mathbf{u}_R), \mathbf w^\star, L^\star \big)$ such that
$\mathbf{u}_R=\mathbf{A}_R \cdot \mathbf{w}^\star \bmod q$. The reduction
handles this query by requesting a challenge ciphertext for the IBE security game with the messages $\mathbf{m}_0=\mathsf{vdec}_{n,q-1}(\mathbf h_{\mathsf U,b})$, for some random bit $b \sample U(\bit)$
and $\mathbf{m}_1 \leftarrow U(\{0,1\}^m)$. In return, $\bdv$ obtains
handles this query by requesting a challenge ciphertext for the IBE security game with the messages $\mathbf{m}_0=\mathsf{vdec}_{n,q-1}(\mathbf h_{\mathsf U,b})$, for some random bit $b \sample \U(\bit)$
and $\mathbf{m}_1 \sample \U(\{0,1\}^m)$. In return, $\bdv$ obtains
a challenge ciphertext $\mathbf c^\star_\OA$ under identity $\vk^\star$, which is embedded in $\adv$'s challenge ciphertext. Namely,
$\mathbf{\Psi}^\star = (\vk^\star, \mathbf c_\rec^\star, \mathbf c_\OA^\star, \Sigma^\star)$ is obtained by computing $\mathbf c_\rec^\star$ as an ABB encryption
of the witness $\mathbf w^\star$ using the matrix $\mathbf{B}_{\mathsf U,b} \in \ZZ_q^{n \times \bar{m}}$ as in~\eqref{eq:c-recipient}
@ -1216,18 +1216,18 @@ we can assess % corresponds to \SFGame 3.
\[ \mathsf{PP} = \big(\bar{\mathbf A}, \mathbf B, \mathbf U \big) \in \Zq^{n \times m} \times \Zq^{n \times \bar{m}} \times \Zq^{n \times m} \]
from its real-or-random (ROR) challenger.
Our reduction uses $\mathsf{PP}$ to compute public parameters for our $\mathsf{GE}$ scheme. To this end, it samples $\mathbf F \sample U(\Zq^{2n \times n \bar{m}k})$,
$\mathbf V \sample U(\Zq^{n\times m})$ as in the real $\mathsf{SETUP}_\mathsf{init}$ algorithm.
Our reduction uses $\mathsf{PP}$ to compute public parameters for our $\mathsf{GE}$ scheme. To this end, it samples $\mathbf F \sample \U(\Zq^{2n \times n \bar{m}k})$,
$\mathbf V \sample \U(\Zq^{n\times m})$ as in the real $\mathsf{SETUP}_\mathsf{init}$ algorithm.
The reduction $\bdv$ also computes $\mathbf B_\OA =\bar{\mathbf{A}} \cdot \mathbf T_\OA \bmod q $,
where the small-norm matrix $\mathbf{T}_\OA$ is sampled from $D_{\ZZ,\sigma}^{m \times \bar{m}}$, and sends $\adv$ the parameters
\[ \mathsf{param}= \big\{\lambda, n, q, k, m, \sigma, \beta, \ell, \kappa, \mathcal{OTS}, \compar, \mathsf{FRD}, \bar{\mathbf{A}}, \mathbf{G}, \mathbf{F}, \mathbf{U}, \mathbf{V} \big\}, \]
where $\bar{\mathbf{A}}$ is taken from $\mathsf{PP}$,
along with $\pk_\OA = \mathbf B_\OA$. The rest of the keys are generated as in Game $4$.
The reduction $\bdv$ then tosses a coin $b \sample U(\bit)$. When the adversary $\adv$ triggers an execution of the join protocol,
The reduction $\bdv$ then tosses a coin $b \sample \U(\bit)$. When the adversary $\adv$ triggers an execution of the join protocol,
$\bdv$ generates the public keys $(\pk_i)_{i\in \bit}$ by defining $\pk_{\USR,b} = \mathbf B$ using the matrix $\mathbf B \in \ZZ_q^{n \times \bar{m}}$ supplied by
the ROR challenger as part of $\mathsf{PP}$ and generates $(\pk_{\USR,1-b},\sk_{1-b}) =(\mathbf{B}_{\USR,1-b} = \bar{\mathbf{A}} \cdot \mathbf T_{1-b}, \mathbf{T}_{1-b})$ for
a secret key $\mathbf{T}_{1-b} \leftarrow D_{\ZZ^m,\sigma}^{\bar{m}}$ of its own.
a secret key $\mathbf{T}_{1-b} \sample D_{\ZZ^m,\sigma}^{\bar{m}}$ of its own.
The two public keys $(\pk_{\USR,i})_{i\in \bit}$ are then certified by the adversarially-controlled $\GM$.
Notice that in the adversary's view, both public keys $\pk_{\USR,b}$ and $\pk_{\USR,1-b}$ are identically distributed.
@ -1341,7 +1341,7 @@ of the ABB scheme, which would contradict the $\LWE$ assumption, as established
At the very beginning of the IND-sID-CPA game, the reduction $\bdv$ generates a one-time signature key pair $(\sk^\star, \vk^\star)$ and hands $\vk^\star$ to its selective security challenger as the target identity under which the challenge ciphertext will later be computed. In response, $\bdv$ receives the public parameters
$$\mathsf{PP} = (\bar{\mathbf A}, \mathbf B, \mathbf U) \in \Zq^{n \times m} \times \Zq ^{n \times \bar m} \times \Zq^{n \times m}$$ from its IBE challenger.
The reduction then runs the missing steps of the actual $\Setup_{\mathsf{init}}$ algorithm: namely, $\bdv$ samples $\mathbf F \leftarrow U(\Zq^{2m \times n\bar{m}k}), \mathbf V \leftarrow U(\Zq^{n \times m})$ and generates $\compar$ before sending the common public parameters
The reduction then runs the missing steps of the actual $\Setup_{\mathsf{init}}$ algorithm: namely, $\bdv$ samples $\mathbf F \sample \U(\Zq^{2m \times n\bar{m}k}), \mathbf V \sample \U(\Zq^{n \times m})$ and generates $\compar$ before sending the common public parameters
$$\param = \bigl\{\lambda, n, q, k, m, B, \chi, \sigma, \beta, \ell, \kappa, \mathcal{OTS}, \compar, \mathsf{FRD}, \bar{\mathbf{A}}, \mathbf{G}, \mathbf{F}, \mathbf{U}, \mathbf{V} \bigr\}$$
to the adversary $\adv$.
@ -1357,7 +1357,7 @@ of the ABB scheme, which would contradict the $\LWE$ assumption, as established
At some point, the adversary $\adv$ queries a challenge ciphertext by outputting a triple $((\mathbf A_R, \mathbf u_R), \mathbf w, L)$ such that $\mathbf{w} \in \{0,1\}^m$ satisfies
$\mathbf u_R = \mathbf A_R \cdot \mathbf{w} \bmod q$. Then, the reduction $\bdv$ requests a challenge ciphertext $\mathbf c^\star_\rec$ to its IBE
challenger by sending it the messages $\mathbf m_1 = \mathbf{w} \in \{0,1\}^m$ and $\mathbf m_0 \leftarrow U(\{0,1\}^m)$. The resulting ciphertext $\mathbf c^\star_\rec$
challenger by sending it the messages $\mathbf m_1 = \mathbf{w} \in \{0,1\}^m$ and $\mathbf m_0 \sample \U(\{0,1\}^m)$. The resulting ciphertext $\mathbf c^\star_\rec$
is embedded in $\mathbf{\Psi}^\star = (\vk^\star, \mathbf c_\rec^\star, \mathbf c_\OA^\star, \Sigma^\star)$ by faithfully computing
$\mathbf c_\OA^\star$ and $\Sigma^\star$ as in the actual $\textsf{Enc}$ algorithm.

14
chap-GS-LWE.tex
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@ -1,10 +1,10 @@
In this chapter, we present the first dynamic group signature scheme that relies on lattice assumptions.
This construction relies on a signature scheme with efficient protocols as in~\cref{ch:sigmasig}, and it is used in a similar fashion.
As a consequence, it is possible to construct lattice-based anonymous credential from this building block.
The group signature scheme relies on the Gentry, Peikert and Vaikuntanathan identity-based encryption~\cite{GPV08} with the Canetti, Halevi and Katz~\cite{CHK04} in order to obtain a CCA2-secure public key encryption scheme which will be used to provide full-anonymity.
In this chapter, we present the first dynamic group signature scheme based on lattice assumptions.
This construction relies on a signature scheme with efficient protocols as in~\cref{ch:sigmasig}, which is used in a similar manner.
As a consequence, it is possible to design lattice-based anonymous credentials from this building block.
The group signature scheme relies on the Gentry, Peikert and Vaikuntanathan identity-based encryption~\cite{GPV08} with the Canetti, Halevi and Katz~\cite{CHK04} transform to obtain a CCA2-secure public key encryption scheme which will be used to provide full-anonymity.
The group signature security is proven secure in the \ROM under the \SIS and \LWE assumptions, which are fixed-size and well studied assumptions.
For security parameter $\lambda$ and for group of up to $\Ngs$ members, the scheme features public key size $\softO(\lambda^2) \cdot \log \Ngs$, user's secret key size $\softO(\lambda)$ and signature size $\softO(\lambda) \cdot \log \Ngs$.
The group signature is proven secure in the \ROM under the \SIS and \LWE assumptions, which are fixed-size and well studied assumptions.
As of the security parameter $\lambda$ and groups of up to $\Ngs$ members, the scheme features public key size $\softO(\lambda^2) \cdot \log \Ngs$, user's secret key size $\softO(\lambda)$ and signature size $\softO(\lambda) \cdot \log \Ngs$.
Our scheme thus achieves a level of efficiency comparable to recent proposals based on standard (i.e. non-ideal) lattices~\cite{LLLS13,NZZ15,LNW15,LLNW16} in the static setting as depicted in \cref{table:lattice-gs-comparison}.
In particular, the cost of moving to dynamic group is reasonable: while using the scheme from~\cite{LNW15} as a building block, our construction lengthens the signature size only by a (small) constant factor.
@ -29,7 +29,7 @@ In particular, the cost of moving to dynamic group is reasonable: while using th
\label{table:lattice-gs-comparison}
\end{table}
The signature scheme with efficient protocols is here built upon the $\SIS$-based signature of Böhl \textit{et al.}~\cite{BHJ+15}, which is itself a variant of Boyen's signature~\cite{Boy10}.
The signature scheme with efficient protocols is built upon the $\SIS$-based signature of Böhl \textit{et al.}~\cite{BHJ+15}, which is itself a variant of Boyen's signature~\cite{Boy10}.
The latter scheme involves a public key containing matrices $\mathbf{A}, \mathbf{A}_0, \ldots, \mathbf{A}_\ell \in \Zq^{n \times m}$ and signs an $\ell$-bit message $\mathfrak m \in \bit^\ell$ by computing a short vector $\mathbf{v} \in \ZZ^{2m}_{}$ such that ${[\mathbf{A} \mid \mathbf{A}_0 + \sum_{j=1}^\ell \mathfrak m[j] \mathbf{A}_j ]} \cdot \mathbf{v} = \mathbf 0^n \bmod q$.
The variant proposed by Böhl \textit{et al.} only uses a constant number of matrices $\mathbf{A}, \mathbf{A}_0, \mathbf{A}_1 \in \Zq^{n \times m}$ where each signature is assigned with a single-use tag $\tau$ and the public key involves an extra matrix $\mathbf{D} \in \Zq^{n \times m}$ and a vector $\mathbf{u} \in \Zq^n$.
A message $\mathfrak m$ is then signed by first applying a chameleon hash function $\mathbf{h} = \mathcal{H}(\mathfrak m, \mathbf{s}) \in \bit^m_{}$ and signing $\mathbf{h}$ by computing a short $\mathbf{v} \in \ZZ^{2m}_{}$ such that ${[\mathbf{A} \mid \mathbf{A}_0 + \tau \mathbf{A}_1 ]} \cdot \mathbf{v} = \mathbf{u} + \mathbf{D} \cdot \mathbf{h} \bmod q$.