Add sigmasig
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\chapter{Lattice-Based Dynamic Group Signatures}
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\addcontentsline{tof}{chapter}{\protect\numberline{\thechapter} Signatures de groupe dynamique à base de réseaux euclidiens}
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\label{ch:gs-lwe}
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% TODO: remove
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\clearpage
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@ -11,20 +11,33 @@ This construction has been the first fully secure group signature scheme from la
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Before describing those scheme, let us recall in this Chapter the definition of a dynamic group signature and its related security definitions.
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\section{State of the art of ZK proofs} \label{sse:gs-definitions}
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\addcontentsline{tof}{section}{\protect\numberline{\thesection} Définition formelle et correction}
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\section{Background} \label{sse:gs-background}
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\addcontentsline{tof}{section}{\protect\numberline{\thesection} Historique}
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Dynamic group signatures are a solution to allow a user to authenticate a message on behalf of a set of users it belongs to (the \textit{group}) that can be publicly verified while remaining anonymous inside its group.
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On the other hand, the user remains accountable for the signatures it provides as there exists an authority, the \textit{opening authority}, that can lift the anonymity of a given signature using its own secret key.
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In the dynamic setting, a group signature scheme has a second authority: the \textit{group manager}, that allows a user to enroll the group after an interaction with it.
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This primitive was introduced by Bellare, Micciancio and Warinschi~\cite{BMW03} in 2003 and was extended to dynamic groups by Bellare, Shi and Zhang ({BSZ}) in 2005~\cite{BSZ05}.
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The \cite{BMW03}~model summarize the security of a group signature scheme in two notions: \textit{anonymity} and \textit{traceability}.
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The former notions models the fact that without the opening authority's secret, even if everyone colludes, no one can trace back a user from a signature; the latter sum up the fact that even if everyone is corrupted (even the opening authority), it is impossible to forge a valid signature that does not open to a valid user.
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One application of this primitive can be to handle anonymous access control for public transportation systems.
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In order to commute, a person should prove the possession of a valid subscription to the transportation service.
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Thus, at registration to the service, the commuter joins the group of \emph{users with a valid subscription} and when it takes the transport, it is asked to sign the timestamp of its entry in the name of the group.
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In case of misbehaviour, another entity --\,let say the police\,-- is able to lift the anonymity of the signatures logged by the reading machine.
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Then, the public transportation company is unable to learn anything from seeing the signatures, except the validity of the subscription of a user, and on the other hand, the police does not have access to the logs except if the public transportation company hands them to it.
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Other applications of group signatures can be advocated as authentication of low-range communications for intelligent cars or anonymous access control of a building.
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As we can see, most applications necessitate the use of \textit{dynamically growing} groups in order to be meaningful.
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Bootle, Cerulli, Chaidos, Ghadafi and Groth~\cite{BCC+16} rise the problem of revocation and proposed a model that handle the issues that arose from the introduction of revocation called ``\textit{fully-dynamic}'' group signatures, but as the main difficulty is to allow users to dynamically enroll to the group ---\,as revocation has been known to be implemented in a modular manner~\cite{LLNW14}\,--- we do not consider this approach.
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\section{Formal Definition and Correctness} \label{sse:gs-definitions}
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\addcontentsline{tof}{section}{\protect\numberline{\thesection} Définition formelle et correction}
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\addcontentsline{tof}{section}{\protect\numberline{\thesection} Définition formelle et correction}
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This section recalls the syntax and the security definitions of dynamic group signatures based on the model of Kiayias and Yung~\cite{KY06}.
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A \emph{group signature} allows a group member to
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attest that a message was provided by a member of a \emph{group} without being
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altered during the process and preserving the \emph{anonymity} of the users.
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This primitive was introduced by Bellare, Micciancio and Warinschi~\cite{BMW03}
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in 2003 and was extended to dynamic groups by Bellare, Shi and Zhang
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({BSZ}) in 2005~\cite{BSZ05}.
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%A \emph{group signature} allows a group member to attest that a message was provided by a member of a \emph{group} without being altered during the process and preserving the \emph{anonymity} of the users.
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\begin{figure}
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@ -155,20 +168,20 @@ provides $\join_{\user}$ with $\langle i,\scr_{i },\crt_{i } \rangle$.
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%
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\item If
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$[\join_{\user}(\lambda,\mathcal{Y}),\join_{\GM}(\lambda,St,\mathcal{Y},\mathcal{S}_{\GM})]$
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is run by two honest parties following the protocol and
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$\langle i, \crt_{i }, \scr_{i } \rangle$ is obtained by $\join_{\user}$, then
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is run by two honest parties following the protocol and
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$\langle i, \crt_{i }, \scr_{i } \rangle$ is obtained by $\join_{\user}$, then
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we have $\crt_{i } \leftrightharpoons_{\mathcal{Y}} \scr_{i }$.
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%
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\item For each %revocation period $t$ and any
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$\langle i, \crt_{i }, \scr_{i } \rangle$ such that $\crt_{i }
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\leftrightharpoons_{\mathcal{Y}} \scr_{i }$, satisfying condition 2, we have
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\leftrightharpoons_{\mathcal{Y}} \scr_{i }$, satisfying condition 2, we have
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$ \mathsf{Verify}\big(\mathsf{Sign}(\mathcal{Y}, \crt_{i }, \scr_{i
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},M),M,\mathcal{Y}\big)=1$.
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},M),M,\mathcal{Y}\big)=1$.
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%
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\item For any outcome $\langle i, \crt_{i }, \scr_{i } \rangle$ of $[\join_{\user}(.,. ),\join_{\GM}(.,St,.,. )]$ for some valid
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\item For any outcome $\langle i, \crt_{i }, \scr_{i } \rangle$ of $[\join_{\user}(.,. ),\join_{\GM}(.,St,.,. )]$ for some valid
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$St$,
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if $\sigma =\mathsf{Sign}(\mathcal{Y},\crt_{i }, \scr_{i},M)$, then
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$\mathsf{Open}(M,\sigma,\mathcal{S}_{\OA},\mathcal{Y},St')=i.$
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$\mathsf{Open}(M,\sigma,\mathcal{S}_{\OA},\mathcal{Y},St')=i.$
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%
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\end{enumerate}
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%
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@ -249,7 +262,7 @@ following oracles:
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certificate $\crt_{i }$ and a membership secret $\scr_{i }$. If no such elements $(\crt_i,\scr_i)$ exist or if $i \not\in U^b$, the
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interface returns $\bot$. Otherwise, it outputs a signature $\sigma$ on
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behalf of user
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$i$
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$i$
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and also sets $\mathsf{Sigs} \leftarrow \mathsf{Sigs} || (i,M,\sigma)$.
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%
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\item $Q_{\mathsf{open}}$: when this oracle is invoked on input of a valid
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@ -272,7 +285,7 @@ following oracles:
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\end{itemize}
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\noindent Based on the above syntax, the
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\noindent Based on the above syntax, the
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security properties are formalized as follows.
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\subsection{Security Against Misidentification Attacks}
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@ -300,7 +313,7 @@ security properties are formalized as follows.
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In a misidentification attack, the adversary can corrupt the opening authority
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using the $Q_{\mathsf{keyOA}}$ oracle and introduce
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malicious users in the group via $Q_{\ajoin}$-queries.
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malicious users in the group via $Q_{\ajoin}$-queries.
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It aims at producing a valid signature $\sigma^\star$ that does not open to any
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adversarially-controlled user.
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@ -309,11 +322,11 @@ adversarially-controlled user.
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A dynamic group signature scheme is secure against \emph{misidentification
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attacks} if, for any $\ppt$ adversary $\adv$ involved in Experiment~$\Exp{\textrm{mis-id}}{\adv}(\lambda)$
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described in Figure~\ref{exp:mis-id}, we have:
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\[\advantage{\adv}{\mathrm{mis}\textrm{-}\mathrm{id}}(\lambda) \triangleq
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\[\advantage{\adv}{\mathrm{mis}\textrm{-}\mathrm{id}}(\lambda) \triangleq
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\Proba{\,\Exp{\mathrm{mis}\textrm{-}\mathrm{id}}{\adv}(\lambda)=1} =
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\negl[\lambda].\]
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\end{definition}
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\subsection{Non-Frameability}
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@ -334,7 +347,7 @@ adversarially-controlled user.
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\pcif i=\mathsf{Open}(M^\star,\sigma^\star,\mathcal{S}_{\OA},
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\mathcal{Y},St') \not \in U^b \pcthen\\
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\pcind \pcreturn 0\\
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\pcif
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\pcif
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\bigwedge_{j \in U^b \textrm{ s.t. } j=i~} (j,M^\star,\ast)
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\not\in \mathsf{Sigs} \pcthen \\
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\pcind \pcreturn 1\\
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@ -425,7 +438,7 @@ to query $Q_{\mathsf{open}}$ for $(M^\star,\sigma^\star)$.
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%
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A dynamic group signature scheme is fully anonymous if, for any $\ppt$ adversary
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$\adv$
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in the experiment~$\Exp{\mathrm{anon}}{\adv, d}(\lambda)$ described in Figure~\ref{exp:anon}, the following distance is negligible:
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in the experiment~$\Exp{\mathrm{anon}}{\adv, d}(\lambda)$ described in Figure~\ref{exp:anon}, the following distance is negligible:
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\[\advantage{\adv}{\mathrm{anon}}\left( \lambda \right) \triangleq
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\left| \Proba{\,\Expt_{\adv, 1}^{\mathrm{anon}}(\lambda) = 1} -\Proba{\,\Expt_{\adv, 0}^{\mathrm{anon}}(\lambda) = 1} \right|\]
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50
chap-ZK.tex
50
chap-ZK.tex
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@ -128,24 +128,24 @@ An example of commitment scheme that will prove useful in \cref{sse:stern} is th
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This construction relies on the following hash function:
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\begin{definition}[$\SIS$-based hash function] \label{de:sis-hash}
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Let $n,\ell,q \in \ZZ$ be parameters such that the $\SIS_{n,\ell,q, \sqrt \ell}$ assumption holds.
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Let $\mathbf A \in \Zq^{n \times \ell}$, and let $f_{\mathbf A}: \bit^\ell \to \Zq^n$ be the function that maps its input string $x$ into a binary vector $\mathbf x \in \Zq^n$ and outputs $\mathbf A \mathbf x \bmod q \in \Zq^n$.
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Let $\mathbf{A} \in \Zq^{n \times \ell}$, and let $f_{\mathbf{A}}: \bit^\ell \to \Zq^n$ be the function that maps its input string $x$ into a binary vector $\mathbf{x} \in \Zq^n$ and outputs $\mathbf{A} \mathbf{x} \bmod q \in \Zq^n$.
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One can notice that $f_{\mathbf A}$ is indeed a collision resistant one-way function under the $\SIS$ assumption, as finding two inputs $x \neq \tilde{x}$ such that $\mathbf A \cdot \mathbf x = \mathbf A \cdot \tilde{\mathbf x} \bmod q$ leads to a non-zero vector $\mathbf x' =\mathbf x - \tilde{\mathbf x} \in \ZZ$ such that $\|\mathbf x'\|_2 \leq \sqrt \ell$.
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One can notice that $f_{\mathbf{A}}$ is indeed a collision resistant one-way function under the $\SIS$ assumption, as finding two inputs $x \neq \tilde{x}$ such that $\mathbf{A} \cdot \mathbf{x} = \mathbf{A} \cdot \tilde{\mathbf{x}} \bmod q$ leads to a non-zero vector $\mathbf{x}' =\mathbf{x} - \tilde{\mathbf{x}} \in \ZZ$ such that $\|\mathbf{x}'\|_2 \leq \sqrt \ell$.
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It is thus possible to apply the \textit{Merkle-Damg{\aa}rd construction}~\cite{Mer79,Mer89,Dam89} on $f_{\mathbf A}$ to obtain a \textit{collision resistant hash function} $h_{\mathbf A}: \bit^\star \to \Zq^n$ that is secure under $\SIS_{n,\ell, q, \sqrt \ell}$.
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It is thus possible to apply the \textit{Merkle-Damg{\aa}rd construction}~\cite{Mer79,Mer89,Dam89} on $f_{\mathbf{A}}$ to obtain a \textit{collision resistant hash function} $h_{\mathbf{A}}: \bit^\star \to \Zq^n$ that is secure under $\SIS_{n,\ell, q, \sqrt \ell}$.
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\end{definition}
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It is then possible to use this hash function $h_{\mathbf A}$ to construct the following string commitment scheme.
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It is then possible to use this hash function $h_{\mathbf{A}}$ to construct the following string commitment scheme.
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\begin{definition}[\SIS-based commitment scheme] \label{de:sis-commitment}
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Given parameters $n,m,q \in \ZZ$, let us define the following commitment scheme due to~\cite{KTX08}.
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\begin{description}
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\item[\textsf{Setup}$(1^\lambda)$:] Pick two random matrices $\mathbf A_M, \mathbf A_\rho \in \U(\Zq^{n \times m})$ and define the public parameters as the matrix $\mathbf A = [ \mathbf A_M \mid \mathbf A_\rho]$.
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\item[$\textsf{Commit}(\mathbf A, M; \rho)$:] To commit to a string $M \in \{0,1\}^\star$ under randomness $\rho \in \{0,1\}^{m}$, first parse $\mathbf A \in \Zq^{n \times 2m}$ as $[\mathbf A_M \mid \mathbf A_\rho]$ as in the \textsf{\textbf{Setup}} algorithm,
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then compute $\com = h_{\mathbf A_M}(M) + f_{\mathbf A_\rho}(\rho) \in \Zq^n$,
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where $h_{\mathbf A_M}$ and $f_{\mathbf A_\rho}$ are the hash function and the one-way collision resistant function defined in \cref{de:sis-hash}.
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\item[\textsf{Setup}$(1^\lambda)$:] Pick two random matrices $\mathbf{A}_M, \mathbf{A}_\rho \in \U(\Zq^{n \times m})$ and define the public parameters as the matrix $\mathbf{A} = [ \mathbf{A}_M \mid \mathbf{A}_\rho]$.
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\item[$\textsf{Commit}(\mathbf{A}, M; \rho)$:] To commit to a string $M \in \{0,1\}^\star$ under randomness $\rho \in \{0,1\}^{m}$, first parse $\mathbf{A} \in \Zq^{n \times 2m}$ as $[\mathbf{A}_M \mid \mathbf{A}_\rho]$ as in the \textsf{\textbf{Setup}} algorithm,
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then compute $\com = h_{\mathbf{A}_M}(M) + f_{\mathbf{A}_\rho}(\rho) \in \Zq^n$,
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where $h_{\mathbf{A}_M}$ and $f_{\mathbf{A}_\rho}$ are the hash function and the one-way collision resistant function defined in \cref{de:sis-hash}.
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The opening corresponds to the randomness $\rho$ used in the computation.
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\item[$\textsf{Verify}(\mathbf A, \com, \open, M)$:] First parse $\mathbf A$ as in the \textsf{\textbf{Setup}} algorithm. Then accept if and only if $\open \in \bit^m$ and $\com = h_{\mathbf A_M}(M) + f_{\mathbf A_\rho}(\rho)$.
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\item[$\textsf{Verify}(\mathbf{A}, \com, \open, M)$:] First parse $\mathbf{A}$ as in the \textsf{\textbf{Setup}} algorithm. Then accept if and only if $\open \in \bit^m$ and $\com = h_{\mathbf{A}_M}(M) + f_{\mathbf{A}_\rho}(\rho)$.
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\end{description}
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\end{definition}
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@ -231,29 +231,29 @@ In the protocol described in Figure~\ref{fig:schnorr-dlog}, the underlying commi
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Given its effiency, Schnorr's protocol is used along with Fiat-Shamir heuristic in the pairing-based group signature described in~\cref{ch:sigmasig}.
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This methodology has also been adapted in the ideal lattice-setting by Lyubashevsky~\cite{Lyu08, Lyu09} along with a technique called \textit{rejection sampling} in order to construct zero-knowledge proofs from ideal lattice assumptions and is described in Figure~\ref{fig:schnorr-lwe}.
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In this description $D_y$ and $D_c$ are the distributions from which $\mathbf y_1, \mathbf y_2$ and $\mathbf c$ have to be sampled respectively, and $G$ describes the set of \textit{good} responses $\mathbf z_1, \mathbf z_2$ in order not to leak informations about $\mathbf s_1, \mathbf s_2$.
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The part under square braces is called the \textit{rejection} phase, and ensure that the transmitted $\mathbf z_1, \mathbf z_2$ will not leak any information about $\mathbf s_1, \mathbf s_2$ to V.
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In this description $D_y$ and $D_c$ are the distributions from which $\mathbf{y}_1, \mathbf{y}_2$ and $\mathbf{c}$ have to be sampled respectively, and $G$ describes the set of \textit{good} responses $\mathbf{z}_1, \mathbf{z}_2$ in order not to leak informations about $\mathbf{s}_1, \mathbf{s}_2$.
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The part under square braces is called the \textit{rejection} phase, and ensure that the transmitted $\mathbf{z}_1, \mathbf{z}_2$ will not leak any information about $\mathbf{s}_1, \mathbf{s}_2$ to V.
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This part induced a noticeable error-rate where the prover aborts the proof. As the protocol is proven \textit{witness indistinguishable}~\cite{Lyu09}, one can run the protocol multiple times in parallel and hope that one of them will not abort~\cite{FS90}.
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\begin{figure}
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\textbf{Common input:} A public element $\mathbf a \in R$ where $R = \ZZ_p[\mathbf x]/\langle \mathbf x^n + 1 \rangle$.
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\textbf{Common input:} A public element $\mathbf{a} \in R$ where $R = \ZZ_p[\mathbf{x}]/\langle \mathbf{x}^n + 1 \rangle$.
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\bigskip
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\centering
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\procedure{Schnorr's Protocol for Ring-SIS}{%
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P(\mathbf t = \mathbf a \cdot \mathbf s_1 + \mathbf s_2, (\mathbf s_1, \mathbf s_2)) \> \> V(\mathbf t) \\
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\mathbf y_1, \mathbf y_2 \sample D_y \in R \> \> \\
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\mathbf w = \mathbf a \cdot \mathbf y_1 + \mathbf y_2 \in R \\
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\> \sendmessageright*{\mathbf w} \> \\
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\> \> \mathbf c \sample D_c \in R \mbox{ (small)} \\
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\> \sendmessageleft*{\mathbf c} \> \\
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\mathbf z_1 \gets \mathbf s_1 \mathbf c + \mathbf y_1 \in R\\
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\mathbf z_2 \gets \mathbf s_2 \mathbf c + \mathbf y_2 \in R\\{}
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[\pcif \mathbf z_1, \mathbf z_2 \notin G^2 \pcthen\\
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\pcind \mathbf z_1, \mathbf z_2 \gets \bot, \bot ]\\
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\> \sendmessageright*{\mathbf z_1, \mathbf z_2} \> \\
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\> \> \pcif \mathbf z_1 \in G \wedge \mathbf z_2 \in G \wedge\\
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\>\> \pcind \mathbf a \cdot \mathbf z_1 + \mathbf z_2 = \mathbf t \mathbf c + \mathbf w \pcthen\\
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P(\mathbf{t} = \mathbf{a} \cdot \mathbf{s}_1 + \mathbf{s}_2, (\mathbf{s}_1, \mathbf{s}_2)) \> \> V(\mathbf{t}) \\
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\mathbf{y}_1, \mathbf{y}_2 \sample D_y \in R \> \> \\
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\mathbf{w} = \mathbf{a} \cdot \mathbf{y}_1 + \mathbf{y}_2 \in R \\
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\> \sendmessageright*{\mathbf{w}} \> \\
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\> \> \mathbf{c} \sample D_c \in R \mbox{ (small)} \\
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\> \sendmessageleft*{\mathbf{c}} \> \\
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\mathbf{z}_1 \gets \mathbf{s}_1 \mathbf{c} + \mathbf{y}_1 \in R\\
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\mathbf{z}_2 \gets \mathbf{s}_2 \mathbf{c} + \mathbf{y}_2 \in R\\{}
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[\pcif \mathbf{z}_1, \mathbf{z}_2 \notin G^2 \pcthen\\
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\pcind \mathbf{z}_1, \mathbf{z}_2 \gets \bot, \bot ]\\
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\> \sendmessageright*{\mathbf{z}_1, \mathbf{z}_2} \> \\
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\> \> \pcif \mathbf{z}_1 \in G \wedge \mathbf{z}_2 \in G \wedge\\
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\>\> \pcind \mathbf{a} \cdot \mathbf{z}_1 + \mathbf{z}_2 = \mathbf{t} \mathbf{c} + \mathbf{w} \pcthen\\
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\>\> \pcind \pcreturn 1\\
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\>\> \pcelse \\
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\>\> \pcind \pcreturn 0
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@ -233,7 +233,7 @@ Two examples of security game are given in Figure~\ref{fig:sec-game-examples}: t
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\caption{Some security games examples} \label{fig:sec-game-examples}
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\end{figure}
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\index{Reduction!Advantage}
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\index{Reduction!Advantage} \index{Encryption!IND-CPA}
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The \indcpa{} game is an \emph{indistinguishability} game. Meaning that the goal for the adversary $\mathcal A$ against this game is to distinguish between two messages from different distributions.
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To model this, for any adversary $\adv$, we define a notion of \emph{advantage} for the $\indcpa$ game as
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\[
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@ -255,6 +255,7 @@ The goal of the adversary is not to distinguish between two distributions, but t
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Those signature queries are provided by an oracle \oracle{sign}{sk,\cdot}, which on input $m$ returns the signature $\sigma = \Sigma.\mathsf{sign}(sk, m)$ and add $\sigma$ to $\ensemble{sign}$. The initialization of these sets and the behaviour of oracle may be omitted in the rest of this thesis for the sake of readability.
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\index{Signatures!EU-CMA}
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For EU-CMA, the advantage of an adversary $\adv$ is defined as
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\[
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\advantage{\textrm{EU-CMA}}{\adv}(\lambda)
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\chapter*{List of Publications}
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\chapter*[Publication List]{List of Publications}
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\addcontentsline{toc}{chapter}{List of publications}
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\addcontentsline{tof}{chapter}{Liste des publications}
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Available at \url{https://hal.inria.fr/hal-01622197v1/}.\\
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\doi{10.1007/978-3-319-70694-8_19}.
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\end{description}
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\addcontentsline{tof}{chapter}{\protect\numberline{\thechapter} Signatures de groupe dynamique à base de couplages}
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\label{ch:sigmasig}
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This section present the result of~\cite{LMPY16}
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%-----------------------------------------------------------------------
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\section{Building blocks}
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We use bilinear maps $e:\GG \times \Gh \to \GT$ over
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groups of prime order $p$ and we rely on the assumed security of the \SDL and \SXDH problems defined in \cref{se:pairings}. All these definitions are recalled below.
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\defPairings*
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\defSXDH*
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\defSDL*
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\addcontentsline{tof}{section}{\protect\numberline{\thechapter} Briques de base}
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\subsection{Quasi-Adaptive NIZK Arguments for Linear Subspaces} \label{sse:sigmasig-qa-nizk}
|
||||
\addcontentsline{tof}{section}{\protect\numberline{\thechapter} Argument NIZK quasi-adaptatif pour un sous-espace linéaire}
|
||||
|
||||
Quasi-Adaptive NIZK (QA-NIZK) proofs \cite{JR13} are NIZK proofs where the common reference string (CRS)
|
||||
may depend on the language for which proofs have to be generated.
|
||||
Formal definitions are given in \cite{JR13,LPJY14,KW15}. %Appendix~\ref{QA-NIZK}.
|
||||
|
||||
This section recalls the QA-NIZK argument of \cite{KW15} for proving membership in the row space of a matrix.
|
||||
In the description below, we assume that all
|
||||
algorithms take as input the description of common public parameters $\mathsf{cp}$ consisting of asymmetric
|
||||
bilinear groups $(\GG,\Gh,\GG_T,p)$ of prime order $p>2^\lambda$, where $\lambda$ is the security parameter.
|
||||
In this setting the problem is to convince that $\boldsymbol v$ is a linear combination of the rows of a given
|
||||
$\mathbf{M}\in\GG^{t\times n}$.
|
||||
|
||||
Kiltz and Wee \cite{KW15} suggested the following construction which simplifies \cite{LPJY14} and remains secure under \SXDH.
|
||||
We stress that $\mathsf{cp}$ is independent of the matrix $\mathbf{M} = (\vec{M}_1\cdots\vec{M}_t)^T$.
|
||||
|
||||
\begin{description}
|
||||
\item[\boldmath$\mathsf{Keygen}(\mathsf{cp},\mathbf{M})$:]
|
||||
Given public parameters $\mathsf{cp}=(\GG,\Gh,\GG_T,p)$ and the matrix $\mathbf{M}=(M_{i,j})\in\GG^{t\times n}$.
|
||||
Then, choose $\hat{g_z} \sample \Gh$. Pick $\mathsf{tk}=(\chi_1,\ldots,\chi_n) \sample \Zp^n$
|
||||
and compute $\hat{g}_j=\hat{g_z}^{\chi_j}$, for all $j=1$ to $n$.
|
||||
Then, for $i=1$ to $t$, compute $z_i=\prod_{j=1}^n M_{i,j}^{-\chi_j}$ and
|
||||
output $\mathsf{crs}=\big(\{ z_i \}_{i=1}^t,~ \hat{g}_z,~\{ \hat{g}_j \}_{j=1}^n \big)
|
||||
\in \GG^t\times\Gh^{n+1}$.
|
||||
|
||||
\item[\boldmath$\mathsf{Prove}(\mathsf{crs}, {\boldsymbol v}, \{\omega_i\}_{i=1}^t)$:]
|
||||
To prove that ${\boldsymbol v}=\vec{M}_1^{\omega_1}\cdots\vec{M}_t^{\omega_t}$,
|
||||
for some witness $\omega_1,\ldots,\omega_t \in \Zp$,
|
||||
where $\vec{M}_i$ denotes the $i$-th row of $\mathbf{M}$,
|
||||
parse $\mathsf{crs}$ as above
|
||||
and return $\pi=\prod_{i=1}^t z_{i}^{\omega_i}$.
|
||||
|
||||
\item[\boldmath$\mathsf{Sim}(\mathsf{tk}, {\boldsymbol v})$:]
|
||||
In order to simulate a proof for a vector ${\boldsymbol v} \in \GG^n$ using $\mathsf{tk}= \{ \chi_i \}_{i=1}^n $,
|
||||
output $\pi = \prod_{j=1}^n v_j^{-\chi_j} $.
|
||||
|
||||
\item[\boldmath$\mathsf{Verify}(\mathsf{crs}, {\boldsymbol v}, \pi)$:]
|
||||
Given $\pi \in \GG$ and ${\boldsymbol v}=(v_1,\dotsc,v_n)$,
|
||||
return $1$ if and only if $(v_1,\dotsc,v_n)\neq (1_{\GG},\dotsc,1_{\GG})$ and $\pi$ satisfies
|
||||
$ 1_{\GG_T} = e(\pi,\hat{g_z}) \cdot \prod_{j=1}^n e(v_j,\hat{g}_j) . $
|
||||
\end{description}
|
||||
|
||||
The proof of the soundness of this QA-NIZK argument system requires the matrix $\mathbf{M}$ to be witness-samplable.
|
||||
This means that the reduction has to know the discrete logarithms of the group elements of $\mathbf{M}$.
|
||||
This requirement is compatible with our security proofs.
|
||||
|
||||
\section{A Randomizable Signature on Multi-Block Messages} \label{scal-sig}
|
||||
|
||||
In \cite{LPY15}, Libert \textit{et al.} described an F-unforgeable signature based on the SXDH assumption. We show that their scheme
|
||||
implies an efficient ordinary digital signature which makes it possible to efficiently sign multi-block messages in $\Zp^{\ell}$ while keeping the scheme
|
||||
compatible with efficient protocols. In order to keep the signature length independent of the number of blocks, we exploit the property that the underlying QA-NIZK argument \cite{KW15} has constant size, regardless of the dimensions of the considered linear subspace.
|
||||
Moreover, we show that their scheme remains unforgeable under the SXDH assumption.
|
||||
|
||||
\begin{description}
|
||||
\item[\textsf{Keygen}$(\lambda,\ell):$] Choose bilinear groups $\mathsf{cp}=(\GG,\Gh,\GT,p)$
|
||||
of prime order $p>2^{\lambda}$ with $g \sample \GG$, $\hat{g} \sample \Gh$.
|
||||
\end{description}
|
||||
\begin{enumerate}
|
||||
\item Choose $\omega,a \sample \Zp$,
|
||||
and set $h=g^a$,
|
||||
$\Omega=h^{\omega}$.
|
||||
\item Choose $\vec{v}=(v_1,\ldots,v_\ell,w) \sample \GG^{\ell+1}$.
|
||||
\item Define a matrix $\mathbf{{M}}=(M_{j,i})_{j,i} \in {\GG}^{ (\ell+2) \times (2\ell+4) }$
|
||||
\begin{equation}\label{matrix-scal-sig}
|
||||
\mathbf{{M}} = %\big({M}_{i,j} \big)_{i,j} =
|
||||
\setlength{\arraycolsep}{0.3em}\def\arraystretch{1.3}
|
||||
\left(\begin{array}{c|c|c|c}
|
||||
g & \mathbf{1}_{{}_{\ell+1}} & \mathbf{1}_{{}_{\ell+1}} & h \\ \hline
|
||||
\vec{v}^\top & g^{\mathbf{I_{\ell+1}}} & h^{\mathbf{I_{\ell+1}}}
|
||||
& \mathbf{1}_{{}_{\ell+1}}^\top
|
||||
\end{array}\right) ,
|
||||
\end{equation}
|
||||
where $\mathbf{1}_{{}_{\ell+1}}=(1_{\GG},\ldots,1_{\GG})\in\GG^{\ell+1}$.
|
||||
\item Run $\mathsf{Keygen}(\mathsf{cp},M)$ of the QA-NIZK argument of Section~\ref{sse:sigmasig-qa-nizk}
|
||||
to get $\mathsf{crs}=(\{ z_i \}_{i=1}^{\ell+2},~ \hat{g}_z,~\{ \hat{g}_j \}_{j=1}^{2\ell+4} )$.
|
||||
\bigskip
|
||||
\item[]
|
||||
The private key is $ \mathsf{sk}:=\omega $ and the public key is
|
||||
\begin{align*}
|
||||
\mathsf{pk}=\Bigl(
|
||||
\mathsf{cp},~g,~h,~\hat{g}, ~\vec{v}%=(v_1,\ldots,v_\ell,w)
|
||||
,~\Omega=h^\omega,~\mathsf{crs}
|
||||
\Bigr).
|
||||
\end{align*}
|
||||
\end{enumerate}
|
||||
|
||||
\begin{description}
|
||||
\item[\textsf{Sign}$(\mathsf{sk},\vec{m}=(m_1,\ldots,m_\ell)):$] given
|
||||
the private key $\mathsf{sk}=\omega$ and a message
|
||||
$\vec{m}\in \Zp^\ell$, choose $s \sample \Zp$ to compute
|
||||
\begin{align*}
|
||||
\sigma_1 &
|
||||
= g^\omega\cdot (v_1^{m_1}\cdots v_\ell^{m_\ell}\cdot w)^{s}, &
|
||||
\sigma_2 & = g^{s}, & \sigma_3 & = h^{s} .
|
||||
\end{align*}
|
||||
Then, run $\mathsf{Prove}$ of the QA-NIZK argument to prove that
|
||||
the following vector of $\GG^{2\ell+4}$
|
||||
\begin{align} \label{eq:vector}
|
||||
(\sigma_1,\sigma_2^{m_1},\ldots,\sigma_2^{m_\ell},\sigma_2,
|
||||
\sigma_3^{m_1},\ldots,\sigma_3^{m_\ell},\sigma_3,\Omega)
|
||||
\end{align}
|
||||
is in the row space of $\mathbf{M}$. This QA-NIZK proof $\pi\in\GG$ consists of $\pi = z_1^\omega \cdot (z_2^{m_1}\cdots z_{\ell+1}^{m_\ell} \cdot
|
||||
z_{\ell+2})^{s}.$
|
||||
|
||||
Return the signature $\sigma=\big(\sigma_1,\sigma_2,\sigma_3, \pi \big)\in\GG^{4}$.
|
||||
|
||||
\item[\textsf{Verify}$(\mathsf{pk},\sigma,\vec{m}):$]
|
||||
parse $\sigma$ as above and $\vec{m}$ as a tuple $(m_1,\ldots,m_\ell)$ in $\Zp^\ell$ and return $1$
|
||||
if and only if
|
||||
\begin{align} \label{sig-ver-1}
|
||||
e(\Omega,\hat{g}_{2\ell+4})^{-1} =
|
||||
&~ e(\pi,\hat{g}_z) \cdot e(\sigma_1,\hat{g_1}) \\ \nonumber
|
||||
&~ \cdot e(\sigma_2,\hat{g}_{2}^{m_1}\cdots\hat{g}_{\ell+1}^{m_\ell} \cdot \hat{g}_{\ell+2} ) \\ \nonumber
|
||||
&~~~ \cdot e(\sigma_3,\hat{g}_{\ell+3}^{m_1}\cdots\hat{g}_{2\ell+2}^{m_\ell} \cdot \hat{g}_{2\ell+3}) .
|
||||
\end{align}
|
||||
|
||||
\end{description}
|
||||
|
||||
The signature on $\ell$ scalars thus only consists of 4 elements in $\GG$
|
||||
while the verification equation only involves a computation of 5 pairings.
|
||||
|
||||
\begin{theorem} \label{th:eu-cma-1}
|
||||
The above signature scheme is existentially unforgeable under chosen-message attacks (\textsf{eu-cma}) if the SXDH assumption holds in $(\GG, \Gh, \GG_T)$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{proof}
|
||||
We will proceed as in~\cite{LPY15} to prove that the scheme of
|
||||
section~\ref{scal-sig} is secure under chosen-message attacks. Namely we will consider a sequence of hybrid games involving two
|
||||
kinds of signatures. \vspace{-0.1 cm}
|
||||
|
||||
\begin{description}
|
||||
\item[Type A signatures:] These are real signatures:
|
||||
\begin{equation} \label{eq:rel-sig-A}
|
||||
\begin{aligned}
|
||||
\sigma_1 &= g^\omega \cdot ( v_1^{m_1} \cdots v_\ell^{m_\ell} \cdot w)^s, &
|
||||
\sigma_2 &= g^s, \\
|
||||
\pi &= z_1^\omega \cdot (z_2^{m_1}\cdots z_{\ell+1}^{m_\ell} \cdot
|
||||
z_{\ell+2})^{s} ,&
|
||||
\sigma_3 &= h^s.
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
Since $(\sigma_1,\sigma_2^{m_1},\ldots,\sigma_2^{m_\ell},\sigma_2, \sigma_3^{m_1},\ldots,\sigma_3^{m_\ell},\sigma_3,\Omega)$
|
||||
is in the row space of $\mathbf{M}$, the QA-NIZK proof $\pi$ has the same distribution as if it were computed as
|
||||
\begin{equation}
|
||||
\label{eq:rel-sim-A}
|
||||
\begin{aligned}
|
||||
\pi &= \sigma_1^{-\chi_1} \cdot \left( \prod_{i=2}^{\ell+1} \sigma_2^{-\chi_i m_{i-1}} \right) \cdot \sigma_2^{-\chi_{\ell + 2}} \cdot \quad \\ \quad &
|
||||
\left( \prod_{i=\ell + 3}^{2 \ell + 2} \sigma_3^{-\chi_i m_{i - \ell - 2}} \right) \cdot
|
||||
\sigma_3^{-\chi_{2\ell+3}} \cdot \Omega^{-\chi_{2 \ell + 4}} .
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
\end{description} \smallskip
|
||||
|
||||
\noindent We also define \textbf{Type $\mathbf{A'}$} signatures as a generalization of
|
||||
Type A signatures where only condition~\eqref{eq:rel-sig-A} are imposed and no
|
||||
restriction is given on $\pi$ beyond the fact that it should be a valid
|
||||
homomorphic signature on vector~\eqref{eq:vector}.
|
||||
\smallskip
|
||||
|
||||
\begin{description}
|
||||
\item[Type B signatures:] These use a random value $\omega' \in_R \Zp$ instead of the secret key $\omega$. We pick random $\omega', s, s_1 \sample \Zp$ and
|
||||
compute:
|
||||
\begin{equation*}
|
||||
\begin{gathered}
|
||||
(\sigma_1,\sigma_2,\sigma_3) =( g^{\omega'} \cdot ( v_1^{m_1} \cdots v_\ell^{m_\ell} \cdot w)^s, ~ g^s, ~ h^{s+s_1}),
|
||||
\end{gathered}
|
||||
\label{eq:rel-sig-B}
|
||||
\end{equation*}
|
||||
The QA-NIZK proof $\pi$ is
|
||||
computed as in \eqref{eq:rel-sim-A} by using $\mathsf{tk}=\{\chi_i \}_{i=1}^{2\ell+4}$. Note that Type B signatures can be generated without using $\omega \in \Zp$.
|
||||
\end{description}
|
||||
\smallskip
|
||||
|
||||
|
||||
We consider a sequence of games.
|
||||
In Game $i$, $S_i$ denotes the event that $\adv$
|
||||
produces a valid signature $\sigma^\star$ on $M^\star$ such that
|
||||
$(M^\star, \sigma^\star)$ was not queried before, and by $E_i$ the event that
|
||||
$\adv$ produces a Type $\mathrm{A}'$ signature.
|
||||
|
||||
\begin{description}
|
||||
\item[Game 0:] This is the real game. The challenger $\bdv$ produces
|
||||
a key pair $(\mathsf{sk}, \mathsf{pk})$ and sends $\mathsf{pk}$ to $\adv$. Then $\adv$
|
||||
makes $Q$ signature queries: $\adv$ sends messages $M_i$ to $\bdv$, and $\bdv$
|
||||
answers by sending $\sigma_i = \Sign(\mathsf{sk}, M_i)$ to $\adv$. Finally $\adv$
|
||||
sends a pair $(M^\star, \sigma^\star) \notin \{ (M_i, \sigma_i) \}_{i=1}^Q$
|
||||
and wins if $\Verify(\mathsf{pk}, \sigma^\star, M^\star) = 1$.
|
||||
|
||||
\item[Game 1:] We change the way $\bdv$ answers signing queries.
|
||||
The QA-NIZK proofs $\pi$ are then computed as simulated QA-NIZK proofs
|
||||
using $\mathsf{tk}$
|
||||
as in~\eqref{eq:rel-sim-A}. These QA-NIZK proofs are thus simulated
|
||||
proofs for true statements, and then their distribution remains unchanged.
|
||||
We have $\Pr[S_1] = \Pr[S_1 \wedge E_1] + \Pr[S_1 \wedge
|
||||
\neg E_1]$.
|
||||
Lemma~\ref{le:type-a-sig} states
|
||||
that the event $S_1 \wedge
|
||||
\neg E_1$ happens with all but negligible probability: $\Pr[S_1 \wedge
|
||||
\neg E_1] \leq \advantage{\DDH}{\Gh}(\lambda) - 1/p$. Thus our task is now
|
||||
to upper-bound the probability $\Pr[S_1 \wedge E_1]$.
|
||||
|
||||
\item[Game 2.$\boldsymbol{k~(0 \leq k \leq Q)}$:] In Game $2.k$, the
|
||||
challenger returns a Type B signature for the first $k$ queries. At the
|
||||
last $Q - k$ signature queries, the challenger answers a type $A$
|
||||
signature. \cref{le:type-b-sig} ensures that
|
||||
\[\left| \Pr\Bigl[S_{2.k} \wedge E_{2.k}\Bigr] - \Pr\Bigl[S_{2.(k-1)} \wedge E_{2.(k-1)}\Bigr] \right|\]
|
||||
is smaller than $\advantage{\DDH}{\GG}(\lambda) + 1/p$.
|
||||
\end{description}
|
||||
|
||||
In Game $2.Q$, we know that if SXDH holds, $\adv$ can only output a type $\mathrm{A}'$
|
||||
forgery even if it only obtains type B signatures during the game.
|
||||
Nevertheless, lemma~\ref{le:final-forgery} shows
|
||||
that a type $\mathrm{A}'$ forgery in Game
|
||||
$2.Q$ contradicts the DDH assumptions in $\GG$. Therefore we have
|
||||
$\Pr[S_{2.Q} \wedge E_{2.Q}] \leq \advantage{\DDH}{\GG}(\lambda)$. Putting the above altogether, the probability $\Pr[S_0]$ is upper-bounded by
|
||||
\begin{multline*}
|
||||
\advantage{\DDH}{\Gh}(\lambda) + \frac{1}{p} + Q \left( \advantage{\DDH}{\GG}(\lambda) + \frac{1}{p} \right) + \advantage{\DDH}{\GG}(\lambda) \\
|
||||
< (Q + 2) \cdot \left( \advantage{\mathrm{SXDH}{\GG, \Gh}}(\lambda) + \frac{1}{p} \right).
|
||||
\end{multline*}
|
||||
\end{proof}
|
||||
|
||||
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\begin{lemma} \label{le:type-a-sig}
|
||||
In \textbf{Game 1}, if the DDH assumption holds in $\Gh$, $\adv$ can only output a type $A'$
|
||||
forgery.
|
||||
\end{lemma}
|
||||
|
||||
\begin{proof}
|
||||
Let $\adv$ be an attacker that does not
|
||||
output a type $\mathrm{A}'$ forgery. We will build an attacker $\bdv$ against the soundness of the
|
||||
Quasi-Adaptive NIZK (QA-NIZK) scheme, which security is implied from the double-pairing
|
||||
problem that reduces from DDH as explained in~\cite{LPJY13}.
|
||||
Let us define the vector $\ssigma \in \GG^{2\ell+4}$ as
|
||||
\[
|
||||
\ssigma \triangleq (\sigma_1^\star, \sigma_2^{\star m_1}, \ldots, \sigma_2^{\star m_\ell}, \sigma_2^\star, \sigma_3^{\star m_1}, \ldots, \sigma_3^{\star m_\ell}, \sigma_3^\star, \Omega)
|
||||
\in \GG^{2\ell + 4}.
|
||||
\]
|
||||
If $(M^\star, \sigma^\star)$ is not a type $\mathrm{A}'$ forgery, $\ssigma$ is then not in the row
|
||||
space of $\mathbf{M}$.
|
||||
|
||||
Our reduction $\bdv$ receives as input $\mathsf{cp}=(\GG,\hat\GG,\GG_T,p)$, a matrix ${\mathbf{M}}$ as in
|
||||
(\ref{matrix-scal-sig}) and a common
|
||||
reference string $\mathsf{crs}$ (depending on the matrix) for an instance of the
|
||||
QA-NIZK scheme allowing to prove that vectors of dimension $2\ell + 4$ are in the row space of ${\mathbf{M}}$.
|
||||
The generation of the matrix ${\mathbf{M}}$ fixes $g$, $h$ and $\vec{v}=(v_1,\ldots,v_\ell,w)\in\GG^{\ell+1}$.
|
||||
After that, $\bdv$ picks $\omega \sample Z_p$ and $\hat g \sample \Gh$, and set $\Omega = h^\omega$.
|
||||
Then, the reduction $\bdv$ sends to $\adv$ $\mathsf{cp}$ and the verification key:
|
||||
\begin{align*}
|
||||
\mathsf{pk} = \bigl( g,h,\hat g, \vec{v}, \omega,\mathsf{crs} \bigr).
|
||||
\end{align*}
|
||||
|
||||
Since $\bdv$ knows the secret key $\omega \in \Zp$, it can answer all signing queries by honestly
|
||||
running the $\Sign$ algorithm, in particular, it does not need to know $\mathsf{tk}$ to do this.
|
||||
|
||||
When $\adv$ halts, it outputs $(M^\star, \sigma^\star)$ where $\sigma^\star$ is not a Type $\mathrm{A}'$ forgery, so that $\ssigma$ is not in the row space of $\mathbf{M}$.
|
||||
Therefore, outputting $\pi^\star$ constitutes a valid proof against the soundness property of the
|
||||
scheme, and thus implies an algorithm against DDH as in~\cite{KW15} since the matrix can be
|
||||
witness-samplable.
|
||||
\end{proof}
|
||||
|
||||
\begin{lemma} \label{le:type-b-sig}
|
||||
If DDH holds in $\GG$, for each $k \in
|
||||
\{1,\ldots, Q \}$, $\adv$ produces a type $A'$ forgery with negligibly different probabilities in \textbf{Game $\boldsymbol{2.k}$} and \textbf{Game $\boldsymbol{2.(k-1)}$}.
|
||||
\end{lemma}
|
||||
%
|
||||
\begin{proof}
|
||||
Let us assume there exists an index $k \in \{1, \ldots, Q\}$ and an adversary $\adv$ that outputs a
|
||||
Type $\mathrm{A}'$ forgery with smaller probability in Game $2.k$ than in Game
|
||||
$2.(k-1)$. We build a DDH distinguisher $\bdv$. \medskip
|
||||
\\
|
||||
Algorithm $\bdv$ takes in $(g^a, g^b, \eta) \in \GG^3$, where $\eta =
|
||||
g^{a(b+c)}$, and decides if $c=0$ or $c \in_R \Zp$. To do this, $\bdv$ sets $h = g^a$. It
|
||||
picks $\omega, a_{v_1}, b_{v_1}, \ldots, a_{v_\ell}, b_{v_\ell}, a_{w}, b_{w} \sample \Zp$
|
||||
and sets $\Omega = h^\omega$ as well as:
|
||||
\[ \forall i \in \{1,\dots, \ell \}:~~ v_i = g^{a_{v_i}} \cdot h^{b_{v_i}}, \quad w = g^{a_w} \cdot h^{b_w}. \]
|
||||
% in order to have the discrete logs of $v_i$ and $w$. \medskip
|
||||
% \\
|
||||
|
||||
The reduction $\bdv$ also chooses $\mathsf{tk} = \{ \chi_i \}_{i=1}^{2\ell + 4}$ and
|
||||
computes $\mathsf{crs} = ( \{z_j\}_{j=1}^{2\ell+4}, \hat g_z, \{ \hat g_i \}_{i=1}^{2\ell + 4})$
|
||||
as in steps 3-4 of \textsf{Keygen}. It then outputs $\mathsf{pk}=(g,h,\hat g, \vec{v}, \omega,\mathsf{crs})$.
|
||||
\smallskip
|
||||
|
||||
Then, queries are answered depending on their index~$j$:\\
|
||||
\textbf{Case $\boldsymbol{j < k}$:} $\bdv$ computes a Type B signature, $\sigma = (\sigma_1, \sigma_2,
|
||||
\sigma_3, \pi)$, using $\mathsf{tk}=\{ \chi_i \}_{i=1}^{2\ell + 4}$ with the QA-NIZK simulator
|
||||
to computes $\pi$.
|
||||
|
||||
\noindent\textbf{Case $\boldsymbol{j > k}$:} The last $Q - k - 1$ signing queries are computed as
|
||||
Type A signatures, which $\bdv$ is able to generate using the secret key $\omega \in \Zp$ he knows
|
||||
and $\mathsf{crs}$ or $\mathsf{tk}=\{ \chi_i \}_{i=1}^{2\ell + 4}$ to produces valid proofs.
|
||||
|
||||
\noindent\textbf{Case $\boldsymbol{j = k}$:} In the $k$-th signing query $(m_1,\dots,m_\ell)$, $\bdv$
|
||||
embeds the DDH instance in the signature and simulates either Game $2.k$ or Game $2.(k-1)$
|
||||
depending on whether $\eta = g^{ab}$ or $\eta = g^{a(b+c)}$ for some $c \in_R \Zp$. Namely, $\bdv$ computes $\sigma_2 = g^b$, $\sigma_3 = \eta$,
|
||||
and
|
||||
$ \sigma_1 = g^\omega \sigma_2^{a_w + \sum_{i=1}^\ell a_{v_i} m_i} \sigma_3^{b_w + \sum_{i=1}^\ell b_{v_i} m_i}. $
|
||||
Then $\bdv$ simulates QA-NIZK proofs $\pi$ as recalled in \eqref{eq:rel-sim-A}, and sends $\sigma = (\sigma_1, \sigma_2, \sigma_3, \pi)$ to $\adv$.
|
||||
\smallskip
|
||||
|
||||
If $\eta = g^{ab}$, the $k$-th signature $\sigma$ is
|
||||
a Type A signature with $s=b$. If $\eta = g^{a(b+c)}$ for some $c
|
||||
\in_R \Zp$, we have:
|
||||
\begin{align*}
|
||||
\sigma_1 & = g^\omega g^{ac\cdot(b_w + \sum_{i=1}^\ell b_{v_i} m_i)} (v_1^{m_1} \cdots v_\ell^{m_\ell} w)^b\\
|
||||
& = g^{\omega'} (v_1^{m_1} \cdots v_\ell^{m_\ell} w)^b \\
|
||||
\sigma_2 &= g^b, \qquad \qquad \qquad \qquad \qquad
|
||||
\sigma_3 = h^{b+c}
|
||||
\end{align*}
|
||||
Where $\omega' = \omega + ac\cdot(b_w + \sum_{i=1}^\ell b_{v_i} m_i)$. Since the term $b_w +
|
||||
\sum_{i=1}^\ell b_{v_i}m_i$ is uniform and independent of $\adv$'s view, $\sigma$ is
|
||||
distributed as a Type B signature if $\eta = g^{a(b+c)}$.
|
||||
|
||||
When $\adv$ terminates, it outputs a couple $(m_1^\star\cdots m_\ell^\star, \sigma^\star)$ that has not been queried
|
||||
during the signing queries. Now the reduction $\bdv$ has to determine whether $\sigma^\star$ is a
|
||||
Type $\mathrm{A}'$ forgery or not. To this end, it tests if the equality:
|
||||
\begin{equation} \label{eq:verif-proof}
|
||||
\sigma_1^\star = g^\omega \sigma_2^{\star a_w + \sum_{i=1}^\ell a_{v_i} m_i^\star} \sigma_3^{\star b_w + \sum_{i=1}^\ell b_{v_i} m_i^\star}
|
||||
\end{equation}
|
||||
is satisfied. If it is, $\bdv$ outputs $1$ to indicate that $\eta = g^{ab}$. Otherwise it outputs
|
||||
$0$ and rather bets that $\eta \in_R \GG$.
|
||||
|
||||
To see why this test allows recognizing Type $\mathrm{A}'$ forgeries,
|
||||
we remark that $\sigma^\star$ is of the form:
|
||||
\begin{align*}
|
||||
\sigma^\star_2 & = g^s , &
|
||||
\sigma^\star_3 & = h^{s + s_1} , &
|
||||
\sigma^\star_1 & = g^{\omega + s_0} (v_1^{m^\star_1} \cdots v_\ell^{m^\star_\ell} w)^s ,
|
||||
\end{align*}
|
||||
and the goal of $\bdv$ is to decide whether $(s_0, s_1) = (0, 0)$ or not. We notice that
|
||||
$s_0 = a\cdot s_1 \cdot (b_w + \sum_{i=1}^\ell b_{v_i} \cdot m_i^\star)$ if the forgery fulfills
|
||||
relation~\eqref{eq:verif-proof} and we show this to only happen with probability $1/p$ for any $s_1\neq 0$
|
||||
meaning that Type $\mathrm{B}$ forgery passes the test with the same probability.
|
||||
|
||||
%\noindent And the goal of $\bdv$ is to decide if $(s_0, s_1) = (0, 0)$ or not. We notice that if
|
||||
%$s_0 \neq 0$ and $s_1 = 0$, then the relation~\eqref{eq:verif-proof} cannot be satisfied. We then
|
||||
%have the case $s_1 \neq 0$ left which implies that $s_0 = a\cdot s_1 \cdot (b_w + \sum_{i=1}^\ell
|
||||
%b_{v_i} \cdot m_i^\star)$ to satisfy~\eqref{eq:verif-proof}, which can only happen with
|
||||
%probability $1/p$.
|
||||
|
||||
From the entire game, and assuming a forgery which passes the test, we have the following linear system:
|
||||
%On the other hand, the information that $\adv$ can infer about
|
||||
%$(a_{v_1},\ldots, a_{v_\ell}, a_w, b_{v_1}, \ldots, b_{v_\ell}, b_w) \in \Zp^{2 \ell + 2}$
|
||||
%during the game amounts to the first
|
||||
%$\ell + 2$ rows of the right-hand-side member in the following linear system:
|
||||
\[
|
||||
\left(
|
||||
\bgroup
|
||||
\def\arraystretch{1.5}
|
||||
\begin{array}{c|c}
|
||||
\mathbf{I}_{\ell+1} & a \cdot \mathbf{I}_{\ell + 1}\\ \hline
|
||||
\boldsymbol{0}_{\ell + 1}^{\top} & ac \cdot( m_1 | \cdots | m_\ell | 1) \\ \hline
|
||||
\boldsymbol{0}_{\ell + 1}^{\top} & a s_1 \cdot( m_1^\star | \cdots | m_\ell^\star | 1)
|
||||
\end{array}
|
||||
\egroup
|
||||
\right) \cdot
|
||||
% \begin{pmatrix}
|
||||
% 1 & & & a & & \\
|
||||
% & \ddots & & & \ddots & \\
|
||||
% & & 1 & & & & a \\
|
||||
% & & & a c \cdot m_1 & \cdots & a c \cdot m_\ell & ac \\
|
||||
% & & & a c \cdot m_1^\star & \cdots & a c \cdot m_\ell^\star & ac
|
||||
% \end{pmatrix} \cdot
|
||||
\begin{pmatrix}
|
||||
a_{v_1} \\ \vdots \\ a_{v_\ell} \\ a_w\\
|
||||
b_{v_1} \\ \vdots \\ b_{v_\ell} \\ b_w
|
||||
\end{pmatrix}
|
||||
=
|
||||
\begin{pmatrix}
|
||||
\log_g(v_1) \\ \vdots \\ \log_g(v_\ell) \\ \log_g(w) \\
|
||||
\omega' - \omega \\ s_0
|
||||
\end{pmatrix}
|
||||
\]
|
||||
where, $\boldsymbol{0}_{\ell + 1}$ denotes the zero vector of length $\ell + 1$ and $m_1, \ldots, m_\ell$
|
||||
is the message involved in the $k$-th signing query. Note that the $(l+2)$-th equation is meaningless when
|
||||
$c=0$ since then $\omega' = \omega$. However, even if $c\neq 0$ the information that $\adv$ can infer about
|
||||
$(a_{v_1},\ldots, a_{v_\ell}, a_w, b_{v_1}, \ldots, b_{v_\ell}, b_w) \in \Zp^{2 \ell + 2}$
|
||||
during the game amounts to the first $\ell+2$ equations of the system which is of full rank. It means that
|
||||
this vector is unpredictable since all the solutions of this linear system live in a sub-space of dimension
|
||||
at least one (actually $\ell=(2\ell+2) -(\ell+2)$). Finally, as long as $s_1\neq 0$, the right value $s_0$
|
||||
can only be guessed with probability $1/p$ since the last row of the matrix is independent of the others
|
||||
as soon as $(m_1, \ldots, m_\ell) \neq (m^\star_1, \ldots, m^\star_\ell) \neq 0$.
|
||||
|
||||
To conclude the proof, since $\bdv$ is able the tell apart the type of the forgery, if $\adv$'s probability to
|
||||
output a forgery of some Type in Game $k-1$ (\textit{i.e.}, $c=0$) was significantly different than in Game $k$
|
||||
(\textit{i.e.}, $c\neq0$) then $B$ would be able to solve the DDH problem with non-negligible advantage.
|
||||
|
||||
\end{proof}
|
||||
|
||||
\begin{lemma} \label{le:final-forgery}
|
||||
In \textbf{Game $\boldsymbol{2.Q}$}, a PPT adversary outputting a type $A'$ forgery would contradict
|
||||
the DDH assumption in $\GG$:
|
||||
$ \Pr[S_{2.Q} \wedge E_{2.Q}] \leq \advantage{\DDH}{\GG}(\lambda).$
|
||||
\end{lemma}
|
||||
|
||||
\begin{proof}
|
||||
We will build an algorithm $\bdv$ for solving the Computational Diffie Hellman problem~(CDH) which is at
|
||||
least as hard as the DDH problem. The reduction $\bdv$ takes as input a tuple $(g, h, \Omega =
|
||||
h^\omega)$ and computes $g^\omega$. To generate $\mathsf{pk}$, $\bdv$ picks $\hat g
|
||||
\sample \Gh$, $a_{v_1}, \ldots, a_{v_\ell}, a_w \sample \Zp$ and computes
|
||||
$ v_1 = g^{a_{v_1}},$ \ldots, $ v_\ell = g^{a_{v_\ell}}$, and $w = g^{a_w}.$ Then $\bdv$ generates
|
||||
$\mathsf{tk} = \{ \chi_i \}_{i=1}^{2\ell + 4}$,
|
||||
$\mathsf{crs} = (\{ z_j\}_{j=1}^{\ell + 2}, \hat g_z, \{\hat g_i\}_{i=1}^{2\ell + 4})$
|
||||
as in step 3-4 of the key generation algorithm, then sends the public key
|
||||
$ pk = \bigl(g, h, \hat g, \boldsymbol{v} , \Omega=h^\omega, \mathsf{crs}\bigr) $ to $\adv$.
|
||||
%\begin{multline*}
|
||||
% pk = \Bigl(g, h, \hat g, \boldsymbol{v} , \Omega=h^\omega,
|
||||
% \mathsf{pk}_{hsps}, \{ (z_j, r_j) \}_{j=1}^{\ell + 2} \Bigr)
|
||||
%\end{multline*}
|
||||
|
||||
\noindent $\bdv$ also retains $\mathsf{tk} = \{ \chi_i \}_{i=1}^{2\ell + 4}$ to handle
|
||||
signing queries. We recall that during the game, signing queries are answered by returning a
|
||||
Type B signature so that, using $\mathsf{tk}$, $\bdv$ can answer all queries without knowing the
|
||||
$\omega = \log_h(\Omega)$ which is part of the CDH challenge.
|
||||
|
||||
The results of Lemma~\ref{le:type-b-sig} implies that even if $\adv$ only obtains Type B signatures,
|
||||
it will necessarily output a Type $\mathrm{A}'$ forgery
|
||||
$\sigma^\star = (\sigma^\star_1, \sigma^\star_2, \sigma^\star_3, \pi^\star)$
|
||||
unless the DDH assumption does not hold in $\GG$.
|
||||
This event thus allows $\bdv$ to compute
|
||||
\[g^\omega = \sigma_1^\star \cdot {\sigma_2^\star}^{-a_w - \sum_{i=1}^\ell a_{v_i} m_i^\star}_{},\]
|
||||
which contradicts the DDH assumption in $\GG$.
|
||||
\end{proof}
|
||||
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
|
||||
|
|
|
@ -18,7 +18,7 @@ In this chapter, we describe the different structures on which the cryptography
|
|||
|
||||
\section{Pairing-Based Cryptography}
|
||||
\addcontentsline{tof}{section}{\protect\numberline{\thesection} Cryptographie à base de couplage}
|
||||
\label{se:pairing}
|
||||
\label{se:pairings}
|
||||
|
||||
\input sec-pairings
|
||||
|
||||
|
|
|
@ -11,9 +11,9 @@
|
|||
\draw[matA, xshift=-2cm, yshift=.5cm] (0,0) rectangle node {$\mathbf{A}$} (1.5,1);
|
||||
\node at (-.2, .75) {$,$};
|
||||
\draw[matA] (0,0) rectangle node {$\mathbf{A}^T_{}$} (1,1.5);
|
||||
\draw[|-|, vecS] (1.2, 1.5) -- node[right] {$\mathbf s$} ++(0, -1);
|
||||
\draw[|-|, vecS] (1.2, 1.5) -- node[right] {$\mathbf{s}$} ++(0, -1);
|
||||
\node at (1.8, .75) {$+$};
|
||||
\draw[|-|, vecE] (2.1, 1.5) -- node[right] {$\mathbf e$} ++ (0, -1.5);
|
||||
\draw[|-|, vecE] (2.1, 1.5) -- node[right] {$\mathbf{e}$} ++ (0, -1.5);
|
||||
\end{tikzpicture}
|
||||
\right)$\\[.5em]
|
||||
$\in \Zq^{n \times m} \times \Zq^{m}$,
|
||||
|
@ -22,15 +22,15 @@
|
|||
\begin{minipage}[t]{.4\textwidth}
|
||||
\textbf{$\SIS_{n,m,q,\beta}$ problem:}\\Given\\[.5em]
|
||||
$\tikz[baseline=.3cm]{ \draw[fill=blue!10] (0,0) rectangle node {$\mathbf{A}$} (1.5,1); } \in \Zq^{n \times m},$
|
||||
find $\textcolor{red!70!black}{\mathbf x} \in \ZZ^m_{}$ such that\\[.5em]
|
||||
find $\textcolor{red!70!black}{\mathbf{x}} \in \ZZ^m_{}$ such that\\[.5em]
|
||||
$\begin{tikzpicture}[baseline=.25cm]
|
||||
\tikzstyle{matA}=[fill=blue!10]
|
||||
\tikzstyle{vecX}=[color=red!70!black]
|
||||
\draw[matA] (0,0) rectangle node {$\mathbf{A}$} (1.5,1);
|
||||
\draw[|-|, vecX] (1.7, 1) -- node[right] {$\mathbf x$} ++ (0, -1.5);
|
||||
\draw[|-|, vecX] (1.7, 1) -- node[right] {$\mathbf{x}$} ++ (0, -1.5);
|
||||
\node at (2.4, .25) {$=$};
|
||||
\draw[|-|] (2.8, 1) -- node[right] {$\mathbf 0^n$} ++ (0, -1);
|
||||
\end{tikzpicture},$ and\\$0< \|\textcolor{red!70!black}{\mathbf x}\| \leq \beta$.
|
||||
\end{tikzpicture},$ and\\$0< \|\textcolor{red!70!black}{\mathbf{x}}\| \leq \beta$.
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\medskip
|
||||
|
|
|
@ -1,7 +1,7 @@
|
|||
\makeatletter
|
||||
\newcommand\frenchtableofcontents{%
|
||||
\selectlanguage{french}%
|
||||
\chapter*{\contentsname
|
||||
\chapter*[\contentsname]{\contentsname
|
||||
\@mkboth{%
|
||||
\MakeUppercase\contentsname}{\MakeUppercase\contentsname}}%
|
||||
\@starttoc{tof}%
|
||||
|
|
|
@ -14,6 +14,7 @@
|
|||
\newcommand{\redto}{\ensuremath{\preceq_P}}
|
||||
%% Primitives
|
||||
\newcommand{\ZK}{\textsf{ZK}\xspace}
|
||||
\newcommand{\ZKAoK}{\textsf{ZKAoK}\xspace}
|
||||
\newcommand{\NIZK}{\textsf{NIZK}\xspace}
|
||||
\newcommand{\PKE}{\textsf{PKE}\xspace}
|
||||
\newcommand{\OT}{\textsf{OT}\xspace}
|
||||
|
@ -105,6 +106,7 @@
|
|||
\newcommand{\bjoin}{\mathsf{b}\textrm{-}\mathsf{join}}
|
||||
\newcommand{\pjoin}{\mathsf{p}\textrm{-}\mathsf{join}}
|
||||
\newcommand{\interface}{\mathcal{I}}
|
||||
\newcommand{\ssigma}{\boldsymbol{\sigma}\xspace}
|
||||
|
||||
% Other
|
||||
\newcommand{\TODO}{\textbf{\textcolor{red}{TODO}}}
|
||||
|
|
6
main.tex
6
main.tex
|
@ -76,6 +76,7 @@
|
|||
À \ldots
|
||||
\end{flushright}
|
||||
\vspace*{\stretch{2}}
|
||||
%%%%%%%%%%%%%
|
||||
|
||||
\input abstract
|
||||
|
||||
|
@ -86,12 +87,15 @@
|
|||
\cleardoublepage
|
||||
\tableofcontents
|
||||
|
||||
\cleardoublepage
|
||||
\input symbols
|
||||
|
||||
\mainmatter
|
||||
\pagestyle{ruled}
|
||||
|
||||
\input chap-introduction
|
||||
|
||||
\cleardoublepage
|
||||
{\let\newpage\relax
|
||||
\part{Background}
|
||||
\label{pa:background}
|
||||
|
@ -104,6 +108,7 @@
|
|||
|
||||
\input chap-ZK
|
||||
|
||||
\cleardoublepage
|
||||
{\let\newpage\relax
|
||||
\part{Group Signatures and Anonymous Credentials}
|
||||
\label{pa:gs-ac}
|
||||
|
@ -116,6 +121,7 @@
|
|||
|
||||
\input chap-GS-LWE
|
||||
|
||||
\cleardoublepage
|
||||
{\let\newpage\relax
|
||||
\part{Group Encryption and Adaptive Oblivious Transfer}
|
||||
\label{pa:ge-ot}
|
||||
|
|
|
@ -67,13 +67,13 @@ In order to define the $\SIVP$ problem and assumption, let us first define the s
|
|||
\begin{definition}[Successive minima] \label{de:lattice-lambda}
|
||||
For a lattice $\Lambda$ of dimension $n$, let us define for $i \in \{1,\ldots,n\}$ the $i$-th successive minimum as
|
||||
\[ \lambda_i(\Lambda) = \inf \bigl\{ r \mid \dim \left( \Span\left(\lambda \cap \mathcal B\left(\mathbf 0, r \right) \right) \right) \geq i \bigr\}, \]
|
||||
where $\mathcal B(\mathbf c, r)$ denotes the ball of radius $r$ centered in $\mathbf c$.
|
||||
where $\mathcal B(\mathbf{c}, r)$ denotes the ball of radius $r$ centered in $\mathbf{c}$.
|
||||
\end{definition}
|
||||
|
||||
This leads us to the $\SIVP$ problem, which is finding a set of sufficiently short linearly independent vectors given a lattice basis.
|
||||
|
||||
\begin{definition}[$\SIVP$] \label{de:sivp}
|
||||
For a dimension $n$ lattice described by a basis $\mathbf B \in \RR^{n \times m}$, and a parameter $\gamma > 0$, the shortest independent vectors problem is to find $n$ linearly independent vectors $v_1, \ldots, v_n$ such that $\| v_1 \| \leq \| v_2 \| \leq \ldots \leq \| v_n \|$ and $\|v_n\| \leq \gamma \cdot \lambda_n(\mathbf B)$.
|
||||
For a dimension $n$ lattice described by a basis $\mathbf{B} \in \RR^{n \times m}$, and a parameter $\gamma > 0$, the shortest independent vectors problem is to find $n$ linearly independent vectors $v_1, \ldots, v_n$ such that $\| v_1 \| \leq \| v_2 \| \leq \ldots \leq \| v_n \|$ and $\|v_n\| \leq \gamma \cdot \lambda_n(\mathbf{B})$.
|
||||
\end{definition}
|
||||
|
||||
As explained before, the hardness of this assumption for worst-case lattices implies the hardness of the following two assumptions in their average-case setting, which are illustrated in Figure~\ref{fig:lwe-sis}.
|
||||
|
@ -84,7 +84,7 @@ In other words, it means that no polynomial time algorithms can solve those prob
|
|||
Let~$m,q,\beta$ be functions of~$n \in \mathbb{N}$.
|
||||
The \textit{Short Integer Solution} problem $\SIS_{n,m,q,\beta}$ is, given~$\mathbf{A} \sample \U(\Zq^{n \times m})$, find~$\mathbf{x} \in \Lambda_q^{\perp}(\mathbf{A})$ with~$0 < \|\mathbf{x}\| \leq \beta$.
|
||||
|
||||
The \textit{Inhomogeneous Short Integer Solution}~$\ISIS_{n,m,q,\beta}$ problem is, given~$\mathbf{A} \sample \U(\Zq^{n \times m})$ and $\mathbf u \in \Zq^n$, find~$\mathbf{x} \in \Lambda_q^{\mathbf u}(\mathbf A)$ with~$0 < \| \mathbf x \| \leq \beta$.
|
||||
The \textit{Inhomogeneous Short Integer Solution}~$\ISIS_{n,m,q,\beta}$ problem is, given~$\mathbf{A} \sample \U(\Zq^{n \times m})$ and $\mathbf{u} \in \Zq^n$, find~$\mathbf{x} \in \Lambda_q^{\mathbf{u}}(\mathbf{A})$ with~$0 < \| \mathbf{x} \| \leq \beta$.
|
||||
\end{definition}
|
||||
|
||||
Evidences of the hardness of the $\SIS$ and $\ISIS$ assumptions are given by the following Lemma, which reduced these problems from $\SIVP$.
|
||||
|
@ -168,10 +168,10 @@ We also make use of an algorithm that extends a trapdoor for~$\mathbf{A} \in \ZZ
|
|||
In some of our security proofs, analogously to \cite{Boy10,BHJ+15} we also use a technique due to Agrawal, Boneh and Boyen~\cite{ABB10} that implements an all-but-one trapdoor mechanism (akin to the one of Boneh and Boyen \cite{BB04}) in the lattice setting.
|
||||
|
||||
\begin{lemma}[{\cite[Th.~19]{ABB10}}]\label{lem:sampler}
|
||||
There exists a $\ppt$ algorithm $\SampleR$ that takes as inputs matrices $\mathbf A, \mathbf C \in \ZZ_q^{n \times m}$, a low-norm matrix $\mathbf R \in \ZZ^{m \times m}$,
|
||||
a short basis $\mathbf{T_C} \in \ZZ^{m \times m}$ of $\Lambda_q^{\perp}(\mathbf{C})$, a vector $\mathbf u \in \ZZ_q^{n}$ and a rational $\sigma$ such that $\sigma \geq \|
|
||||
\widetilde{\mathbf{T_C}}\| \cdot \Omega(\sqrt{\log n})$, and outputs a short vector $\mathbf{b} \in \ZZ^{2m}$ such that $\left[ \begin{array}{c|c} \mathbf A ~ &~ \mathbf A
|
||||
\cdot \mathbf R + \mathbf C \end{array} \right]\cdot \mathbf b = \mathbf u \bmod q$ and with distribution statistically close to $D_{L,\sigma}$ where $L$ denotes the shifted
|
||||
lattice $\Lambda^\mathbf{u}_q \left( \left[ \begin{array}{c|c} \mathbf A ~&~ \mathbf A \cdot \mathbf R + \mathbf C \end{array} \right] \right)$.
|
||||
%$\{ \mathbf x \in \ZZ^{2 m} : \left[ \begin{array}{c|c} \mathbf A ~&~ \mathbf A \cdot \mathbf R + \mathbf C \end{array} \right] \cdot \mathbf x = \mathbf u \bmod q \}$.
|
||||
There exists a $\ppt$ algorithm $\SampleR$ that takes as inputs matrices $\mathbf{A}, \mathbf{C} \in \ZZ_q^{n \times m}$, a low-norm matrix $\mathbf{R} \in \ZZ^{m \times m}$,
|
||||
a short basis $\mathbf{T_C} \in \ZZ^{m \times m}$ of $\Lambda_q^{\perp}(\mathbf{C})$, a vector $\mathbf{u} \in \ZZ_q^{n}$ and a rational $\sigma$ such that $\sigma \geq \|
|
||||
\widetilde{\mathbf{T_C}}\| \cdot \Omega(\sqrt{\log n})$, and outputs a short vector $\mathbf{b} \in \ZZ^{2m}$ such that $\left[ \begin{array}{c|c} \mathbf{A} ~ &~ \mathbf{A}
|
||||
\cdot \mathbf{R} + \mathbf{C} \end{array} \right]\cdot \mathbf{b} = \mathbf{u} \bmod q$ and with distribution statistically close to $D_{L,\sigma}$ where $L$ denotes the shifted
|
||||
lattice $\Lambda^\mathbf{u}_q \left( \left[ \begin{array}{c|c} \mathbf{A} ~&~ \mathbf{A} \cdot \mathbf{R} + \mathbf{C} \end{array} \right] \right)$.
|
||||
%$\{ \mathbf{x} \in \ZZ^{2 m} : \left[ \begin{array}{c|c} \mathbf{A} ~&~ \mathbf{A} \cdot \mathbf{R} + \mathbf{C} \end{array} \right] \cdot \mathbf{x} = \mathbf{u} \bmod q \}$.
|
||||
\end{lemma}
|
||||
|
|
|
@ -11,14 +11,14 @@ In the following, we rely on the black-box definition of cryptographic pairings
|
|||
|
||||
|
||||
%\subsection{Bilinear maps}
|
||||
\begin{definition}[Pairings~\cite{BSS05}] \label{de:pairings} \index{Pairings}
|
||||
\begin{restatable}[Pairings~\cite{BSS05}]{definition}{defPairings} \label{de:pairings} \index{Pairings}
|
||||
A pairing is a map $e: \GG \times \Gh \to \GT$ over cyclic groups of order $p$ that verifies the following properties for any $g \in \GG, \hat{g} \in \Gh$:
|
||||
\begin{enumerate}[\quad (i)]
|
||||
\item bilinearity: for any $a, b \in \Zp$, we have $e(g^a, \hat{g}^b) = e(g^b, \hat{g}^a) = e(g, \hat{g})^{ab}$.
|
||||
\item non-degeneracy: $e(g,\hat{g}) = 1_{\GT} \iff g = 1_{\GG}$ or $\hat{g} = 1_{\Gh}$.
|
||||
\item the map is computable in polynomial time in the size of the input.
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
\end{restatable}
|
||||
|
||||
For cryptographic purpose, pairings are usually defined over elliptic curves, hence $\GT$ is a multiplicative subgroup of the multiplicative group of a finite field.
|
||||
|
||||
|
@ -29,9 +29,9 @@ described in \cref{de:DDH} and recalled here.
|
|||
|
||||
This hypothesis, from which the Diffie-Hellman key exchange relies its security on, is then used to defined the $\SXDH$ assumption.
|
||||
|
||||
\begin{definition}[{$\SXDH$~\cite[As.~1]{BGdMM05}}] \index{Pairings!SXDH} \label{de:SXDH}
|
||||
\begin{restatable}[{$\SXDH$~\cite[As.~1]{BGdMM05}}]{definition}{defSXDH} \index{Pairings!SXDH} \label{de:SXDH}
|
||||
The \emph{Symmetric eXternal Diffie-Hellman} ($\SXDH$) assumption holds if the $\DDH$ assumption holds both in $\GG$ and $\Gh$.
|
||||
\end{definition}
|
||||
\end{restatable}
|
||||
|
||||
In \cref{ch:sigmasig}, the security of the group signature scheme relies on the $\SXDH$ assumption, which is a well-studied assumption.
|
||||
Moreover, this assumption is static, meaning that the size of the assumption is independent of any parameters, and is non-interactive, in the sense that it does not involve any oracle.
|
||||
|
@ -41,12 +41,12 @@ For instance, Cheon gave an attack against $q$-Strong Diffie-Hellmann problem fo
|
|||
|
||||
In the aforementioned chapter, we also rely on the following assumption, which generalizes the Discrete Logarithm problem to asymmetric groups.
|
||||
|
||||
\begin{definition}[$\SDL$]
|
||||
\begin{restatable}[$\SDL$]{definition}{defSDL}
|
||||
\label{de:SDL} \index{Pairings!SDL}
|
||||
In bilinear groups $\bigl(\GG,\Gh,\GT^{}\bigr)$ of prime order $p$, the \emph {Symmetric Discrete Logarithm} ($\SDL$) problem consists in, given
|
||||
$\bigl(g,\hat{g},g^a_{},\hat{g}^a_{}\bigr) \in \bigl(\GG \times \Gh\bigr)^2_{}$
|
||||
where $a \sample \ZZ_p^{}$, computing $a \in \ZZ_p^{}$.
|
||||
\end{definition}
|
||||
\end{restatable}
|
||||
|
||||
This assumption is still a static and non-interactive assumption.
|
||||
|
||||
|
|
|
@ -5,16 +5,16 @@
|
|||
|
||||
On the other hand, Stern's protocol has been originally introduced in the context of code-base cryptography~\cite{Ste96}.
|
||||
\index{Syndrome Decoding Problem}
|
||||
Initially, it was introduced for Syndrome Decoding Problem (\SDP): given a matrix $\mathbf M \in \FF_2^{n \times m}$ and a syndrome $\mathbf v \in \FF_2^n$, the goal is to find a binary vector $\mathbf w \in \FF_2^m$ with fixed hamming weight $w$ such that $\mathbf M \cdot \mathbf w = \mathbf v \bmod 2$.
|
||||
Initially, it was introduced for Syndrome Decoding Problem (\SDP): given a matrix $\mathbf{M} \in \FF_2^{n \times m}$ and a syndrome $\mathbf{v} \in \FF_2^n$, the goal is to find a binary vector $\mathbf{w} \in \FF_2^m$ with fixed hamming weight $w$ such that $\mathbf{M} \cdot \mathbf{w} = \mathbf{v} \bmod 2$.
|
||||
|
||||
This problem shows similarities with the $\ISIS$ problem defined in \cref{de:sis} where the constraints on the norm of $\mathbf x$ is a constraint on Hamming weight, and operations are in $\FF_2$ instead of $\Zq$.
|
||||
This problem shows similarities with the $\ISIS$ problem defined in \cref{de:sis} where the constraints on the norm of $\mathbf{x}$ is a constraint on Hamming weight, and operations are in $\FF_2$ instead of $\Zq$.
|
||||
|
||||
After the first works of Kawachi, Tanaka and Xagawa~\cite{KTX08} that extended Stern's proofs to statements $\bmod q$, the work of Ling, Nguyen, Stehlé and Wang~\cite{LNSW13} enables the use of Stern's protocol to prove general $\SIS$ or $\LWE$ statements (meaning the knowledge of a solution to these problems).
|
||||
These advances in the expressivity of Stern-like protocols has been used to further improve it and therefore enable privacy-based primitives for which no constructions existed in the post-quantum world, such as dynamic group signatures~\cite{LLM+16}, group encryption~\cite{LLM+16a}, electronic cash~\cite{LLNW17}, etc.
|
||||
|
||||
Unlike the Schnorr-like proof we saw in the previous section, Stern's proof is mainly combinatorial and relies on the fact that every permutation on a binary vector $\mathbf w \in \bit^m_{}$ leaves its Hamming weight $w$ invariant. As a consequence, for $\pi \in \permutations_m$, $\mathbf w$ satisfies these conditions if and only if $\pi(\mathbf x)$ also does.
|
||||
Unlike the Schnorr-like proof we saw in the previous section, Stern's proof is mainly combinatorial and relies on the fact that every permutation on a binary vector $\mathbf{w} \in \bit^m_{}$ leaves its Hamming weight $w$ invariant. As a consequence, for $\pi \in \permutations_m$, $\mathbf{w}$ satisfies these conditions if and only if $\pi(\mathbf{x})$ also does.
|
||||
Therefore, the randomness of $\pi$ is used to verify these two constraints (being binary and the hamming weight) in a zero-knowledge fashion.
|
||||
We can notice that this can be extended to vectors $\mathbf w \in \nbit^m$ of fixed numbers of $-1$ and $1$ that allowed~\cite{LNSW13} to propose the generalization of this protocol to any $\ISIS_{n,m,q,\beta}$ statements.
|
||||
We can notice that this can be extended to vectors $\mathbf{w} \in \nbit^m$ of fixed numbers of $-1$ and $1$ that allowed~\cite{LNSW13} to propose the generalization of this protocol to any $\ISIS_{n,m,q,\beta}$ statements.
|
||||
In \cref{sse:stern-abstraction}, we describes these permutations while abstracting the set of ZK-provable statements as the set $\mathsf{VALID}$ that satisfies conditions~\eqref{eq:zk-equivalence}.
|
||||
|
||||
It is worth noticing that this argument on knowledge does not form a $\Sigma$-protocol, as the challenge space is ternary as described in \cref{sse:stern-abstraction}.
|
||||
|
@ -31,8 +31,8 @@ In this Section, we describe in a high-level view how Stern's protocol works, an
|
|||
%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{figure}[h]
|
||||
\begin{itemize}
|
||||
\item $\mathsf B^{2}_{\mathfrak m}$: the set of vectors in $\bit^{2\mathfrak m}$ with Hamming weight $\mathfrak m$.
|
||||
\item $\mathsf B^{3}_{\mathfrak m}$: the set of vectors in $\nbit^{3\mathfrak m}$ which has exactly $\mathfrak m$ coordinates equal to $j$ for each $j \in \nbit$.
|
||||
\item $\mathsf{B}^{2}_{\mathfrak m}$: the set of vectors in $\bit^{2\mathfrak m}$ with Hamming weight $\mathfrak m$.
|
||||
\item $\mathsf{B}^{3}_{\mathfrak m}$: the set of vectors in $\nbit^{3\mathfrak m}$ which has exactly $\mathfrak m$ coordinates equal to $j$ for each $j \in \nbit$.
|
||||
\end{itemize}
|
||||
\caption{Notations for Stern-like protocols.}
|
||||
\label{fig:stern-notations}
|
||||
|
@ -40,7 +40,7 @@ In this Section, we describe in a high-level view how Stern's protocol works, an
|
|||
|
||||
The original Stern protocol was designed to prove knowledge of a SDP preimage. That is, to prove the knowledge of a vector $\mathbf{w} \in \bit^m$ that verifies
|
||||
\begin{equation} \label{eq:sdp-statement}
|
||||
\mathbf M \cdot \mathbf{w} = \mathbf v \bmod 2.
|
||||
\mathbf{M} \cdot \mathbf{w} = \mathbf{v} \bmod 2.
|
||||
\end{equation}
|
||||
|
||||
A first improvement by~\cite{KTX08} was to extend this protocol using a statistically hiding SIS-based commitment scheme as described in~\ref{fig:Interactive-Protocol} to prove in (statistical) zero-knowledge that
|
||||
|
@ -51,7 +51,7 @@ A first improvement by~\cite{KTX08} was to extend this protocol using a statisti
|
|||
The details of this proof is given in \cref{sse:stern-abstraction}, but it can be summarized in the following Lemma.
|
||||
|
||||
\begin{lemma}[{\cite[Se. 4]{KTX08}}] \label{le:zk-ktx}
|
||||
There exists a statistical \textsf{ZKAoK} with perfect completeness and soundness error 2/3 to prove the knowledge of a secret vector $\mathbf{w} \in \bit^m$ that verifies relation~\eqref{eq:isis-stern-relation} for public input $(\mathbf M, \mathbf v) \in \Zq^{n \times m} \times \Zq^{n}$.
|
||||
There exists a statistical \textsf{ZKAoK} with perfect completeness and soundness error 2/3 to prove the knowledge of a secret vector $\mathbf{w} \in \bit^m$ that verifies relation~\eqref{eq:isis-stern-relation} for public input $(\mathbf{M}, \mathbf{v}) \in \Zq^{n \times m} \times \Zq^{n}$.
|
||||
\end{lemma}
|
||||
|
||||
Ling, Nguyen, Stehlé and Wang~\cite{LNSW13} noticed that the \textsf{ZKAoK} of \cref{le:zk-ktx} straightforwardly works to prove knowledge of a vector in $\nbit^m$.
|
||||
|
@ -59,12 +59,12 @@ Ling, Nguyen, Stehlé and Wang~\cite{LNSW13} noticed that the \textsf{ZKAoK} of
|
|||
A technique to relax the constraints on the \textsf{ZKAoK} of \cref{le:zk-ktx} is the so-called ``Decomposition-Extension'' technique~\cite{LNSW13}\cite{LSS14,ELL+15,LLNW16} as explained in the introduction of this Section (\ref{sse:stern}).
|
||||
|
||||
\index{Lattices!Inhomogeneous \SIS}
|
||||
To prove the knowledge of an \ISIS preimage, i.e.
|
||||
the knowledge of a bounded vector $\mathbf{w} \in [-B,B]^m$ that satisfies relation~\eqref{eq:isis-stern-relation}, the goal is to rewrite $\mathbf w$ as $\bar{\mathbf w} = \mathbf K \cdot \mathbf w \bmod q$ with a public transfer matrix $\mathbf K$ such that $\bar{\mathbf w} \in \nbit^{m'}$ and of known numbers of elements equal to $j$ for $j \in \nbit$.
|
||||
This reduces to use \cref{le:zk-ktx} to prove the knowledge of $\bar{\mathbf w} \in \nbit^{m'}$ for public input $(\mathbf M \cdot \mathbf K, \mathbf v)$.
|
||||
To prove the knowledge of an \ISIS preimage, i.e.
|
||||
the knowledge of a bounded vector $\mathbf{w} \in [-B,B]^m$ that satisfies relation~\eqref{eq:isis-stern-relation}, the goal is to rewrite $\mathbf{w}$ as $\bar{\mathbf{w}} = \mathbf{K} \cdot \mathbf{w} \bmod q$ with a public transfer matrix $\mathbf{K}$ such that $\bar{\mathbf{w}} \in \nbit^{m'}$ and of known numbers of elements equal to $j$ for $j \in \nbit$.
|
||||
This reduces to use \cref{le:zk-ktx} to prove the knowledge of $\bar{\mathbf{w}} \in \nbit^{m'}$ for public input $(\mathbf{M} \cdot \mathbf{K}, \mathbf{v})$.
|
||||
|
||||
To construct such a transfer matrix $\mathbf K$, \cite{LNSW13} showed that \textit{decomposing} a vector $\mathbf x \in [-B,B]^m$ as a vector $\tilde{\mathbf x} \in \nbit^{m \cdot \delta_B}$ and \textit{extending} the resulting vector into $\bar{\mathbf x} \in \mathsf B^3_{m \delta_B}$ leads to a new statement that can be proven using the~\cite{KTX08} variant of Stern's protocol.
|
||||
The resulting matrix $\mathbf{K}= \left[\mathbf{K}_{m,B}^{} \mid \mathbf 0^{m \times 2m\delta_B}\right] \in \ZZ^{m \times 3m\delta_B}$, where $\mathbf{K}_{m,B}^{}$ is the \nbit-decomposition matrix $\mathbf{K}_{m,B} = \mathbf I_m \otimes \left[B_1 \mid \cdots \mid B_{\delta_B} \right]$ with $B_j^{} = \left\lfloor \frac{B + 2^{j-1}}{2^j} \right\rfloor$ for all $j \in \{1,\ldots,j\}$ can be computed from public parameters.
|
||||
To construct such a transfer matrix $\mathbf{K}$, \cite{LNSW13} showed that \textit{decomposing} a vector $\mathbf{x} \in [-B,B]^m$ as a vector $\tilde{\mathbf{x}} \in \nbit^{m \cdot \delta_B}$ and \textit{extending} the resulting vector into $\bar{\mathbf{x}} \in \mathsf{B}^3_{m \delta_B}$ leads to a new statement that can be proven using the~\cite{KTX08} variant of Stern's protocol.
|
||||
The resulting matrix $\mathbf{K}= \left[\mathbf{K}_{m,B}^{} \mid \mathbf 0^{m \times 2m\delta_B}\right] \in \ZZ^{m \times 3m\delta_B}$, where $\mathbf{K}_{m,B}^{}$ is the \nbit-decomposition matrix $\mathbf{K}_{m,B} = \mathbf{I}_m \otimes \left[B_1 \mid \cdots \mid B_{\delta_B} \right]$ with $B_j^{} = \left\lfloor \frac{B + 2^{j-1}}{2^j} \right\rfloor$ for all $j \in \{1,\ldots,j\}$ can be computed from public parameters.
|
||||
|
||||
|
||||
\subsection{Abstraction of Stern's Protocol} \label{sse:stern-abstraction}
|
||||
|
@ -189,7 +189,7 @@ The proof of the theorem relies on standard simulation and extraction techniques
|
|||
|
||||
\noindent
|
||||
\item[\textsf{Case} $\overline{Ch}=2$:] $\mathsf{SIM}$ samples $\mathbf{w}' \hookleftarrow \U(\mathsf{VALID})$, $\mathbf{r} \hookleftarrow \U(\mathbb{Z}_q^D)$, $\pi \hookleftarrow \U(\mathcal{S})$, and randomnesses $\rho_1, \rho_2, \rho_3$ for $\mathsf{COM}$.
|
||||
|
||||
|
||||
Then it sends the commitment $\mathrm{CMT}= \big(C'_1, C'_2, C'_3\big)$ to $\widehat{\mathcal{V}}$, where
|
||||
\begin{gather*}
|
||||
C'_1 = \mathsf{COM}(\pi, \mathbf{M}\cdot \mathbf{r}; \rho_1), \qquad
|
||||
|
@ -209,7 +209,7 @@ The proof of the theorem relies on standard simulation and extraction techniques
|
|||
|
||||
\noindent
|
||||
\item[Case $\overline{Ch}=3$:] $\mathsf{SIM}$ samples $\mathbf{w}' \hookleftarrow \U(\mathsf{VALID})$, $\mathbf{r} \hookleftarrow \U(\mathbb{Z}_q^D)$, $\pi \hookleftarrow \U(\mathcal{S})$, and randomnesses $\rho_1, \rho_2, \rho_3$ for $\mathsf{COM}$.
|
||||
|
||||
|
||||
Then it sends the commitment $\mathrm{CMT}= \big(C'_1, C'_2, C'_3\big)$ to $\widehat{\mathcal{V}}$, where
|
||||
\[ C'_2 = \mathsf{COM}(\Gamma_{\pi}(\mathbf{r}); \rho_2), \qquad C'_3 = \mathsf{COM}(\Gamma_{\pi}(\mathbf{w}' + \mathbf{r}); \rho_3)\]
|
||||
as in the previous two cases, while
|
||||
|
@ -231,7 +231,7 @@ The proof of the theorem relies on standard simulation and extraction techniques
|
|||
\medskip
|
||||
|
||||
\noindent
|
||||
\scbf{Argument of Knowledge.} Let us assume that
|
||||
\scbf{Argument of Knowledge.} Let us assume that
|
||||
\begin{gather*}
|
||||
\mathrm{RSP}_1 = (\mathbf{t}_x, \mathbf{t}_r, \rho_{2}^{(1)}, \rho_{3}^{(1)}), \qquad
|
||||
\mathrm{RSP}_2 = (\phi_2, \mathbf{y}, \rho_{1}^{(2)}, \rho_{3}^{(2)}),\\
|
||||
|
|
85
symbols.tex
85
symbols.tex
|
@ -1,41 +1,54 @@
|
|||
\chapter*{List of Symbols}
|
||||
\chapter*[List of Symbols]{List of Symbols}
|
||||
\addcontentsline{toc}{chapter}{List of Symbols}
|
||||
\addcontentsline{tof}{chapter}{Liste des symboles et abréviations}
|
||||
|
||||
\begin{longtable}{ll}
|
||||
\multicolumn{2}{l}{\scbf{General Notations}} \\
|
||||
TM & Turing Machine \\
|
||||
$\ppt$ & Probabilistic Polynomial Time \\
|
||||
$\epsilon$ & empty word \\
|
||||
$\mathbf A$ & bold uppercase letters represent matrices\\
|
||||
$\mathbf b$ & bold lowercase letters represent column vectors\\
|
||||
$\widetilde{\mathbf A}$ & Gram-Schmidt orthogonalization of matrix $\mathbf A$\\
|
||||
$\mathbf{A}^T_{}, \mathbf{u}^T_{}$ & the transpose of a matrix or a vector respectively\\
|
||||
$\U(S)$ & If $S$ is a finite set, $\U(S)$ denotes the uniform distribution over $S$ \\
|
||||
[1ex] \multicolumn{2}{l}{\scbf{Usual sets}} \\
|
||||
$\QQ$ & the set of rational numbers \\
|
||||
$\RR$ & the set of real numbers \\
|
||||
$\ZZ$ & the set of relative integers \\
|
||||
$\ZZ_q$ & the field $\ZZ_{/q\ZZ}$, with $q$ prime \\
|
||||
$\FF_2$ & the field $\ZZ_{/2\ZZ}$ \\
|
||||
[1ex] \multicolumn{2}{l}{\scbf{Protocols}} \\
|
||||
$\PKE$ & Public Key Encryption \\
|
||||
$\ZK$ & Zero-Knowledge \\
|
||||
$\NIZK$ & Non-Interactive Zero-Knowledge \\
|
||||
$\OT$ & Oblivious Transfer \\
|
||||
[1ex] \multicolumn{2}{l}{\scbf{Security Models}} \\
|
||||
$\ROM$ & Random-Oracle Model \\
|
||||
$\UC$ & Universal Composability \\
|
||||
[1ex] \multicolumn{2}{l}{\scbf{Security Assumptions}} \\
|
||||
[.5ex] \multicolumn{2}{l}{\quad\textbf{Lattice-based}} \\
|
||||
$\SIS$ & Short Integer Solution \\
|
||||
$\ISIS$ & Inhomogeneous Short Integer Solution \\
|
||||
$\LWE$ & Learning with Errors \\
|
||||
$\SIVP$ & Shortest Independent Vectors Problem \\
|
||||
[.5ex] \multicolumn{2}{l}{\quad\textbf{Cyclic groups}} \\
|
||||
$\DLP$ & Discrete Logarithm Problem \\
|
||||
$\DDH$ & Decisional Diffie-Hellman \\
|
||||
[.5ex] \multicolumn{2}{l}{\quad\textbf{Bilinear groups}} \\
|
||||
$\SXDH$ & Symmetric eXternal Diffie-Hellman \\
|
||||
$\SDL$ & Symmetric Discrete Logarithm
|
||||
\multicolumn{2}{l}{\scbf{General Notations}} \\
|
||||
TM & Turing Machine \\
|
||||
$\ppt$ & Probabilistic Polynomial Time \\
|
||||
$\epsilon$ & empty word \\
|
||||
$\mathbf{A}$ & bold uppercase letters represent matrices \\
|
||||
$\mathbf{b}$ & bold lowercase letters represent column vectors \\
|
||||
$\widetilde{\mathbf{A}}$ & Gram-Schmidt orthogonalization of matrix $\mathbf{A}$ \\
|
||||
$\mathbf{A}^T_{}, \mathbf{u}^T_{}$ & the transpose of a matrix or a vector respectively \\
|
||||
$\U(S)$ & If $S$ is a finite set, $\U(S)$ denotes the uniform distribution over $S$\\
|
||||
$\Pr[E]$ & Probability that an event $E$ occurs \\
|
||||
[1ex] \multicolumn{2}{l}{\scbf{Usual sets}} \\
|
||||
$\QQ$ & the set of rational numbers \\
|
||||
$\RR$ & the set of real numbers \\
|
||||
$\ZZ$ & the set of relative integers \\
|
||||
$\ZZ_q$ & the field $\ZZ_{/q\ZZ}$, with $q$ prime \\
|
||||
$\FF_2$ & the field $\ZZ_{/2\ZZ}$ \\
|
||||
$\mathbb{S}^d$ & the set of vectors of dimension $d$ in the set $\mathbb{S}$ \\
|
||||
$\mathbb{S}^{n \times m}$ & the set of matrices with $n$ rows and $m$ columns in the set $\mathbb{S}$ \\
|
||||
[1ex] \multicolumn{2}{l}{\scbf{Protocols}} \\
|
||||
$\PKE$ & Public Key Encryption \\
|
||||
$\ZK$ & Zero-Knowledge \\
|
||||
$\ZKAoK$ & Zero-Knowledge Argument of Knowledge \\
|
||||
$\NIZK$ & Non-Interactive Zero-Knowledge \\
|
||||
$\OT$ & Oblivious Transfer \\
|
||||
[1ex] \multicolumn{2}{l}{\scbf{Security Notions}} \\
|
||||