Update biblio + add proofs for group signature
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chap-GS-LWE.tex
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chap-GS-LWE.tex
@ -192,11 +192,6 @@ as $\mathbf{u} = \bar{\mathbf{A}} \cdot \mathbf{e}_u \in \Zq^n$. The pu
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\{\mathbf{A}_j \}_{j=0}^{\ell}, ~ \{\mathbf{D}_k\}_{k=0}^{N},~\mathbf{D}, ~\mathbf{u} \big)$$
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is given to $\adv$.
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%Hence,
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% $\bdv$ is able to compute a trapdoor $\mathbf{T}_{\tau^{(i)}} \in \ZZ^{2m \times 2m}$ for each matrix $\{\mathbf{A}_{\tau^{(i)}} \}_{i=1}^Q $ (see~\cite[Se.~4.2]{ABB1},
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% using the basis~$\mathbf{T}_{\mathbf{C}}$ of~$\Lambda_q^{\perp}(\mathbf{C})$.
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At the $i$-th signing query $\mathsf{Msg}^{(i)}=(\mathfrak{m}_1^{(i)},\ldots,\mathfrak{m}_N^{(i)}) \in (\{0,1\}^{2m})^N$, $\bdv$ can use the trapdoor $\mathbf{T}_{\mathbf{C}} \in \ZZ^{m \times m}$ to generate a signature.
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To do this, $\bdv$ first samples $\mathbf{s}^{(i)} \sample D_{\ZZ^{2m},\sigma_1}$ and computes a vector $\mathbf{u}_M \in \Zq^m$ as
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$$\mathbf{u}_M = \mathbf{u} + \mathbf{D} \cdot \bit \bigl( \sum_{k=1}^N \mathbf{D}_k \cdot {\mathfrak{m}_k^{(i)} } + \mathbf{D}_{0} \cdot {\mathbf{s}^{(i)} } \bigr) ~~ \bmod q.$$
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@ -742,7 +737,7 @@ The scheme provides anonymity under the $\mathsf{LWE}_{n,q,\chi}$ assumption.
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\smallskip
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\item[\textsf{Game} 1:] This game is like \textsf{Game} $0$ with the difference that, at each execution of the $\mathsf{Prove}$ protocol, the challenger runs the zero-knowledge
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simulator of the interactive proof system. The latter simulator uses either a trapdoor hidden in the common reference string (if Damg\aa rd's technique \cite{Damg00} is used) or
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simulator of the interactive proof system. The latter simulator uses either a trapdoor hidden in the common reference string (if Damg\aa rd's technique \cite{Dam00} is used) or
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proceeds by programming the random oracle which allows implementing the Fiat-Shamir heuristic. In either case, the statistical zero-knowledge property ensures that the
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adversary cannot distinguish \textsf{Game} $1$ from \textsf{Game} $0$ and $|\Pr[W_1] - \Pr[W_0] | \in \mathsf{negl}(\lambda)$.
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\smallskip
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@ -758,6 +753,806 @@ The scheme provides anonymity under the $\mathsf{LWE}_{n,q,\chi}$ assumption.
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produced without any witness.
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\end{proof}
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\section{A Dynamic Lattice-Based Group Signature} \label{see:lwe-gs-desc}
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In this section, the signature scheme of Section \ref{se:gs-lwe-sigep} is used to design a group signature for dynamic groups using the syntax and the security model of Kiayias and Yung \cite{KY06}, which is recalled in \cref{sse:gs-definitions}.
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In the notations hereunder, for any positive integers $\mathfrak{n}$, and $q \geq 2$, we define the ``powers-of-2'' matrix $\mathbf{H}_{\mathfrak{n} \times \mathfrak{n}\lceil\log q\rceil} \in \ZZ_q^{\mathfrak{n} \times \mathfrak{n}\lceil\log q\rceil}$ to be:
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\begin{eqnarray*}
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\mathbf{H}_{\mathfrak{n} \times \mathfrak{n} \lceil\log q\rceil } &=& \mathbf{I}_{\mathfrak{n}} \otimes [1 \mid 2 \mid 4 \mid \ldots \mid 2^{\lceil\log q\rceil-1} ] .
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%\\ &=& \begin{bmatrix} 1 ~2~4 ~ \ldots ~2^{\lceil\log q\rceil-1} & & & & \\
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% & & & \ddots & \\
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% & & & & 1 ~2~4 ~ \ldots ~2^{\lceil\log q\rceil-1} \\
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%\end{bmatrix}.
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\end{eqnarray*}
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Also, for each vector $\mathbf{v} \in \ZZ_q^{\mathfrak{n}}$, we define $\bit(\mathbf{v}) \in \{0,1\}^{\mathfrak{n}\lceil\log q\rceil}$ to be the vector obtained by replacing each entry of $\mathbf{v}$ by its binary expansion.
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Hence, we have $\mathbf{v}=\mathbf{H}_{\mathfrak{n} \times \mathfrak{n}\lceil\log q\rceil} \cdot \bit(\mathbf{v})$ for any $\mathbf{v} \in \ZZ_q^{\mathfrak{n}}$. \\
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\indent
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In our scheme, each group membership certificate is a
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signature generated by the group manager on the user's public key. Since the group manager only needs to sign known (rather than committed) messages, we can
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use a simplified version of the signature, where the chameleon hash function does not need to choose
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the discrete Gaussian vector $\mathbf{s}$ with a larger standard deviation than other vectors. \\
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\indent
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A key component of the scheme is the two-message joining protocol whereby the group manager admits new group members by signing their public key. The first message is sent by
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the new user $\mathcal{U}_i$ who samples a membership secret consisting of a short vector $\mathbf{z}_i \sample D_{\ZZ^{4m},\sigma}$ (where $m= 2n \lceil\log q\rceil$), which is used to compute a
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syndrome $\mathbf{v}_i = \mathbf{F} \cdot \mathbf{z}_i \in \ZZ_q^{4n}$ for some public matrix $\mathbf{F} \in \ZZ_q^{4n \times 4m} $. This syndrome $\mathbf{v}_i \in \ZZ_q^{4n}$ must be signed by $\mathcal{U}_i$ using his long term secret key $\mathsf{usk}[i]$ (as in
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\cite{KY06,BSZ05}, we assume that each user has a long-term key $\mathsf{upk}[i]$ for a digital signature, which is registered in some PKI) and will uniquely
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identify $\mathcal{U}_i$.
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In order to generate a membership certificate for $\mathbf{v}_i \in \ZZ_q^{4n}$, the group manager $\mathsf{GM}$ signs its binary expansion
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$\mathsf{bin}(\mathbf{v}_i) \in \{0,1\}^{4n \lceil \log q \rceil }$ using the scheme of Section \ref{se:gs-lwe-sigep}. \\ \indent Equipped with his membership
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certificate $(\tau,\mathbf{d},\mathbf{s}) \in \{0,1\}^\ell \times \ZZ^{2m} \times \ZZ^{2m}$, the new group member $\mathcal{U}_i$ can sign a message using a Stern-like protocol for
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demonstrating his knowledge of
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a valid certificate for which he also knows the secret key associated with the certified public key $\mathbf{v}_i \in \ZZ_q^{4n}$. This boils down to
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providing evidence that the membership certificate is a valid signature on some binary message $\mathsf{bin}(\mathbf{v}_i) \in \{0,1\}^{4n \lceil \log q \rceil }$
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for which he also knows a short $\mathbf{z}_i \in \ZZ^{4m}$
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such that
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$ \mathbf{v}_i = \mathbf{H}_{4n \times 2m} \cdot \bit(\mathbf{v}_i) = \mathbf{F} \cdot \mathbf{z}_i \in \mathbb{Z}_q^{4n}$. \\
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\indent Interestingly, the process does not require any proof of knowledge of the membership secret $\mathbf{z}_i$ during the joining phase, which is round-optimal. Analogously to the Kiayias-Yung technique \cite{KY05} and constructions based on structure-preserving signatures
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\cite{AFG+10}, the joining protocol thus remains secure in environments where many users want
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to register at the same time in concurrent sessions. \\
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\indent We remark that a similar Stern-like protocol could also be directly used to prove knowledge of a Boyen signature \cite{Boy10} on a binary expansion of the
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user's syndrome $\mathbf{v}_i \in \ZZ_q^{4n}$ while preserving the user's ability to prove knowledge of a short $\mathbf{z}_i \in \ZZ^{4m}$ such that $\mathbf{F} \cdot \mathbf{z}_i =
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\mathbf{v}_i \bmod q$. However, this would require considerably longer private keys containing $ 4n \cdot \log q$ matrices $\{\mathbf{A}_j\}_{j=0}^\ell$ of dimension $n \times
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m$ each (i.e., we would need $\ell= \Theta(n \cdot \log q)$). In contrast, by using the signature scheme of Section \ref{se:gs-lwe-sigep}, we only need the group public key
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$\mathcal{Y}$ to contain $\ell=\log N_{\mathsf{gs}}$ matrices in $\ZZ_q^{n \times m}$. Since the number of users $N_{\mathsf{gs}}$ is polynomial, we have $\log
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N_{\mathsf{gs}} \ll n$, which results in a much more efficient scheme.
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\subsection{Description of the Scheme}
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\begin{description}
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\item[\textsf{Setup}$(1^\lambda,1^{N_{\mathsf{gs}}})$:] Given a security parameter $\lambda>0$
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and the maximal expected number of group members ${N_{\mathsf{gs}}}=2^{\ell} \in
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\mathsf{poly}(\lambda)$, choose lattice parameter
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$n = \mathcal{O}(\lambda)$; prime modulus $q = \widetilde{\mathcal{O}}(\ell n^3)$; dimension $m =2 n\lceil \log q\rceil$; Gaussian parameter $\sigma = \Omega(\sqrt{n\log q}\log n)$; infinity norm bounds $\beta = \sigma\omega({\log m})$ and $B = \sqrt{n} \omega(\log n)$. Let $\chi$ be a $B$-bounded distribution.
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Choose a hash function $H:\{0,1\}^*
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\rightarrow \{1,2,3\}^t$ for some $t = \omega(\log n)$,
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which will be modeled as a random oracle in the security analysis.
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Then, do the following. \smallskip \smallskip
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% \vspace{-0.3 cm}
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\begin{itemize}
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\item[1.] Generate a key pair for the signature of Section \ref{desc-sig-protoc} for signing single-block messages. Namely, run $\TrapGen(1^n,1^m,q)$ to get~$\mathbf{A} \in
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\ZZ_q^{n \times m}$ and a short basis $\mathbf{T}_{\mathbf{A}}$ of
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$\Lambda_q^{\perp}(\mathbf{A})$, which allows computing short vectors in $\Lambda_q^{\perp}(\mathbf{A})$ with Gaussian parameter $\sigma$.
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% $\sigma \geq \| \widetilde{\mathbf{T}_{\mathbf{A}}} \| \cdot \omega (\sqrt{\log m})$.
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Next, choose matrices
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$\mathbf{A}_0,\mathbf{A}_1,\ldots,\mathbf{A}_{\ell},\mathbf{D} \sample U(\ZZ_q^{n \times m})$, $ \mathbf{D}_0,\mathbf{D}_1 \sample U(\ZZ_q^{2n \times 2m})$ and a vector $\mathbf{u} \sample U(\ZZ_q^n)$.
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\item[2.] Choose an additional random matrix $\mathbf{F} \sample U(\ZZ_q^{4n \times 4m})$ uniformly. Looking ahead, this matrix will be used to ensure security against framing attacks.
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\item[3.]
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Generate a master key pair for the Gentry-Peikert-Vaikuntanathan IBE scheme
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in its multi-bit variant. This key pair consists of a statistically uniform matrix
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$\mathbf{B} \in \ZZ_q^{n \times m}$ and a short basis $\mathbf{T}_{\mathbf{B}} \in
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\ZZ^{m \times m}$ of $\Lambda_q^{\perp}(\mathbf{B})$. This basis will allow us to compute GPV private keys with a
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Gaussian parameter $\sigma_{\mathrm{GPV}} \geq \| \widetilde{\mathbf{T}}_{\mathbf{B}} \| \cdot
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\sqrt{\log m}$.
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\item[4.] Choose a one-time signature scheme $\Pi^\mathrm{OTS}=(\mathcal{G},\mathcal{S},\mathcal{V})$ and a hash function $H_0:\{0,1\}^* \rightarrow \ZZ_q^{ n \times 2m}$,
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that will be modeled as random oracles.
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\end{itemize}
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The group public key is defined
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as $$\mathcal{Y}:=\big( \mathbf{A}, ~
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\{\mathbf{A}_j \}_{j=0}^{\ell},~\mathbf{B}, ~\mathbf{D},~ \mathbf{D}_0,~\mathbf{D}_1,~\mathbf{F}, ~\mathbf{u} , ~\Pi^\mathrm{OTS}, ~ H,~H_0 \big).$$
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The opening authority's private key is $\mathcal{S}_{\OA}:=
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\mathbf{T}_{\mathbf{B}} $ and the private key of the group manager consists of $\mathcal{S}_{\GM}:= \mathbf{T}_{\mathbf{A}}$. The algorithm outputs
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$\big( \mathcal{Y},\mathcal{S}_{\GM},\mathcal{S}_{\OA} \big)$.
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\bigskip
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\item[\textsf{Join}$^{(\mathsf{GM},\mathcal{U}_i)}$:] the group manager $\GM$ and the prospective user $\mathcal{U}_i$ run the following interactive protocol: \smallskip
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$\left\langle \mathsf{J}_{\user}(\lambda,\mathcal{Y}),\mathsf{J}_{\GM}(\lambda,St,\mathcal{Y},\mathcal{S}_{\GM}) \right\rangle$
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\begin{itemize}
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\item[1.] $\mathcal{U}_i$ samples a discrete Gaussian vector $\mathbf{z}_{i} \leftarrow D_{\ZZ^{4m},\sigma}$ and computes $\mathbf{v}_{i} = \mathbf{F} \cdot \mathbf{z}_{i} \in \ZZ_q^{ 4n}$.
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He sends the vector $\mathbf{v}_{i} \in \ZZ_q^{4n}$, whose binary representation $\mathsf{bin}(\mathbf{v}_i)$ consists of $4n\lceil\log q\rceil = 2m$ bits, together with an ordinary digital signature $sig_i = \mathrm{Sign}_{\usk[i]}(\mathbf{v}_i)$ to $\GM$.
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\item[2.] $\mathsf{J}_{\GM}$ verifies that $\mathbf{v}_i$ was not previously used by a registered user and that
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$sig_i$ is a valid signature on $ \mathbf{v}_i $ w.r.t. $\upk[i]$. It aborts if this is not the case. Otherwise, $\GM$ chooses a fresh $\ell$-bit identifier $\mathsf{id}_i=\mathsf{id}_i[1]\ldots \mathsf{id}_i[\ell]
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\in \{0,1\}^{\ell}$ and
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uses $\mathcal{S}_{\GM}=\mathbf{T}_{\mathbf{A}}$ to certify
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$\mathcal{U}_i$ as a new group member. To this end, $\GM$
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defines the matrix
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\begin{eqnarray} \label{matr}
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\mathbf{A}_{\mathsf{id}_i}= \left[ \begin{array}{c|c} \mathbf{A} ~& ~ \mathbf{A}_0 +
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\sum_{j=1}^\ell \mathsf{id}_i[j] \mathbf{A}_j
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\end{array} \right] \in \ZZ_q^{ n \times 2m}.
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\end{eqnarray}
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Then, $\GM$ runs $\mathbf{T}_{\mathsf{id}_i}' \leftarrow
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\ExtBasis(\mathbf{A}_{\mathsf{id}_i},\mathbf{T}_{\mathbf{A}})$ to obtain a short delegated basis
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$\mathbf{T}_{\mathsf{id}_i}'$ of $\Lambda_q^{\perp}(\mathbf{A}_{\mathsf{id}_i}) \in \ZZ^{ 2m \times 2m }$.
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Finally, $\GM$ samples a short vector $\mathbf{s}_i \sample D_{\ZZ^{2m},\sigma }$ and uses the obtained delegated basis $\mathbf{T}_{\mathsf{id}_i}' $ to compute a short vector
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$\mathbf{d}_i = \begin{bmatrix} \mathbf{d}_{i,1} \\ \hline \mathbf{d}_{i,2} \end{bmatrix} \in \ZZ^{2m}$ such that
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\begin{eqnarray} \nonumber
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\mathbf{A}_{\mathsf{id}_i} \cdot \mathbf{d}_i &=& \left[ \begin{array}{c|c} \mathbf{A} ~& ~ \mathbf{A}_0 +
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\sum_{j=1}^\ell \mathsf{id}_i[j] \mathbf{A}_j
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\end{array} \right] \cdot \mathbf{d}_i\\
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\label{rel-cert} &=& \mathbf{u} + \mathbf{D} \cdot \bit \bigl( \mathbf{D}_0 \cdot \bit(\mathbf{v}_i) + \mathbf{D}_1 \cdot \mathbf{s}_i \bigr) \bmod q. \quad
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\end{eqnarray}
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The triple $(\mathsf{id}_i,\mathbf{d}_i,\mathbf{s}_i)$ is sent to $\mathcal{U}_i$. Then,
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$\mathsf{J}_{\user}$ verifies that the received $(\mathsf{id}_i,\mathbf{d}_i,\mathbf{s}_i)$ satisfies (\ref{rel-cert}) and that
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$\| \mathbf{d}_i \|_\infty \leq \beta$, $\| \mathbf{s}_i \|_\infty \leq \beta $. If these conditions are not satisfied, $\mathsf{J}_{\user}$ aborts.
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Otherwise,
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$\mathsf{J}_{\user}$ defines the membership
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certificate as
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$ \crt_{i }=( \mathsf{id}_i, \mathbf{d}_i,\mathbf{s}_i )$.
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The membership secret $\scr_{i }$ is defined to be $\scr_i=\mathbf{z}_i \in \ZZ^{4m}$. $\mathsf{J}_{\GM}$ stores
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$\transcript_i=(\mathbf{v}_i, \crt_i, i,\mathsf{upk}[i],sig_i)$
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in the database $St_{trans}$ of joining transcripts. \smallskip \smallskip
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\end{itemize}
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\item[\textsf{Sign}$(\mathcal{Y},\crt_i,\scr_i ,M)$:] To sign $M \in
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\{0,1\}^*$ using $\crt_i=(\mathsf{id}_i,\mathbf{d}_i,\mathbf{s}_i)$, where $\mathbf{d}_i=[ \mathbf{d}_{i,1}^T \mid \mathbf{d}_{i,2}^T ]^T \in \ZZ^{2m}$ and $\mathbf{s}_i \in \ZZ^{2m}$, as
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well as the membership secret $\scr_i=\mathbf{z}_i \in \ZZ^{4m}$, the group
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member $\mathcal{U}_i$ generates a one-time signature key pair $(\mathsf{VK},\mathsf{SK}) \leftarrow \mathcal{G}(n)$ and conducts the following steps. \smallskip
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\begin{itemize}
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\item[1.] Compute $\mathbf{G}_0=H_0(\mathsf{VK}) \in \ZZ_q^{ n \times 2m}$ and use it as an IBE public key to encrypt
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$\bit(\mathbf{v}_i) \in \{0,1\}^{2m}$, where $\mathbf{v}_i=\mathbf{F} \cdot \mathbf{z}_i \in \ZZ_q^{4n}$ is the syndrome of
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$\scr_i=\mathbf{z}_i \in \mathbb{Z}^{4m}$ for the matrix $\mathbf{F}$. Namely, compute $ \mathbf{c}_{\mathbf{v}_i} \in \ZZ_q^m \times \ZZ_q^{2m}$ as
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\begin{eqnarray} \label{enc1}
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\mathbf{c}_{\mathbf{v}_i}=(\mathbf{c}_1,\mathbf{c}_2) &=& \big( \mathbf{B}^T \cdot \mathbf{e}_0 + \mathbf{x}_1 ,~ \mathbf{G}_0^T \cdot \mathbf{e}_0 + \mathbf{x}_2 + \bit(\mathbf{v}_i) \cdot \lfloor q/2 \rfloor \big) \qquad
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%\\ \nonumber && \hspace{4cm}\in \ZZ_q^m \times \ZZ_q^{2m}
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\end{eqnarray}
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for randomly chosen $\mathbf{e}_0 \sample \chi^n$, $\mathbf{x}_1 \sample \chi^m, \mathbf{x}_2 \sample \chi^{2m} $.
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Notice that, as in the construction of \cite{LNW15}, the columns of $\mathbf{G}_0$ can be interpreted as public keys for the multi-bit version
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of the dual Regev encryption scheme.
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\item[2.] Run the protocol in Section~\ref{subsection:zk-for-group-signature} to prove the knowledge of $\mathsf{id}_i
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\in \{0,1\}^{\ell}$,
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vectors $\mathbf{s}_i \in \ZZ^{2m}, \mathbf{d}_{i,1},\mathbf{d}_{i,2} \in \ZZ^{m},\mathbf{z}_i \in \ZZ^{4m}$ with infinity norm bound $\beta $; $\mathbf{e}_0 \in \ZZ^n$, $\mathbf{x}_1 \in \ZZ^m, \mathbf{x}_2 \in \ZZ^{2m} $ with infinity norm bound $B$
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and $\bit(\mathbf{v}_i) \in \{0,1\}^{2m}, \mathbf{w}_{i} \in \{0,1\}^m$, that satisfy
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\eqref{enc1} as well as
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\begin{eqnarray} \label{rel-deux}
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\mathbf{A} \cdot \mathbf{d}_{i,1} + \mathbf{A}_0 \cdot \mathbf{d}_{i,2} + \sum_{j=1}^{\ell} ( \mathsf{id}_i[j] \cdot \mathbf{d}_{i,2}) \cdot \mathbf{A}_j
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- \mathbf{D} \cdot \mathbf{w}_i = \mathbf{u} \in \ZZ_q^n
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\end{eqnarray}
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and
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\vspace*{-0.75cm}
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\begin{eqnarray} \label{eq:rel-3}
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\left\{
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\begin{array}{l}
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\mathbf{H}_{2n \times m} \cdot \mathbf{w}_{i} = \mathbf{D}_0 \cdot \bit(\mathbf{v}_i) + \mathbf{D}_1 \cdot \mathbf{s}_i \in \ZZ_q^{2n} \\
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\mathbf{F} \cdot \mathbf{z}_i = \mathbf{H}_{4n \times 2m} \cdot \bit(\mathbf{v}_i) \in \ZZ_q^{4n}.
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\end{array}
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\right.
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\end{eqnarray}
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The protocol is repeated $t = \omega(\log n)$ times in parallel to achieve negligible soundness error, and then made non-interactive using the Fiat-Shamir
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heuristic~\cite{FS86} as a triple $\pi_K=(
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\{\mathsf{Comm}_{K,j}\}_{j=1}^t,\mathsf{Chall}_K,\{\mathsf{Resp}_{K,j}\}_{j=1}^t)$,
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where $\mathsf{Chall}_K = H(M, \vk, \mathbf{c}_{\mathbf{v}_i},
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\{ \mathsf{Comm}_{K,j}\}_{j=1}^t) \in \{1,2,3\}^t$
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\item[3.] Compute a one-time signature $sig=\mathcal{S}(\mathsf{SK},(\mathbf{c}_{\mathbf{v}_i} , \pi_K))$. \smallskip
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\end{itemize}
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Output the signature that consists of
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\begin{equation} \label{eq:sig-final} \Sigma=\big( \mathsf{VK} ,\mathbf{c}_{\mathbf{v}_i}, \pi_K,sig \big).
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\end{equation}
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\smallskip
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\item[\textsf{Verify}$(\mathcal{Y},M,\Sigma)$:] Parse the signature $\Sigma$ as in
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(\ref{eq:sig-final}). Then, return $1$ if and only if:
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(i) $\mathcal{V}(\mathsf{VK},(\mathbf{c}_{\mathbf{v}_i},\mathbf{c}_{\mathbf{s}_i},\mathbf{c}_{\mathsf{id}},\pi_K),sig)=1$;
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(ii) The proof $\pi_K$ properly verifies. \smallskip %Otherwise, return $0$. \smallskip
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\item[\textsf{Open}$(\mathcal{Y},\mathcal{S}_{\OA},M,\Sigma)$:] Parse~$\mathcal{S}_{\OA}$ as~$
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\mathbf{T}_{\mathbf{B}} \in \ZZ^{m \times m}$ and $\Sigma$ as
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in~(\ref{eq:sig-final}). \smallskip
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\begin{itemize}
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\item[1.]
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Compute $\mathbf{G}_0=H_0(\mathsf{VK}) \in \ZZ_q^{n \times 2m}$. Then, using $\mathbf{T}_{\mathbf{B}}$
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to compute a small-norm matrix
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$\mathbf{E}_{0,\mathsf{VK}} \in \ZZ^{m \times 2m }$ such that $ \mathbf{B} \cdot \mathbf{E}_{0,\mathsf{VK}} = \mathbf{G}_0 \bmod q $.
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\item[2.] Using $\mathbf{E}_{0,\mathsf{VK}}$, decrypt $\mathbf{c}_{\mathbf{v}_i}$ to obtain a string $\bit(\mathbf{v} ) \in \{0,1\}^{2m}$
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(i.e., by computing $\lfloor (\mathbf{c}_2 - \mathbf{E}_{0,\mathsf{VK}}^T \cdot \mathbf{c}_1) / (q/2) \rceil$). \smallskip
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\item[3.] Determine if the $\bit(\mathbf{v} ) \in \{0,1\}^{2m} $ obtained at step 2 corresponds to a vector $\mathbf{v} = \mathbf{H}_{4n \times 2m} \cdot \bit(\mathbf{v} ) \bmod q$ that appears in a record $\transcript_i=(\mathbf{v} , \crt_i, i,\mathsf{upk}[i],sig_i)$ of the database $St_{trans}$ for some $i$. If so,
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output the corresponding $i$ (and, optionally, $\mathsf{upk}[i]$). Otherwise, output $\perp$.
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\end{itemize}
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\end{description}
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We remark that the scheme readily extends to provide a mechanism whereby the opening authority can efficiently prove that signatures were correctly opened at each opening operation.
|
||||
The difference between the dynamic group signature models suggested by Kiayias and Yung \cite{KY06} and Bellare \textit{et al.} \cite{BSZ05} is that, in the latter, the opening authority
|
||||
($\mathsf{OA}$) must be able to convince a judge that the $\mathsf{Open}$ algorithm was run correctly.
|
||||
Here, such a mechanism can be realized using the techniques of public-key encryption with non-interactive opening \cite{DHKT08}. Namely, since
|
||||
$\bit(\mathbf{v}_i)$ is encrypted using an IBE scheme for the identity $\vk$, the $\mathsf{OA}$ can simply reveal the decryption matrix $\mathbf{E}_{0,\mathsf{VK}} $,
|
||||
that satisfies $\mathbf{B} \cdot \mathbf{E}_{0,\vk} = \mathbf{G}_0 \bmod q$ (which corresponds to the verification of a GPV signature) and allows the verifier to perform step 2 of the opening
|
||||
algorithm himself. The resulting construction is easily seen to satisfy the notion of opening soundness of Sakai \textit{et al.} \cite{SSE+12}.
|
||||
|
||||
\subsection{Efficiency and Correctness}
|
||||
\textsc{Efficiency.} The given dynamic group signature scheme can be implemented in polynomial time. The group public key has total bit-size $\mathcal{O}(\ell n m \log q) = \widetilde{\mathcal{O}}(\lambda^2)\cdot \log N_\textsf{gs}$. The secret signing key of each user consists of a small constant number of low-norm vectors, and has bit-size $\widetilde{\mathcal{O}}(\lambda)$.
|
||||
|
||||
The size of each group signature is largely dominated by that of the non-interactive argument $\pi_K$, which is obtained from the Stern-like protocol of Section~\ref{subsection:zk-for-group-signature}. Each round of the protocol has communication cost $\widetilde{\mathcal{O}}(m \cdot \log q) \cdot \log N_\textsf{gs}$. Thus, the bit-size of $\pi_K$ is $t\hspace*{-1pt}\cdot\hspace*{-1pt} \widetilde{\mathcal{O}}(m \hspace*{-1pt}\cdot\hspace*{-1pt} \log q) \hspace*{-1pt}\cdot\hspace*{-1pt} \log N_\textsf{gs} = \widetilde{\mathcal{O}}(\lambda)\hspace*{-1pt}\cdot \hspace*{-1pt}\log N_\textsf{gs}$. This is also the asymptotic bound on the size of the group signature.
|
||||
|
||||
|
||||
\smallskip
|
||||
\noindent
|
||||
\textsc{Correctness.} The correctness of algorithm \textsf{Verify}$(\mathcal{Y},M,\Sigma)$ follows from the facts that every certified group member is able to compute valid witness vectors satisfying equations~(\ref{enc1}), (\ref{rel-deux}) and (\ref{eq:rel-3}), and that the underlying argument system is perfectly complete. Moreover, the scheme parameters are chosen so that the GPV IBE~\cite{GPV08} is correct, which implies that algorithm \textsf{Open}$(\mathcal{Y},\mathcal{S}_{\OA},M,\Sigma)$ is also correct.
|
||||
|
||||
|
||||
\subsection{Security Analysis}
|
||||
|
||||
Due to the fact that the number of public matrices $\{\mathbf{A}_j\}_{j=0}^\ell$ is only logarithmic in ${N_{\mathsf{gs}}}=2^\ell$ instead of being linear in the security parameter $\lambda$,
|
||||
the proof of security against misidentification attacks (as defined in \cref{sse:gs-sec-notions}) cannot rely on the security of our signature scheme in a modular manner.
|
||||
The reason is that, at each run of the $\mathsf{Join}$ protocol, the group manager maintains a state and, instead of choosing the $\ell$-bit identifier $\mathsf{id}$ uniformly in
|
||||
$\{0,1\}^{\ell}$, it chooses an identifier that has not been used yet. Since $\ell \ll \lambda$ (given that ${N_{\mathsf{gs}}}=2^\ell$ is polynomial in $\lambda$), we thus have
|
||||
to prove security from scratch. However, the strategy of the reduction is exactly the same as in the security proof of the signature scheme.
|
||||
|
||||
|
||||
\begin{theorem} \label{traceability-thm}
|
||||
The scheme is secure against misidentification attacks under the $\SIS_{n,2m,q,\beta'}$ assumption, for $\beta' \hspace*{-1pt}=\hspace*{-1pt} \mathcal{O}(\ell \sigma^2 m^{3/2})$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{proof}
|
||||
We prove that any adversary $\adv$ with non-negligible success probability $\varepsilon$ implies an algorithm $\bdv$ solving the \textsf{SIS} problem
|
||||
in the random oracle model. \\
|
||||
\indent
|
||||
Let $\adv$ be such a $\ppt$ adversary. We build a $\ppt$
|
||||
algorithm $\bdv$ that uses $\adv$ to
|
||||
solve~$\SIS_{n,2m,q,\beta'}$: specifically, $\bdv$ takes as input~$\bar{\mathbf{A}} = \begin{bmatrix} \bar{\mathbf{A}}_1 | \bar{\mathbf{A}}_2 \end{bmatrix} \in
|
||||
\Zq^{n \times 2m}$, where $\bar{\mathbf{A}}_1,\bar{\mathbf{A}}_2 \in \Zq^{n \times m}$, and finds $\mathbf{w} \in
|
||||
\Lambda_q^{\perp}(\bar{\mathbf{A}})$ with~$0 < \|\mathbf{w}\| \leq \beta'$.
|
||||
\medskip
|
||||
|
||||
|
||||
\noindent \textbf{Initialization.} Algorithm~$\bdv$ first chooses a random $coin \sample
|
||||
U(\{0,1,2\})$ as a guess for the kind of misidentification attack that $\adv$ will mount. Also, $\bdv$
|
||||
chooses a random $\ell$-bit string $\mathsf{id}^\dagger \sample U(\{0,1\}^\ell)$.
|
||||
In
|
||||
addition, $\bdv$
|
||||
samples~$i^\star
|
||||
\sample U([1,Q_a])$. \\
|
||||
\indent
|
||||
Looking ahead, $coin=0$ corresponds to the case where, after repeated executions of $\adv$, the knowledge extractor of the proof system
|
||||
reveals witnesses containing a new identifier $\mathsf{id}^\star \in \{0,1\}^\ell$ that does not belong to any user in $U^a$.
|
||||
In this case, $\bdv$ will be able to exploit $\adv$'s forgery when $\mathsf{id}^\star=\mathsf{id}^\dagger$.
|
||||
The case $coin=1$ corresponds to $\bdv$'s expectation that the knowledge extractor will obtain the identifier $ \mathsf{id}^\star = \mathsf{id}^\dagger$ of a group member in
|
||||
$ U^a$ (i.e., a group member that was legitimately introduced at the $i^\star$-th $\mathcal{Q}_{\ajoin}$-query, for some $i^\star \in \{1,\ldots,Q_a\}$, where the identifier
|
||||
$\mathsf{id}^\dagger$ is used by $\mathcal{Q}_{\ajoin}$),
|
||||
but $\bit ( \mathbf{v}^\star ) \in \{0,1\}^{2m}$ (which is encrypted in in $\mathbf{c}_{\mathbf{v}_i}^\star$ as part of the forgery $\Sigma^\star$) and the extracted $\mathbf{s}^\star \in \ZZ^{2m}$ are such that $ \bit \bigl( \mathbf{D}_0 \cdot \bit ( \mathbf{v}^\star ) + \mathbf{D}_1 \cdot \mathbf{s}^\star \bigr) \in \{0,1\}^m $
|
||||
does not match
|
||||
the string $ \bit \bigl( \mathbf{D}_0 \cdot \bit ( \mathbf{v}_{i^\star} ) + \mathbf{D}_1 \cdot \mathbf{s}_{i^\star} \bigr) \in \{0,1\}^{2m} $ for which
|
||||
user $i^\star$ obtained a membership certificate at the $i^\star$-th $\mathcal{Q}_{\ajoin}$-query. When $coin=1$, the choice of $i^\star$ corresponds to a guess that the knowledge
|
||||
extractor will reveal an $\ell$-bit identifier that coincides with the identifier $\mathsf{id}^\dagger$ assigned to the user introduced at the $i^\star$-th $\mathcal{Q}_{\ajoin}$-query.
|
||||
The last case $coin=2$ corresponds to $\bdv$'s expectation that decrypting $\mathbf{c}_{\mathbf{v}_i}^\star$ (which is part of $\Sigma^\star$) and running
|
||||
the knowledge extractor on $\adv$ will uncover vectors $\bit ( \mathbf{v}^\star ) \in \{0,1\}^{2m}$, $\mathbf{w}^\star \in \{0,1\}^m$ and $\mathbf{s}^\star \in \ZZ^{2m}$
|
||||
such that $\mathbf{w}^\star= \bit(\mathbf{D}_0 \cdot \bit(\mathbf{v}^\star) + \mathbf{D}_1 \cdot \mathbf{s}^\star )$ and
|
||||
\begin{eqnarray} \label{collide}
|
||||
\bit \bigl( \mathbf{D}_0 \cdot \bit ( \mathbf{v}^\star ) + \mathbf{D}_1 \cdot \mathbf{s}^\star \bigr) = \bit \bigl( \mathbf{D}_0 \cdot \bit ( \mathbf{v}_{i^\star} ) + \mathbf{D}_1 \cdot \mathbf{s}_{i^\star} \bigr)
|
||||
\end{eqnarray}
|
||||
but $(\bit ( \mathbf{v}^\star ), \mathbf{s}^\star) \neq ( \bit ( \mathbf{v}_{i^\star} ), \mathbf{s}_{i^\star} ) $, where $ \mathbf{v}_{i^\star} \in \Zq^{4n}$ and $\mathbf{s}_{i^\star} \in \ZZ^{2m}$ are the vectors
|
||||
involved in the $i^\star$-th $\mathcal{Q}_{\ajoin}$-query.
|
||||
\\
|
||||
\indent
|
||||
Depending on $coin \in \{0,1,2\}$, the group public key $\mathcal{Y}$ is
|
||||
generated using different methods. \smallskip
|
||||
|
||||
\noindent $\bullet$ If $coin=0$, algorithm~$\bdv$ first randomly chooses $\mathsf{id}^\dagger \sample U(\{0,1\}^\ell)$ as a guess for the $\ell$-bit string
|
||||
that will be revealed by the knowledge extractor of the proof system after repeated executions of the adversary $\adv$.
|
||||
Then, it runs
|
||||
$\TrapGen(1^n,1^m,q)$ to obtain $\mathbf{C} \in \Zq^{n \times m}$ and a
|
||||
basis $\mathbf{T}_{\mathbf{C}}$ of~$\Lambda_q^{\perp}(\mathbf{C})$ with
|
||||
$\|\widetilde{\mathbf{T}_{\mathbf{C}}}\| \leq \bigO(\sqrt{n \log q})$. Then,
|
||||
it chooses~$\ell+2$ matrices~$ \mathbf{Q}_0,\ldots,\mathbf{Q}_{\ell},\mathbf{Q}_D \in \ZZ^{m \times m}$,
|
||||
each matrix having its columns sampled independently from~$D_{\ZZ^m,\sigma}$. Then, $\bdv$ defines the matrices $\{ \mathbf{A}_i\}_{i=0}^{\ell}$ as
|
||||
\begin{eqnarray*}
|
||||
\left\{
|
||||
\begin{array}{ll}
|
||||
\mathbf{A}_0 = \bar{\mathbf{A}}_1 \cdot \mathbf{Q}_0 + (\sum_{i=1}^{\ell} {\mathsf{id}^\dagger[i]}) \cdot
|
||||
\mathbf{C} \\
|
||||
\mathbf{A}_j = \bar{\mathbf{A}}_1 \cdot \mathbf{Q}_i + (-1)^{\mathsf{id}^{\dagger}[j]} \cdot
|
||||
\mathbf{C}, \quad \text{ for } j \in
|
||||
[1,\ell]. \\
|
||||
\mathbf{D} = \bar{\mathbf{A}}_1 \cdot \mathbf{Q}_D
|
||||
\end{array}
|
||||
\right.
|
||||
\end{eqnarray*}
|
||||
It also defines $\mathbf{A}=\bar{\mathbf{A}}_1$.
|
||||
Next, it samples a vector $\mathbf{e}_u \sample D_{\ZZ,\sigma}^m$ and computes a syndrome $\mathbf{u} = \bar{\mathbf{A}}_1 \cdot \mathbf{e}_u \in \Zq^n$. It picks $\mathbf{D}_0,\mathbf{D}_1
|
||||
\sample U(\Zq^{2n \times 2m})$ at random and also faithfully generates the GPV master key pair $(\mathbf{B},\mathbf{T}_{\mathbf{B}})$ as in Step~3 of the real setup algorithm. The group
|
||||
public key $\mathcal{Y}=\big(\mathbf{A},\{\mathbf{A}_j \}_{j=0}^{\ell}, \mathbf{B}, \mathbf{D},\mathbf{D}_0,\mathbf{D}_1,\mathbf{F}, \mathbf{u},\mathcal{OTS},H,H_0 \big)$
|
||||
is finally given to~$\adv$. \\
|
||||
\indent Note that, for each $\mathsf{id} \neq \mathsf{id}^\dagger$, we have
|
||||
\begin{eqnarray} \nonumber
|
||||
\mathbf{A}_{\mathsf{id}} &=& \left[
|
||||
\begin{array}{c|c} \bar{\mathbf{A}}_1 ~&~ \mathbf{A}_0 +
|
||||
\sum_{i=1}^\ell \mathsf{id}[i] \mathbf{A}_i
|
||||
\end{array} \right] \\ \nonumber & = & \left[
|
||||
\begin{array}{c|c} \bar{\mathbf{A}}_1 ~&~ \bar{\mathbf{A}}_1 \cdot (\mathbf{Q}_0 +
|
||||
\sum_{i=1}^{\ell} \mathsf{id}[i] \mathbf{Q}_i) + (
|
||||
\sum_{i=1}^{\ell} \mathsf{id}^\dagger [i] +(-1)^{\mathsf{id}^\dagger[i]} \mathsf{id}[i])\cdot \mathbf{C}
|
||||
\end{array} \right] \\ \label{sim-matr} &=&
|
||||
\left[
|
||||
\begin{array}{c|c} \bar{\mathbf{A}}_1 ~&~ \bar{\mathbf{A}}_1 + h_{\mathsf{id}} \cdot \mathbf{C}
|
||||
\end{array} \right]
|
||||
% \vspace*{-.1cm}
|
||||
\end{eqnarray}
|
||||
where $h_{\mathsf{id}} \in [1,\ell]$ denotes the Hamming distance between
|
||||
the identifiers $\mathsf{id}$ and $\mathsf{id}^\dagger$. Since $q>\ell$, we have
|
||||
$h_{\mathsf{id}_j} \neq 0 \bmod q$ whenever $\mathsf{id}_j \neq \mathsf{id}^\dagger$, so
|
||||
that algorithm $\bdv$ is able to compute (see~\cite[Se.~4.2]{ABB10},
|
||||
using the basis~$\mathbf{T}_{\mathbf{C}}$ of~$\Lambda_q^{\perp}(\mathbf{C})$ and
|
||||
the refined $\GPVSample$ of Lemma~\ref{le:GPV}) a basis
|
||||
$\mathbf{T}_{\mathsf{id}}$ of $\Lambda_q^{\perp}(\mathbf{A}_{\mathsf{id}})$
|
||||
with~$\|\widetilde{\mathbf{T}_{\mathsf{id}}}\| \leq \Omega(\sqrt{n\log
|
||||
q\log n})$. In contrast,
|
||||
algorithm~$\bdv$ lacks a trapdoor for $\mathbf{A}_{\mathsf{id}^\dagger}$ as the
|
||||
latter only depends on $\mathbf{A}$ and $\{\mathbf{Q}_k\}_{k=0}^{\ell}$.
|
||||
Observe that, since the columns of the matrices~$\{\mathbf{Q}_k\}_{k=0}^\ell$ are sampled
|
||||
from~$D_{\ZZ^m,\sigma}$, the
|
||||
matrices~$ \mathbf{A}_0,\ldots,\mathbf{A}_{\ell}$ are within
|
||||
statistical distance~$2^{-\Omega(m)}$ of~$U(\Zq^{n \times m})$.
|
||||
\smallskip
|
||||
|
||||
|
||||
\noindent $\bullet$ If $coin=1$, algorithm~$\bdv$ sets up $\mathcal{Y}$ by defining
|
||||
$\mathbf{D}=\bar{\mathbf{A}}$. Initially, $\bdv$
|
||||
chooses $Q_a-1$ distinct strings $\mathsf{id}_1, \ldots,\mathsf{id}_{i^\star-1}, \mathsf{id}_{i^\star+1},\ldots,\mathsf{id}_{Q_a} \in \{0,1\}^\ell$ such that, for each $i \in [1,Q_a] \backslash \{i^\star\}$, $\mathsf{id}_i$ will be embedded in the membership certificate
|
||||
returned in the $i$-th $\mathcal{Q}_{\ajoin}$-query. Let also $\mathsf{id}^\dagger=\mathsf{id}_{i^\star}$ be the $\ell$-bit identifier
|
||||
that will be used in the $i^\star$-th query.
|
||||
The reduction $\bdv$ picks random $h_0,h_1,\ldots,h_\ell \in \Zq$ under the constraints
|
||||
\begin{eqnarray*}
|
||||
h_{\mathsf{id}^\dagger} = h_0 + \sum_{j=1}^\ell \mathsf{id}^\dagger[j] \cdot h_j &=& 0 \bmod q \\
|
||||
h_{\mathsf{id}_i} = h_0 + \sum_{j=1}^\ell \mathsf{id}_i[j] \cdot h_j & \neq & 0 \bmod q \qquad \qquad i \in \{1,\ldots,Q_a\} \setminus \{i^\dagger\}
|
||||
\end{eqnarray*}
|
||||
Next, $\bdv$ runs $(\mathbf{C},\mathbf{T}_{\mathbf{C}}) \leftarrow \mathsf{TrapGen}(1^n,1^m,q)$, $(\mathbf{D}_1,\mathbf{T}_{\mathbf{D}_1}) \leftarrow \mathsf{TrapGen}(1^{2n},1^{2m},q)$ so as to obtain statistically random matrices $\mathbf{C} \in \Zq^{n \times m}$, $ \mathbf{D}_1 \in \Zq^{2n \times 2m}$ together with
|
||||
trapdoors $\mathbf{T}_{\mathbf{C}} \in \ZZ^{m \times m} $, $\mathbf{T}_{\mathbf{D}_1} \in \ZZ^{2m \times 2m}$ consisting of short bases of $\Lambda_q^{\perp}(\mathbf{C})$ and $\Lambda_q^{\perp}(\mathbf{D}_1)$, respectively. Then,
|
||||
$\bdv$
|
||||
picks a random $\mathbf{D}_0 \sample U(\Zq^{2n \times 2m})$ and re-randomizes $\mathbf{D}=\bar{\mathbf{A}}_1 \in \Zq^{n \times m}$ using Gaussian matrices
|
||||
$\mathbf{S},\mathbf{S}_0,\mathbf{S}_1,\ldots,\mathbf{S}_{\ell} \sample \ZZ^{m \times m}$ whose columns are sampled from the distribution $D_{\ZZ^m,\sigma}$.
|
||||
Namely, from $\mathbf{D} =\bar{\mathbf{A}}_1 $, $\bdv$
|
||||
defines
|
||||
\begin{eqnarray} \nonumber
|
||||
\mathbf{A} &=& \bar{\mathbf{A}}_1 \cdot \mathbf{S} \\ \label{setup-sig2}
|
||||
\mathbf{A}_0 &=& \bar{\mathbf{A}}_1 \cdot \mathbf{S}_0 + h_0 \cdot \mathbf{C} \\ \nonumber
|
||||
\mathbf{A}_j &=& \bar{\mathbf{A}}_1 \cdot \mathbf{S}_j + h_j \cdot \mathbf{C} \qquad \qquad \forall j \in \{1,\ldots,\ell\} .
|
||||
\end{eqnarray}
|
||||
As part of the generation of
|
||||
$\mathcal{Y}$, the vector $\mathbf{u} \in \Zq^n$ is obtained by picking short discrete Gaussian vectors
|
||||
$ \mathbf{d}_{i^\star,1}, \mathbf{d}_{i^\star,2} \sample D_{\ZZ^m,\sigma} $
|
||||
and computing
|
||||
\begin{eqnarray} \label{def-u}
|
||||
\mathbf{u} = [ \mathbf{A} ~\mid ~ \mathbf{A}_0 +
|
||||
\sum_{j=1}^\ell \mathsf{id}^\dagger[j] \mathbf{A}_j
|
||||
] \cdot
|
||||
\begin{bmatrix}
|
||||
\mathbf{d}_{i^\star,1} \\ \hline \mathbf{d}_{i^\star,2}
|
||||
\end{bmatrix}
|
||||
- \mathbf{D} \cdot \bit(\mathbf{c}_M),
|
||||
\end{eqnarray}
|
||||
where
|
||||
$\mathbf{c}_{M} \sample U(\Zq^{2n})$ is a randomly chosen vector. Observe that, since $\mathbf{A}$ is statistically uniform over $\Zq^{n \times m}$ and $ \mathbf{d}_{i^\star,1}
|
||||
\sample D_{\ZZ^m,\sigma}$, the distribution of
|
||||
$\mathbf{u} $ is statistically close to $U(\Zq^n)$.
|
||||
\medskip
|
||||
|
||||
\noindent $\bullet$ If $coin=2$, $\bdv$ picks $\bar{\mathbf{A}}' \sample U(\Zq^{n \times 2m})$
|
||||
and a random matrix $\mathbf{Q} \sample \ZZ^{2m \times 2m}$ whose columns are sampled from $D_{\ZZ^{2m},\sigma}$. These
|
||||
are used to define $$\mathbf{D}_0= \begin{bmatrix} \bar{\mathbf{A}} \\ \hline \bar{\mathbf{A}}' \end{bmatrix} \in \Zq^{2n \times 2m} ,$$
|
||||
and $\mathbf{D}_1=\mathbf{D}_0 \cdot \mathbf{Q} \bmod q$, which is statistically close to $U(\Zq^{2n \times 2m})$. All other components of $\mathcal{Y}$ are obtained by faithfully running the setup algorithm. \medskip
|
||||
|
||||
|
||||
\indent For each value of $coin \in \{0,1,2\}$, the group public key
|
||||
$$\mathcal{Y}=\big(\mathbf{A},\{\mathbf{A}_j \}_{j=0}^{\ell},\mathbf{B},\mathbf{D},\mathbf{D}_0,\mathbf{D}_1,\mathbf{F}, \mathbf{u},\mathcal{OTS},H,H_0 \big)$$ has a distribution which is statistically close to that of the real scheme and $\mathcal{Y}$ is given to $\adv$.
|
||||
|
||||
\medskip
|
||||
|
||||
|
||||
\noindent \textbf{Queries.} The reduction~$\bdv$ starts interacting
|
||||
with the adversary~$\adv$ and the way it handles~$\adv$'s queries to the $\mathcal{Q}_{\ajoin}$ oracle depends on the value of~$coin \in \{0,1,2\}$. \smallskip \smallskip
|
||||
|
||||
\noindent $\bullet$ If $coin=0$, answers $\mathcal{Q}_{\ajoin}$-queries as follows. When $\adv$ triggers an execution of the joining protocol, it chooses
|
||||
a syndrome $\mathbf{v}_{i} \in \Zq^n$.
|
||||
To answer the query, $\bdv$ chooses a fresh $\ell$-bit identifier $\mathsf{id}_i \in \{0,1\}^\ell$ such that
|
||||
$\mathsf{id}_i \neq \mathsf{id}^\dagger$. If $\adv$ also provides a correct signature $sig_i$ such that
|
||||
$\mathrm{Verify}_{\mathsf{upk}[i]}(\mathbf{v}_{i},sig_i)=1$, $\bdv$ samples $\mathbf{s}_i \sample D_{\ZZ^{2m},\sigma}$ and uses the trapdoor $\mathbf{T}_{\mathbf{C}}$ to compute a short vector
|
||||
$\mathbf{d}_i=[\mathbf{d}_{i,1}^T ~|~\mathbf{d}_{i,2}^T]^T \in \ZZ^{2m}$ such that
|
||||
\begin{eqnarray} \label{sim-cert}
|
||||
\mathbf{A}_{\mathsf{id}_i} \cdot \begin{bmatrix} \mathbf{d}_{i,1} \\ \hline \mathbf{d}_{i,2} \end{bmatrix} = \mathbf{u} + \mathbf{D} \cdot \bit \bigl( \mathbf{D}_0 \cdot \bit(\mathbf{v}_{i}) + \mathbf{D}_1 \cdot \mathbf{s}_i \bigr) ,
|
||||
\end{eqnarray}
|
||||
where $\mathbf{A}_{\mathsf{id}_i} \in \Zq^{n \times 2m}$ is the matrix in (\ref{sim-matr}). Note that $\bdv$ is able to compute such a vector using the $\mathsf{SampleRight}$
|
||||
algorithm of \cite{ABB10} (since the Hamming distance $h_{\mathsf{id}_i}$ between $\mathsf{id}_i$ and $\mathsf{id}^\star$ is non-zero). The membership certificate
|
||||
$\crt_i= (\mathsf{id}_i,\mathbf{d}_i,\mathbf{s}_i)$ is then returned to $\adv$.
|
||||
\smallskip
|
||||
|
||||
\noindent $\bullet$ If $coin=1$, algorithm~$\bdv$ responds each $\mathcal{Q}_{\ajoin}$-query depending on the index $i \in \{1,\ldots,Q_a\}$ of the query. Specifically,
|
||||
we distinguish two cases. \smallskip
|
||||
|
||||
\begin{itemize}
|
||||
\item[-] If $i \neq i^\star$, $\bdv$ proceeds as in the previous case. Namely, it recalls the $\ell$-bit identifier $\mathsf{id}_i \in \{0,1\}^\ell$ (for which $\mathsf{id}_i \neq \mathsf{id}^\dagger$)
|
||||
that was chosen in the setup phase and samples a short vector $\mathbf{s}_{i} \sample D_{\ZZ^{2m},\sigma}$. If $\adv$ also provides a correct signature $sig_i$ such that
|
||||
$\mathrm{Verify}_{\mathsf{upk}[i]}(\mathbf{v}_{i},sig_i)=1$, generates a membership certificate $\crt_i$ for $\adv$ as in the case $coin=0$.
|
||||
Note that
|
||||
\begin{eqnarray} \nonumber
|
||||
\mathbf{A}_{\mathsf{id}_i} &=& \left[
|
||||
\begin{array}{c|c} \bar{\mathbf{A}} \cdot \mathbf{S} ~&~ \bar{\mathbf{A}} \cdot (\mathbf{S}_0 +
|
||||
\sum_{j=1}^{\ell} \mathsf{id}_i[j] \mathbf{S}_j) + h_{\mathsf{id}_i} \mathbf{C}
|
||||
\end{array} \right] \\ \label{sim-matr-coin1} &=&
|
||||
\left[
|
||||
\begin{array}{c|c} \bar{\mathbf{A}} \cdot \mathbf{S} ~&~ \bar{\mathbf{A}} + h_{\mathsf{id}_i} \cdot \mathbf{C}
|
||||
\end{array} \right]
|
||||
% \vspace*{-.1cm}
|
||||
\end{eqnarray}
|
||||
Since $h_{\mathsf{id}_i} \neq 0$, $\bdv$ can use the trapdoor
|
||||
$\mathbf{T}_{\mathbf{C}} \in \ZZ^{m \times m}$ of $\Lambda_q^{\perp}(\mathbf{C})$ to compute a short vector $\mathbf{d}_i = [ \mathbf{d}_{i,1}^T ~|~\mathbf{d}_{i,2}^T ]^T \in \ZZ^{2m}$ such that
|
||||
\begin{eqnarray*}
|
||||
\mathbf{A}_{\mathsf{id}_i} \cdot \begin{bmatrix} \mathbf{d}_{i,1} \\ \hline \mathbf{d}_{i,2} \end{bmatrix} = \mathbf{u} + \mathbf{D} \cdot \bit \bigl( \mathbf{D}_0 \cdot (\bit(\mathbf{v}_{i}) + \mathbf{D}_1 \cdot \mathbf{s}_i \bigr) ,
|
||||
\end{eqnarray*}
|
||||
where $\mathbf{v}_{i} \in \Zq^{4n}$ is the syndrome chosen by $\adv$ at step 1 of the joining protocol.
|
||||
\item[-] If $i = i^\star$, $\bdv$ undertakes to generate a membership certificate $\crt_{i^\star}$ for the $\ell$-bit identifier $\mathsf{id}^\dagger \in \{0,1\}^\ell$ that was
|
||||
chosen at the outset of the game. To this end, $\bdv$ has to compute $\crt_{i^\star}$ without using the trapdoor $\mathbf{T}_{\mathbf{C}}$ since the matrix $\mathbf{A}_{\mathsf{id}^\dagger}$ does no longer
|
||||
depend on $\mathbf{C}$ in (\ref{sim-matr-coin1} ). This can be done by recalling
|
||||
the vector $\mathbf{d}_{i^\star,1},\mathbf{d}_{i^\star,2} \in \ZZ^m$ and $\mathbf{c}_M \in \Zq^{2n}$ that were used to define $\mathbf{u} \in \Zq^n$ in (\ref{def-u}) and using $\mathbf{T}_{\mathbf{D}_1}$. If $\adv$ provides a correct signature
|
||||
$sig_{i^\star}$ such that
|
||||
$\mathrm{Verify}_{\mathsf{upk}[i^\star]}(\mathbf{v}_{i^\star},sig_{i^\star})=1$,
|
||||
$\bdv$ uses the trapdoor $\mathbf{T}_{\mathbf{D}_1}$ of $\Lambda_q^\perp (\mathbf{D}_1)$ to sample a short vector $\mathbf{s}_{i^\star} \in \ZZ^{2m}$ of $D_{\Lambda_q^{\mathbf{c}_{i^\star}}(\mathbf{D}_1),\sigma}$, where $\mathbf{c}_{i^\star} = \mathbf{c}_M - \mathbf{D}_0 \cdot \bit( \mathbf{v}_{i^\star}) \bmod q $,
|
||||
satisfying
|
||||
$$ \mathbf{D}_1 \cdot \mathbf{s}_{i^\star} = \mathbf{c}_M - \mathbf{D}_0 \cdot \bit( \mathbf{v}_{i^\star}) ~\bmod q , $$
|
||||
before returning $\crt_{i^\star}=(\mathsf{id}^\dagger,\mathbf{d}_{i^\star} =[ \mathbf{d}_{i^\star,1}^T \mid \mathbf{d}_{i^\star,2}^T]^T,\mathbf{s}_{i^\star})$
|
||||
to $\adv$. From the definition of $\mathbf{u} \in \Zq^n$ (\ref{def-u}), it is easy to see that $\crt_{i^\star}=(\mathsf{id}^\dagger,\mathbf{d}_{i^\star} ,\mathbf{s}_{i^\star})$ forms a valid membership certificate for
|
||||
any membership secret $\mathbf{z}_{i^\star} \in \ZZ^{4m}$ corresponding to the syndrome $\mathbf{v}_{i^\star} = \mathbf{F} \cdot \mathbf{z}_{i^\star} \bmod q$.
|
||||
%Moreover, the distribution of
|
||||
%$\mathbf{s}_{i^\star}$ is
|
||||
% $D_{\ZZ^m,\sigma}^{\mathbf{c}_{v_{i^\star}}}$, where $\mathbf{c}_{v_{i^\star}} = \mathbf{c}_M - \mathbf{D}_0 \cdot \bit( \mathbf{v}_{i^\star}) \in \Zq^n $, as in \GGame $2$.
|
||||
\end{itemize}
|
||||
|
||||
Regardless of the value of $coin$, queries to the random oracle~$H$
|
||||
are handled by returning a uniformly chosen value in $\{1,2,3\}^t$. For
|
||||
each $\kappa \leq Q_H$, we let~$r_{\kappa}$ denote the answer to the
|
||||
$\kappa$-th $H$-query. Of course, if the adversary makes a given query
|
||||
more than once, then~$\bdv$ consistently returns the previously defined
|
||||
value. Queries to the random oracle $H_0$ are answered in the usual way, by returning a uniformly random value in the appropriate range. \medskip
|
||||
|
||||
\noindent \textbf{Forgery.} When $\adv$ halts, it outputs a
|
||||
signature $ \Sigma^\star=\big( \mathsf{VK}^\star ,\mathbf{c}_{\mathbf{v}_i}^\star, \pi_K^\star,sig^\star \big)$ on some message $M^\star$. At this point, $\bdv$ uses the
|
||||
trapdoor $\mathbf{T}_{\mathbf{B}}$ to decrypt $\mathbf{c}_{\mathbf{v}_i}^\star$ and obtain an $m$-bit string $\bit(\mathbf{v}^\star) \in \{0,1\}^m$.
|
||||
|
||||
%We know that, with probability $\Pr[W_2]$, it holds that
|
||||
%\begin{itemize}
|
||||
%\item[-] The pair $(M^\star,\Sigma^\star)$ results in a successful misidentification attack and, when $\bdv$ runs the $\mathsf{Open}$ algorithm on $\Sigma^\star$, the $\ell$-bit %identifier $\mathsf{id}^\star$ revealed at step 2
|
||||
%coincides with $\mathsf{id}^\dagger$.
|
||||
%\item[-]
|
||||
%If $coin=0$, $\mathsf{id}^\dagger$ did not appear in any membership certificate returned by $\mathcal{Q}_{\ajoin}$ whereas, if $coin=1$, $\mathsf{id}^\dagger$ is the identifier used by
|
||||
%$\mathcal{Q}_{\ajoin}$ at the $i^\star$-th query.
|
||||
%\item[-] If $coin=2$, the opening of $\Sigma^\star$ reveals vectors $\bit(\mathbf{v}^\star)$ and $\mathbf{s}^\star$ that result in a collision (\ref{collide})
|
||||
% with those $(\bit(\mathbf{v}_{i^\star}),\mathbf{s}_{i^\star})$
|
||||
%of the $i^\star$-th joining query.
|
||||
%\end{itemize}
|
||||
%In any other situation, $\bdv$ aborts and reports failure. Note that, in the case $coin=2$, $\bdv$ is done since the collision (\ref{collide}) directly provides a
|
||||
%$\mathsf{SIS}$ solution. We thus assume $coin \in \{0,1\}$.
|
||||
If we parse the proof $\pi_K^\star$ as
|
||||
$(\{\mathsf{Comm}_{K,j}^\star\}_{j=1}^t,\mathsf{Chall}_K^\star,\{\mathsf{Resp}_{K,j}^\star \}_{j=1}^t)$, with high
|
||||
probability, the adversary $\adv$ must have invoked the random oracle~$H$ on the
|
||||
input~$ (M^\star, \mathsf{VK}^\star , \mathbf{c}_{\mathbf{v}_i}^\star, \{\mathsf{Comm}_{K,j}^\star\}_{j=1}^t)$.
|
||||
Otherwise, the probability that
|
||||
$\mathsf{Chall}_K^\star=H (M^\star, \mathsf{VK}^\star , \mathbf{c}_{\mathbf{v}_i}^\star, \{\mathsf{Comm}_{K,j}^\star\}_{j=1}^t)$
|
||||
is negligible (at most~$3^{-t}$). It comes that, with probability at least $ \varepsilon' := \varepsilon-
|
||||
3^{-t} $, $ (M^\star, \mathsf{VK}^\star , \mathbf{c}_{\mathbf{v}_i}^\star, \{\mathsf{Comm}_{K,j}^\star\}_{j=1}^t)$
|
||||
coincides with the $\kappa^\star$-th random oracle query for some $\kappa^\star
|
||||
\leq Q_H$. \\
|
||||
\indent
|
||||
At this stage, the reduction $\bdv$ runs the
|
||||
adversary $\adv$ up to $32 \cdot Q_H / (\varepsilon - 3^{-t})$ times with the \textit{same} random tape and input as in the
|
||||
initial run. All queries are answered as previously with
|
||||
one difference in the treatment of random oracle queries.
|
||||
Namely, the first $\kappa^\star-1$ random oracle queries -- which are
|
||||
identical to those of the first execution since $\adv$ is run with the
|
||||
same random tape as before -- receive the same answers
|
||||
$\mathsf{Chall}_1,\ldots,\mathsf{Chall}_{\kappa^\star-1}$ as in the first run. This implies
|
||||
that the $\kappa^\star$-th query will involve exactly the same tuple
|
||||
$ (M^\star, \mathsf{VK}^\star , \mathbf{c}_{\mathbf{v}_i}^\star, \{\mathsf{Comm}_{K,j}^\star\}_{j=1}^t)$
|
||||
as in the first run. However, from the
|
||||
$\kappa^\star$-th query onwards, $\adv$ obtains fresh random oracle
|
||||
values $\mathsf{Chall}_{\kappa^\star}',\ldots,\mathsf{Chall}_{Q_H}'$ at each new execution. The Improved Forking
|
||||
Lemma of Brickell \textit{et al.}~\cite{BPVY00} guarantees that, with probability at least $1/2$, $\bdv$ can obtain a $3$-fork involving the
|
||||
same tuple $ (M^\star, \mathsf{VK}^\star , \mathbf{c}_{\mathbf{v}_i}^\star, \{\mathsf{Comm}_{K,j}^\star\}_{j=1}^t)$ with
|
||||
pairwise distinct answers
|
||||
$\mathsf{Chall}_{\kappa^\star}^{(1)} ,
|
||||
\mathsf{Chall}_{\kappa^\star}^{(2)}, \mathsf{Chall}_{\kappa^\star}^{(3)} \in \{1,2,3\}^t$. With probability $1-(7/9)^t$ it can be shown that there exists an index $j \in \{1,\ldots,t\}$ for which the $j$-th bits
|
||||
of $\mathsf{Chall}_{\kappa^\star}^{(1)} ,
|
||||
\mathsf{Chall}_{\kappa^\star}^{(2)}, \mathsf{Chall}_{\kappa^\star}^{(3)}$ are $ (\mathsf{Chall}_{\kappa^\star,j}^{(1)} ,
|
||||
\mathsf{Chall}_{\kappa^\star,j}^{(2)}, \mathsf{Chall}_{\kappa^\star,j}^{(3)} )=(1,2,3)$. From the corresponding responses $({\mathsf{Resp}_{K,j}^\star}^{(1)},{\mathsf{Resp}_{K,j}^\star}^{(2)},{\mathsf{Resp}_{K,j}^\star}^{(3)})$,
|
||||
$\bdv$ is able to extract witnesses $(\mathbf{d}_{1}^\star,\mathbf{d}_{2}^\star) \in \ZZ^m \times \ZZ^m$, $\mathsf{id}^\star \in \{0,1\}^\ell$ and $\mathbf{w}^\star \in \{0,1\}^m$ from the proof of knowledge $\pi_K^\star$
|
||||
such that
|
||||
\begin{eqnarray*}
|
||||
\mathbf{A}_{\mathsf{id}^\star } \cdot \begin{bmatrix} \mathbf{d}_{1}^\star \\ \hline \mathbf{d}_{2}^\star \end{bmatrix} &=& \mathbf{u} + \mathbf{D} \cdot \mathbf{w}^\star \\
|
||||
\mathbf{w}^\star &=& \bit \bigl( \mathbf{D}_0 \cdot (\bit(\mathbf{v}^\star) + \mathbf{D}_1 \cdot \mathbf{s}^\star \bigr) ,
|
||||
\end{eqnarray*}
|
||||
At this point, $\bdv$ aborts and
|
||||
declares failure in the following situations:
|
||||
|
||||
\begin{itemize}
|
||||
\item[-] $coin=0$ but $\mathsf{id}^\star \in \{0,1\}^\ell$ is recycled from some output of the $\mathcal{Q}_{\ajoin}$ oracle.
|
||||
\item[-] $coin=0$ and $\mathsf{id}^\star \neq \mathsf{id}^\dagger$.
|
||||
\item[-] $coin=1$ but $\mathsf{id}^\star \in \{0,1\}^\ell$ never appeared in a membership certificate returned by the $\mathcal{Q}_{\ajoin}$ oracle.
|
||||
\item[-] $coin=1$ and $\mathsf{id}^\star \in \{0,1\}^{\ell}$ belongs to some user in $U^a$, but this user is not the one introduced at the $i^\star$-th
|
||||
$\mathcal{Q}_{\ajoin}$-query (i.e., $i^\star \neq i^\dagger$ and $\mathsf{id}^\star \neq \mathsf{id}^\dagger$).
|
||||
\item[-] $coin=1$ and the knowledge extractor revealed vectors $\bit(\mathbf{v}^\star) \in \{0,1\}^{2m}$ and $\mathbf{s}^\star \in \ZZ^{2m}$
|
||||
satisfying the collision (\ref{collide}),
|
||||
where $ \bit(\mathbf{v}_{i^\star})$ and $\mathbf{s}_{i^\star}$ are the vectors
|
||||
involved in the $i^\star$-th $\mathcal{Q}_{\ajoin}$ query.
|
||||
\item[-] $coin=2$ and the knowledge extraction yields vectors $\bit(\mathbf{v}^\star) \in \{0,1\}^{2m}$ and $\mathbf{s}^\star \in \ZZ^{2m}$ such that the collision
|
||||
(\ref{collide}) does not occur.
|
||||
\end{itemize}
|
||||
We call $\mathsf{fail}$ the event that one of the above situations occurs. Given that the choices of $coin \sample U(\{0,1,2\})$ and $i^\star \sample U([1,Q_a])$ are completely independent of $\adv$'s view,
|
||||
the choice of $coin$ is correct with probability $1/3$. If $coin=0$, $\bdv$'s choice of $\mathsf{id}^\dagger \sample U(\{0,1\}^\ell)$ is correct with probability $1/(N_{\mathsf{gs}}-Q_a) \geq 1/N_{\mathsf{gs}}$ and, when
|
||||
$coin=1$, $\bdv$'s correctly guesses $i^\star \in [1,Q_a]$ with probability $1/Q_a$. We find
|
||||
$$\Pr[ \neg \mathsf{fail}] \geq \frac{1}{3 \cdot \max(N_{\mathsf{gs}},Q_a)} =\frac{1}{3 \cdot N_{\mathsf{gs}} } .$$
|
||||
|
||||
Assuming that $\mathsf{fail}$ does not occur, $\bdv$ can solve the problem instance as follows. \smallskip
|
||||
|
||||
|
||||
\noindent $\bullet$ If $coin=0$, we have $\mathsf{id}^\star=\mathsf{id}^\dagger$ and $\bdv$ knows a short vector $\mathbf{e}_u \in \ZZ^m$ such that $\mathbf{u} = \bar{\mathbf{A}}_1 \cdot \mathbf{e}_u \bmod q$. Hence, it can obtain a short integer vector
|
||||
\begin{eqnarray*}
|
||||
\mathbf{h} = {\mathbf{d}_1^\star} + \big( \mathbf{Q}_0 + \sum_{i=1}^\ell \mathsf{id}^\dagger [i] \mathbf{Q}_i \big) \cdot {\mathbf{d}_2^\star} - \mathbf{Q}_D
|
||||
\cdot \bit(\mathbf{v}^\star) - \mathbf{e}_u \in \ZZ^m
|
||||
\end{eqnarray*}
|
||||
such that $ \bar{\mathbf{A}}_1 \cdot \mathbf{h} = \mathbf{0}^m \bmod q$. Moreover,
|
||||
we have $\mathbf{h} \neq \mathbf{0}^m$ w.h.p. since the syndrome $\mathbf{u} \in \Zq^n$ statistically hides
|
||||
$\mathbf{e}_u \in \ZZ^m$
|
||||
in $\Lambda_q^{\mathbf{u}}(\bar{\mathbf{A}}_1)$. Finally, the norm of $\mathbf{h}$ is at most $\| \mathbf{h} \|_2 \leq (\ell+1) \sigma^2 m^{3/2} + \sigma m^{1/2} (m+2)$.
|
||||
This implies that $(\mathbf{h}^T \mid \mathbf{0}^m)^T$ is a short non-zero vector of $\Lambda_q^{\perp}(\bar{\mathbf{A}})$ and solves the initial $\mathsf{SIS}$ instance.
|
||||
\smallskip
|
||||
|
||||
|
||||
\smallskip
|
||||
\noindent $\bullet$ If $coin=1$, the extracted
|
||||
witnesses $(\mathbf{d}_{1}^\star,\mathbf{d}_{2}^\star,\mathbf{s}^\star,\mathsf{id}^\star)$ and the decrypted $\bit(\mathbf{v}^\star)$
|
||||
satisfy $\mathsf{id}^\star=\mathsf{id}^\dagger$, $$\mathbf{w}^\star = \bit( \mathbf{D}_0 \cdot \bit(\mathbf{v}^\star) + \mathbf{D}_1 \cdot \mathbf{s}^\star )
|
||||
\neq \bit( \mathbf{D}_0 \cdot \bit(\mathbf{v}_{i^\star}) + \mathbf{D}_1 \cdot \mathbf{s}_{i^\star} ) = \mathbf{w}_{i^\star} $$
|
||||
(since $\neg \mathsf{fail}$ implies that the collision (\ref{collide}) did not occur if $coin=1$)
|
||||
and
|
||||
\begin{align} \label{rel1}
|
||||
\left[
|
||||
\begin{array}{c|c|c|c|c|c}
|
||||
\mathbf{A} ~&~ \mathbf{A}_0 ~&~ \mathbf{A}_1~ &~ \ldots ~ & ~ \mathbf{A}_{\ell} ~&~ -\mathbf{D}
|
||||
\end{array} \right] \cdot
|
||||
\begin{bmatrix}
|
||||
\mathbf{d}_{1}^\star \\ \hline \mathbf{d}_{2}^\star
|
||||
\\ \hline \mathsf{id}^\dagger[1] \mathbf{d}_{2}^\star \\ \hline \vdots \\ \hline ~~ \mathsf{id}^\dagger[\ell] \mathbf{d}_{2}^\star
|
||||
\\ \hline \mathbf{w}^\star
|
||||
\end{bmatrix}
|
||||
= \mathbf{u} \bmod q.
|
||||
\end{align}
|
||||
Since $\bdv$ already knew short vectors $(\mathbf{d}_{i^\star,1},\mathbf{d}_{i^\star,2}, \mathbf{w}_{i^\star}) \in \ZZ^m \times \ZZ^m \times \ZZ^m $ such that
|
||||
\begin{align} \label{rel2}
|
||||
\left[
|
||||
\begin{array}{c|c|c|c|c|c}
|
||||
\mathbf{A} ~&~ \mathbf{A}_0 ~&~ \mathbf{A}_1~ &~ \ldots ~ & ~ \mathbf{A}_{\ell} ~&~ -\mathbf{D}
|
||||
\end{array} \right] \cdot
|
||||
\begin{bmatrix}
|
||||
\mathbf{d}_{i^\star,1}^\star \\ \hline \mathbf{d}_{i^\star,2}^\star
|
||||
\\ \hline \mathsf{id}^\dagger[1] \mathbf{d}_{i^\star,2}^\star \\ \hline \vdots \\ \hline ~~ \mathsf{id}^\dagger[\ell] \mathbf{d}_{i^\star,2}^\star
|
||||
\\ \hline \mathbf{w}_{i^\star}
|
||||
\end{bmatrix}
|
||||
= \mathbf{u} \bmod q,
|
||||
\end{align}
|
||||
by subtracting (\ref{rel2}) from (\ref{rel1}), we find that
|
||||
\begin{align} \label{the-vec}
|
||||
\mathbf{h} &= \mathbf{S} \cdot (\mathbf{d}_1^\star - \mathbf{d}_{i^\star,1}) + (\mathbf{S}_0 + \sum_{j=1}^\ell {\mathsf{id}^\dagger} [j] \mathbf{S}_j ) \cdot (\mathbf{d}_2^\star - \mathbf{d}_{i^\star,2} )
|
||||
\ + ( \mathbf{w}^\star - \mathbf{w}_{i^\star} ) \quad
|
||||
\end{align}
|
||||
is a small-norm vector $\mathbf{h} \in \ZZ^m$ satisfying $ \bar{\mathbf{A}}_1 \cdot \mathbf{h}=\mathbf{0} \bmod q$. We claim that $\mathbf{h} \neq \mathbf{0}$ with high probability.
|
||||
Indeed, we know that $\mathbf{w}^\star \neq \mathbf{w}_{i^\star}$ if $\neg \mathsf{fail}$ occurs.
|
||||
This implies that the last term of (\ref{the-vec}) is non-zero, which rules out that $(\mathbf{d}_1^\star,\mathbf{d}_2^\star)=(\mathbf{d}_{i^\star,1},\mathbf{d}_{i^\star,2})$.
|
||||
Since the columns of $\mathbf{S}$ and $\{\mathbf{S}_j\}_{j=0}^\ell$ have a lot of entropy conditionally on $\mathcal{Y}$, this implies that we can only have $\mathbf{h}=\mathbf{0}^m$ with negligible probability. Furthermore, the norm of $\mathbf{h}$ can be bounded by $\| \mathbf{h} \|_2 \leq 4 \sigma^2 m^{3/2} (\ell+2) + 2 m^{1/2} $,
|
||||
so that $(\mathbf{h}^T \mid \mathbf{0}^m)^T$ solves the original $\mathsf{SIS}$ instance. \medskip
|
||||
|
||||
\noindent $\bullet$ If $coin=2$, $\bdv$ is done as well since the collision (\ref{collide}) directly provides a vector
|
||||
$$\mathbf{h}=\bit(\mathbf{v}^\star) - \bit(\mathbf{v}_i^\star) + \mathbf{Q} \cdot (\mathbf{s}^\star - \mathbf{s}_i^\star) ~ \in \ZZ^{2m}$$ of $\Lambda_q^{\perp}(\mathbf{D}_0)$ (which is also in
|
||||
the lattice $\Lambda_q^{\perp}(\bar{\mathbf{A}})$ by construction) and has
|
||||
norm $\| \mathbf{h} \|_2 \leq 2(\sigma^2 (2m)^{3/2} + (2m)^{1/2}) $. Moreover, $\mathbf{h} \in \ZZ^{2m}$ is non-zero with overwhelming probability
|
||||
given that $\bit(\mathbf{v}^\star) \neq \bit(\mathbf{v}_i^\star)$ and the large amount of entropy retained by the columns $\mathbf{Q} \in \ZZ^{2m \times 2m}$ given $\mathbf{D}_1= \mathbf{D}_0 \cdot \mathbf{Q}$.
|
||||
\end{proof}
|
||||
|
||||
|
||||
\begin{theorem} \label{non-frame}
|
||||
The scheme is secure against framing attacks under the $\mathsf{SIS}_{4n,4m,q,\beta''}$ assumption, where $\beta'' = 4\sigma \sqrt{m}$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{proof}
|
||||
Let us assume that a PPT adversary $\adv$ can create a
|
||||
forgery $(M^\star,\Sigma^\star)$ that opens to some honest user
|
||||
$i\in U^b$ who did not sign $M^\star$. In the random oracle model, we give a reduction $\bdv$ that uses $\adv$ to solve an instance of the ~$\SIS_{4n,4m,q,\beta''}$ problem:
|
||||
$\bdv$ takes as input~$\bar{\mathbf{A}} \in
|
||||
\Zq^{4n \times 4m}$ and finds a non-zero short vector $\mathbf{w} \in
|
||||
\Lambda_q^{\perp}(\bar{\mathbf{A}})$. % with~$0 < \|\mathbf{w}\| \leq \beta$.
|
||||
\\
|
||||
\indent Algorithm $\bdv$ generates the group public key $\mathcal{Y}$ by faithfully running the real setup algorithm with the sole difference that, at step 2 of $\mathsf{Setup}$,
|
||||
$\bdv$ defines $\mathbf{F}=\bar{\mathbf{A}} \in \Zq^{4n \times 4m}$. However, the distribution of $\mathcal{Y}$ is as in the real scheme.
|
||||
As a result of having generated $\mathcal{Y}$ itself, $\bdv$ knows
|
||||
$\mathcal{S}_{\GM}=\mathbf{T}_{\mathbf{A}}$ and $\mathcal{S}_{\OA}= \mathbf{T}_{\mathbf{B}}$. The adversary $\bdv$ is run on input of the
|
||||
group public key
|
||||
$$ \mathcal{Y}:=\Bigl(\mathbf{A}, \{\mathbf{A}_j\}_{j=0}^\ell,~\mathbf{B},~\mathbf{D},~\mathbf{D}_0,~\mathbf{D}_1,~\mathbf{F}=\bar{\mathbf{A}},~\mathbf{u},~\Pi^{\mathsf{OTS}},~H,~H_0 ) \Bigr). $$
|
||||
|
||||
If $\adv$ chooses
|
||||
to corrupt the group manager or the opening authority during the
|
||||
game, $\bdv$ is able to reveal
|
||||
$\mathcal{S}_{\GM}=\mathbf{T}_{\mathbf{A}}$ and
|
||||
$\mathcal{S}_{\OA}= \mathbf{T}_{\mathbf{B}}$. % At the very beginning of the game, $\bdv$ draws a random index $j^\star \sample \{1,\ldots,Q_b\}$ and
|
||||
Then, $\bdv$ starts interacting with $\adv$ as follows.
|
||||
\begin{itemize}
|
||||
\item[-] $Q_{\mathsf{keyGM}}$-queries: If $\adv$ decides to corrupt the group manager, $\bdv$
|
||||
hands the secret key $\mathcal{S}_{\GM}=\mathbf{T}_{\mathbf{A}}$ to $\adv$.
|
||||
\item[-] $Q_{\bjoin}$-queries: At any time $\adv$ can act as a corrupted group manager and introduce a new honest user $i$ in the group by invoking the $Q_{\bjoin}$ oracle.
|
||||
At each $Q_{\bjoin}$-query, $\bdv$ faithfully
|
||||
runs $\mathsf{J}_{\mathsf{user}}$ on behalf of the honest user in an execution of $\mathsf{Join}$ protocol.
|
||||
|
||||
\item[-] $Q_{\mathsf{pub}}$-queries: These
|
||||
can be answered as in the real game, by having the simulator return
|
||||
$\mathcal{Y}$.
|
||||
\item[-] $Q_{\mathsf{sig}}$-queries: When the adversary $\adv$ requests user $ i \in
|
||||
U^b$ to sign a message $M$, $\bdv$ first generates a one-time key pair $(\mathsf{VK},\mathsf{SK}) \leftarrow \mathcal{G}(n)$ to
|
||||
compute $\mathbf{G}_0=H_0(\mathsf{VK}) \in \Zq^{n \times 2m}$. Next,
|
||||
$\bdv$ recalls the vector $\mathbf{z}_i \in \ZZ^{4m}$ that was chosen to define the syndrome $\mathbf{v}_i = \mathbf{F} \cdot \mathbf{z}_i$ at step 1 of the $\mathsf{Join}$ protocol as well as
|
||||
the identifier $\mathsf{id}_i \in \{0,1\}^\ell$ and the short vectors $(\mathbf{d}_{i,1},\mathbf{d}_{i,2},\mathbf{s}_i) $
|
||||
that were supplied by $\adv$ in an earlier $Q_{\bjoin}$-query. It faithfully computes a signature by IBE-encrypting
|
||||
$\bit(\mathbf{v}_i) \in \{0,1\}^{2m}$ and using $(\mathbf{d}_{i,1},\mathbf{d}_{i,2},\mathbf{s}_i,\mathbf{z}_i,\mathbf{s}_i,\mathsf{id}_i)$ to compute a witness indistinguishable proof $\pi_K=(
|
||||
\{\mathsf{Comm}_{K,j}\}_{j=1}^t,\mathsf{Chall}_K,\{\mathsf{Resp}_{K,j}\}_{j=1}^t)$.
|
||||
Finally, $\bdv$ computes a one-time signature
|
||||
$sig=\mathcal{S}(\mathsf{SK},(\mathbf{c}_{\mathbf{v}_i},\pi_K))$ and returns the signature
|
||||
$\Sigma=\big( \mathsf{VK} ,\mathbf{c}_{\mathbf{v}_i}, \pi_K,sig \big)$ to $\adv$.
|
||||
\end{itemize}
|
||||
When $\adv$ halts, it outputs a signature
|
||||
$ \Sigma^\star = \big( \mathsf{VK}^\star ,\mathbf{c}_{\mathbf{v}}^\star, \pi_K^\star,sig^\star \big)$
|
||||
for
|
||||
some message $M^\star$, which opens to ${i^\star} \in
|
||||
U^b$ although user $i^\star$ did not sign the message $M^\star$ at any time. Since $(M^\star,\Sigma^\star)$ supposedly frames user $i^\star$, the opening of
|
||||
$\Sigma^\star$ must reveal the $m$-bit string $\bit(\mathbf{v}_{i^\star}) \in \{0,1\}^m$. We note that the reduction $\bdv$ has
|
||||
recollection of a short vector $\mathbf{z}_{i^\star} \in \ZZ^{4m}$ (of norm $\| \mathbf{z}_{i^\star} \| < 2\sigma \sqrt{m}$)
|
||||
such that $\mathbf{v}_{i^\star} = \mathbf{F} \cdot \mathbf{z}_{i^\star} \bmod q$ which it
|
||||
chose when running $\mathsf{J}_{\mathsf{user}}$ on behalf of user $i^\star$ when this user was introduced in the group. Hence,
|
||||
$\bdv$
|
||||
would be able to solve its given $\mathsf{SIS}$ instance if it had another short vector $\mathbf{z}' \in \ZZ^{4m}$ satisfying $\mathbf{v}_{i^\star} = \mathbf{F} \cdot {\mathbf{z}'} \bmod q $.
|
||||
To compute such a
|
||||
vector, $\bdv$ proceeds by replaying the adversary $\adv$ sufficiently many times and applying the Improved Forking
|
||||
Lemma of Brickell \textit{et al.}~\cite{BPVY00}. \\
|
||||
\indent
|
||||
If we parse $\pi_K^\star$ as
|
||||
$(\{\mathsf{Comm}_{K,j}^\star\}_{j=1}^t,\mathsf{Chall}_K^\star,\{\mathsf{Resp}_{K,j}^\star \}_{j=1}^t)$, with high
|
||||
probability, $\adv$ must have queried~$H$ on the
|
||||
input~$ (M^\star, \mathsf{VK}^\star ,\mathbf{c}_{\mathbf{v}}^\star, \{\mathsf{Comm}_{K,j}^\star\}_{j=1}^t)$.
|
||||
Otherwise, we would only have
|
||||
$\mathsf{Chall}_K^\star=H (M^\star, \mathsf{VK}^\star ,\mathbf{c}_{\mathbf{v}}^\star, \{\mathsf{Comm}_{K,j}^\star\}_{j=1}^t)$
|
||||
with negligible probability~$3^{-t}$. It comes that, with probability at least $ \varepsilon' := \varepsilon-
|
||||
3^{-t} $, the tuple $ (M^\star, \mathsf{VK}^\star ,\mathbf{c}_{\mathbf{v}}^\star, \{\mathsf{Comm}_{K,j}^\star\}_{j=1}^t)$
|
||||
was the input of the $\kappa^\star$-th random oracle query for some index $\kappa^\star
|
||||
\leq Q_H$. \\
|
||||
\indent
|
||||
At this point, the reduction $\bdv$ runs the
|
||||
adversary $\adv$ up to $32 \cdot Q_H / (\varepsilon - 3^{-t})$ times with the \textit{same} random tape and input as in the
|
||||
first run. All queries are answered as previously with
|
||||
one difference in the way to handle $H$-queries.
|
||||
Namely, the first $\kappa^\star-1$ $H$-queries -- which are
|
||||
the same as in the first execution since $\adv$ is run with the
|
||||
same random tape -- obtain the same answers
|
||||
$\mathsf{Chall}_1,\ldots,\mathsf{Chall}_{\kappa^\star-1}$ as in the original run. This implies
|
||||
that the $\kappa^\star$-th query will also involve exactly the same tuple
|
||||
$ (M^\star, \mathsf{VK}^\star ,\mathbf{c}_{\mathbf{v}}^\star, \{\mathsf{Comm}_{K,j}^\star\}_{j=1}^t)$
|
||||
as in the original run. From the
|
||||
$\kappa^\star$-th query forward, however, the adversary $\adv$ obtains fresh random oracle
|
||||
outputs $\mathsf{Chall}_{\kappa^\star}',\ldots,\mathsf{Chall}_{Q_H}'$ at each new execution. The Improved Forking
|
||||
Lemma of~\cite{BPVY00} ensures that, with probability $>1/2$, $\bdv$ obtains a $3$-fork involving the
|
||||
tuple $ (M^\star, \mathsf{VK}^\star ,\mathbf{c}_{\mathbf{v}}^\star, \{\mathsf{Comm}_{K,j}^\star\}_{j=1}^t)$ of the initial run and with
|
||||
pairwise distinct answers
|
||||
$\mathsf{Chall}_{\kappa^\star}^{(1)} ,
|
||||
\mathsf{Chall}_{\kappa^\star}^{(2)}, \mathsf{Chall}_{\kappa^\star}^{(3)} \in \{1,2,3\}^t$. Since the forgeries of the $3$-fork all correspond to the tuple $ (M^\star, \mathsf{VK}^\star ,
|
||||
\mathbf{c}_{\mathbf{v}}^\star, \{\mathsf{Comm}_{K,j}^\star\}_{j=1}^t)$, they open to the same $m$-bit string $\bit(\mathbf{v}_{i^\star}) \in \{0,1\}^m$ and
|
||||
which is uniquely determined
|
||||
by $\mathbf{c}_{\mathbf{v}}^\star$. In turn, this implies that the three forgeries all reveal the same $\bit(\mathbf{v}_{i^\star})$
|
||||
at the second step of $\mathsf{Open}$.
|
||||
With probability $1-(7/9)^t$ it can be shown that there exists $j \in \{1,\ldots,t\}$ such that the $j$-th bits
|
||||
of $\mathsf{Chall}_{\kappa^\star}^{(1)} ,
|
||||
\mathsf{Chall}_{\kappa^\star }^{(2)}, \mathsf{Chall}_{\kappa^\star }^{(3)}$ are $ (\mathsf{Chall}_{\kappa^\star,j}^{(1)} ,
|
||||
\mathsf{Chall}_{\kappa^\star,j}^{(2)}, \mathsf{Chall}_{\kappa^\star,j}^{(3)} )=(1,2,3)$. From the corresponding responses $({\mathsf{Resp}_{K,j}^\star}^{(1)},{\mathsf{Resp}_{K,j}^\star}^{(2)},{\mathsf{Resp}_{K,j}^\star}^{(3)})$,
|
||||
$\bdv$ is able to extract a short vector $ \mathbf{z}' \in \ZZ^{4m} $ such that $\mathbf{v}_{i^\star} = \mathbf{F} \cdot {\mathbf{z}'} \bmod q $. \\ \indent Due to the statistical witness indistinguishability of
|
||||
the Stern-like proof of knowledge which is used to generate signature, with overwhelming
|
||||
probability, we have $\mathbf{z}' \neq \mathbf{z}_{i^\star}$. Indeed, from the adversary's view, the distribution of
|
||||
$\mathbf{z}_{i^\star}$ is $D_{\Lambda_q^{\mathbf{v}_{i^\star}}(\mathbf{F}),\sigma}$, which means that it has at least $n$ bits of min-entropy.
|
||||
Hence, the difference $\mathbf{h} = \mathbf{z}' - \mathbf{z}_{i^\star} \in \ZZ^{4m}$ is a suitably short non-zero vector of $ \Lambda_q^{\perp}( \bar{\mathbf{A}} ) $.
|
||||
\end{proof}
|
||||
|
||||
\begin{theorem} \label{anonymity-thm}
|
||||
In the random oracle model, the scheme provides \textsf{CCA}-anonymity if
|
||||
the $\LWE_{n,q,\chi}$ assumption holds and if $\Pi^\mathrm{OTS}$ is a strongly unforgeable one-time signature.
|
||||
\end{theorem}
|
||||
|
||||
\begin{proof}
|
||||
We proceed as in~\cite{LNW15} and prove the result via a sequence of games which are computationally indistinguishable.
|
||||
The first game consists of the real anonymity experiment which is parameterized by a bit $d \in \{0,1\}$ that determines the challenger's choice in the challenge phase.
|
||||
The last game is the same regardless of whether $d=0$ or $d=1$. It follows that, under the stated assumptions, no PPT adversary can distinguish $\Expt^\textrm{anon$-0$}_\adv$ from $\Expt^\textrm{anon$-1$}_\adv$ with noticeable advantage.
|
||||
\medskip
|
||||
|
||||
\begin{description}
|
||||
\item[$\textsf{Game}^{(d)}$~0:] This is the real anonymity experiment $\Expt^\textrm{anon$-d$}_\adv(\lambda)$ as described in Definition~\ref{def:anon}.
|
||||
More precisely, the challenger starts by running the algorithm $\mathsf{Setup}(1^\lambda, 1^{\Ngs})$ to obtain $(\gspk, \mathcal{S}_\GM = \mathbf{T_A} \in \ZZ^{m \times m}, \mathcal{S}_\OA = \mathbf{T_B} \in \ZZ^{m \times m})$ along with state information $St$. The challenger next hands the public parameters $\gspk$ and the group manager key $\mathcal{S}_\GM$ to the adversary $\adv$.
|
||||
On the following adversary signature opening queries on signatures $\Sigma = (\vk, \mathbf{c}_{\mathbf{v}_d}, \pi_K, sig)$, the challenger uses the opening authority key $\mathbf{T_A} \in \ZZ^{m \times m}$ he possesses to decrypt the GPV encryption of the signer identity $\mathbf{c}_{\mathbf{v}_d} \in \Zq^m \times \Zq^{2m}$.
|
||||
At some point, the adversary $\adv$ requests a challenge by outputting a target message $M^\star \in \bit^*$ and two user key pairs
|
||||
\[ \bigl(\scr_i^\star = \mathbf{z}^\star_i \in \ZZ^{4m}, \crt_i^\star \in (\mathsf{id}^\star_i, \mathbf{d}^\star_i, \mathbf{s}^\star_i) \in \bit^\ell \times \ZZ^{2m} \times \ZZ^{2m} \bigr)_{i \in \bit} \]
|
||||
which must be valid and distinct (otherwise, the challenger aborts the experiment).
|
||||
This challenge query is answered by having the challenger return a signature of the target message under the identity $id_d$: namely, this challenge signature is computed as $\Sigma^\star = (\vk^\star, \mathbf{c}_{\mathbf{v}_d}^\star, \pi_K^\star, sig^\star) \gets \Sign(\mathcal{Y}, \crt_d^\star, \scr_d^\star, M^\star)$ for the given parameter $d$
|
||||
of the \textsf{Game}.
|
||||
Finally, $\adv$ outputs a bit $d' \in \bit$ which is also the experiment's output. % and the experiment outputs $1$ if $b = b'$ or $0$ otherwise. By assumption, $\adv$ has advantage $\varepsilon$ in this game.
|
||||
\smallskip
|
||||
|
||||
\item[$\textsf{Game}^{(d)}$~1:] In this experiment, we slightly change $\mathsf{Game}^{(d)}~0$ as follows. At the outset of the game, the challenger generates the one-time signature key pair $(\vk^\star, \sk^\star)$ that will be used in the challenge phase.
|
||||
During the game, if the adversary $\adv$ requests the opening of a valid signature $\Sigma = (\vk, \mathbf{c}_{\mathbf{v}_i}, \pi_K, sig)$ where $\vk = \vk^\star$, the challenger returns a random bit and aborts.
|
||||
However, this event $F_1$ would contradict the strong unforgeability of the one-time signature $\Pi^{\mathrm{OTS}}$.
|
||||
Indeed, before the challenge phase $\vk^\star$ is independent of $\adv$'s view and the probability that $\vk^\star$ shows up in $\adv$'s queries is negligible.
|
||||
After seeing the challenge signature $\Sigma^\star$, if $\adv$ comes up with a valid signature $\Sigma = (\vk, \mathbf{c}_{\mathbf{v}_i}, \pi_K, sig)$ such that $\vk = \vk^\star$, then $sig$ is a forged one-time signature, which defeats the strong unforgeability of $\Pi^{\mathrm{OTS}}$.
|
||||
Therefore the probability $\Pr[F_1]$ that the challenger aborts in this experiment is negligible.
|
||||
From here on, we thus assume that $\adv$'s opening queries for valid signatures do not include $\vk^\star$.
|
||||
\smallskip
|
||||
|
||||
\item[$\textsf{Game}^{(d)}$~2:] In this game, we program the random oracle $H_0$ in the following way: at the beginning of the game, we choose
|
||||
a uniformly random matrix $\mathbf{G}_0^\star \sample U(\Zq^{n \times 2m})$ and set $H_0(\vk^\star) = \mathbf{G}^\star_0$. From the adversary's view, the distribution of
|
||||
$\mathbf{G}_0^\star$ is statistically close to the one in the real attack game, as in \cite{GPV08}.
|
||||
As for other queries, for each fresh $H_0$-queries on $\vk$,
|
||||
the challenger samples small-norm matrices $\mathbf{E}_{0,\vk} \sample D_{\ZZ^m, \sigma}^{2m}$ and programs the oracle such that
|
||||
$H_0(\vk) = \mathbf{B} \cdot \mathbf{E}_{0,\vk} \bmod q$. The chosen matrices $\mathbf{E}_{0,\vk}$
|
||||
are retained for later use.
|
||||
Note that the values of $H_0(\vk)$ are statistically close to the uniform.
|
||||
For any query involving a previously queried $\vk$, the challenger consistently returns the previously stored images.
|
||||
The adversary's view remains the same as in $\mathsf{Game}^{(d)}~1$, analogously to the security proof of the GPV IBE~\cite{GPV08}.
|
||||
\smallskip
|
||||
|
||||
\item[$\textsf{Game}^{(d)}$~3:] Here, we will change the behaviour of the opening algorithm.
|
||||
Namely, at each fresh oracle query, we still store the matrices $\mathbf{E}_{0,\vk} \in \Zq^{m \times 2m}$ and, at the beginning of the game, the challenger
|
||||
samples an uniformly random $\mathbf{B^\star} \in \Zq^{n \times m}$ that is later used in place of $\mathbf{B}$ to answer $H_0$-queries.
|
||||
To answer the adversary's queries of the opening of a signature
|
||||
$\Sigma = (\vk, \mathbf{c}_{\mathbf{v}_i}, \ \pi_K, sig)$,
|
||||
the challenger recalls the small-norm matrices $\mathbf{E}_{0,\vk}$ which were defined when $\adv$ first queried $H_0(\vk)$.
|
||||
These matrices are used as ``decryption matrices'' to open $\Sigma$ for the corresponding $\mathbf{G}_0 = H_0(\vk) \in \Zq^{n \times 2m}$.
|
||||
For similar reasons as in the security proof of~\cite{GPV08}, the distribution of $\mathbf{G}_0$ is statistically close to the uniform,
|
||||
which implies that $\mathsf{Game}^{(d)}~2$ and $\mathsf{Game}^{(d)}~3$ are statistically indistinguishable.
|
||||
\smallskip
|
||||
|
||||
|
||||
\item[$\textsf{Game}^{(d)}$~4:] Instead of faithfully generating the
|
||||
NIZKPoK $\pi_K$ of Section~\ref{subsection:zk-for-group-signature}, the challenger simulates the proof without using the witness (note that this is possible since the HVZK property of the underlying proof system is preserved
|
||||
under parallel repetitions). This
|
||||
is done by running the simulator for the underlying interactive protocol for
|
||||
each $j \in \{1,\ldots, t\}$, and then programming the random oracle $H$
|
||||
accordingly. The challenge signature
|
||||
$\Sigma^\star = (\vk^\star, \mathbf{c}_{\mathbf{v}_d}^\star , \pi_K^\star, sig^\star)$
|
||||
is statistically close to the challenge signature of the previous game, because the
|
||||
proof system is statistically zero-knowledge as stated in Lemma~\ref{le:zk-ktx}.
|
||||
Consequently, $\mathsf{Game}^{(d)}~3$ and $\mathsf{Game}^{(d)}~4$ are indistinguishable.
|
||||
\smallskip
|
||||
|
||||
\item[$\textsf{Game}^{(d)}$~5:] In this game, we modify the generation of the challenge ciphertext $\mathbf{c}_{\mathbf{v}_d}^\star$.
|
||||
Instead of using the real encryption algorithm of the GPV IBE to compute $\mathbf{c}_{\mathbf{v}_d}^\star$ as the encryption of $\mathbf{v}_d^\star = \mathbf{F} \cdot \mathbf{z}_d \in \Zq^{4n}$, we return truly random
|
||||
ciphertexts. In other words, we let
|
||||
\[ \mathbf{c}_{\mathbf{v}_d}^\star = \begin{pmatrix}
|
||||
\mathbf{r}_1 \\ \mathbf{r}_2 + \bit(\mathbf{v}_{d}^\star) \lfloor q/2 \rfloor
|
||||
\end{pmatrix}, \]
|
||||
%where $\mathbf{v}_{i_b}^\star= \mathbf{F} \cdot \mathbf{z}_{i_b}^\star $, and
|
||||
where $\mathbf{r}_1 \sample U(\Zq^{m})$, $\mathbf{r}_2 \sample U(\Zq^{2m})$ are uniformly random.
|
||||
The hardness of the decisional $\LWE_{n, q, \chi}$ problem implies that $\mathbf{c}^\star_{\mathbf{v}_d}$ in \ extsf{Game} $4$ and \ extsf{Game} $5$ are computationally indistinguishable.
|
||||
If $\adv$ can distinguish between these two games, it can furthermore distinguish
|
||||
\[ \begin{pmatrix}
|
||||
\mathbf{B}^T \\ \hline {\mathbf{G}_0^\star }^T
|
||||
\end{pmatrix} \mathbf{e}_0 + \begin{pmatrix} \mathbf{x}_1 \\\hline \mathbf{x}_2 \end{pmatrix} \mbox{ from } \begin{pmatrix}
|
||||
\mathbf{r}_1 \\ \hline \mathbf{r}_2
|
||||
\end{pmatrix},\]
|
||||
which would break the decisional $\LWE_{n,q,\chi}$ assumption.
|
||||
|
||||
Therefore, $\mathsf{Game}^{(d)}~4$ and $\mathsf{Game}^{(d)}~5$ are computationally indistinguishable.
|
||||
\smallskip
|
||||
|
||||
\item[\textsf{Game}~6:] We finally make a conceptual modification on the previous game. Namely we sample uniformly random $\mathbf{r}_1^\prime
|
||||
\sample U(\Zq^{m})$, $\mathbf{r}_2^\prime \sample U(\Zq^{2m})$ and assign
|
||||
\[ \mathbf{c}_{\mathbf{v}_d}^\star = \begin{pmatrix}
|
||||
\mathbf{r}_1^\prime \\ \mathbf{r}_2^\prime
|
||||
\end{pmatrix} .\]
|
||||
\end{description}
|
||||
|
||||
Clearly, the distribution of $\mathbf{c}_{\mathbf{v}_i}^\star $ has not changed since $\mathsf{Game}^{(d)}~5$. Since \textsf{Game} $6$ does no longer depend on the
|
||||
challenger's bit $d\in \{0,1\}$, the result follows.
|
||||
\end{proof}
|
||||
|
||||
\section{Subprotocols for Stern-like Argument}
|
||||
\addcontentsline{tof}{section}{\protect\numberline{\thesection} Protocoles pour les preuves à la Stern}
|
||||
\label{se:gs-lwe-stern}
|
||||
@ -1092,5 +1887,47 @@ as the permutation that transforms $\mathbf{z}$ as follows:
|
||||
\end{itemize}
|
||||
It can be check that~(\ref{eq:zk-equivalence}) holds. Therefore, we can obtain a statistical \textsf{ZKAoK} for the given relation by running the protocol in \cref{sse:stern-abstraction}.
|
||||
|
||||
\section{A Dynamic Lattice-Based Group Signature}
|
||||
\input{merge}
|
||||
\subsection{The Underlying ZKAoK for the Group Signature Scheme}\label{subsection:zk-for-group-signature}
|
||||
The argument system upon which our group signature scheme is built can be summarized as follows.
|
||||
\begin{description}
|
||||
\item[Common Input:] Matrices $\mathbf{A}, \{\mathbf{A}_j\}_{j=0}^\ell, \mathbf{B} \in \mathbb{Z}_q^{n \times m}$, $\mathbf{D}_0, \mathbf{D}_1 \in \mathbb{Z}_q^{2n \times 2m}$, $\mathbf{F} \in \mathbb{Z}_q^{4n \times 4m}$, $\mathbf{H}_{2n \times m} \hspace*{-1.5pt}\in\hspace*{-1.5pt} \ZZ_q^{2n \times m}$, $\mathbf{H}_{4n \times 2m} \hspace*{-1.5pt}\in\hspace*{-1.5pt} \ZZ_q^{4n \times 2m}$, $\mathbf{G}_0 \hspace*{-1.5pt}\in\hspace*{-1.5pt} \mathbb{Z}_q^{n \times 2m}$; vectors $\mathbf{u} \hspace*{-1.5pt}\in\hspace*{-1.5pt} \mathbb{Z}_q^n$, $\mathbf{c}_1 \hspace*{-1.5pt}\in\hspace*{-1.5pt} \mathbb{Z}_q^m$, $\mathbf{c}_2 \hspace*{-1.5pt}\in \hspace*{-1.5pt}\mathbb{Z}_q^{2m}$. \smallskip
|
||||
\item [Prover's Input:] $\mathbf{z} \in [-\beta,\beta]^{4m}$, $\mathbf{y} \in \{0,1\}^{2m}$, $\mathbf{w} \in \{0,1\}^m$, $\mathbf{d}_1, \mathbf{d}_2 \in [-\beta, \beta]^m$, $\mathbf{s} \in [-\beta,\beta]^{2m}$, $\mathrm{id} = (\mathrm{id}[1], \ldots, \mathrm{id}[\ell])^T \in \{0,1\}^\ell$,
|
||||
|
||||
$\mathbf{e}_0 \in [-B,B]^n$, $\mathbf{e}_1 \in [-B,B]^m$, $\mathbf{e}_2 \in [-B,B]^{2m}$. \smallskip
|
||||
\item[Prover's Goal:] Convince the verifier in \textsf{ZK} that
|
||||
\end{description}
|
||||
\[
|
||||
\begin{cases}
|
||||
\mathbf{F}\cdot \mathbf{z} = \mathbf{H}_{4n\times 2m}\cdot \mathbf{y}\bmod q; \hspace*{5pt} \mathbf{H}_{2n \times m}\cdot \mathbf{w} = \mathbf{D}_0 \cdot \mathbf{y} + \mathbf{D}_1 \cdot \mathbf{s} \bmod q; \\
|
||||
\mathbf{A}\cdot \mathbf{d}_1 + \mathbf{A}_0 \cdot \mathbf{d}_2 + \sum_{j=1}^\ell \mathbf{A}_j \cdot (\mathrm{id}[j]\cdot \mathbf{d}_2) - \mathbf{D} \cdot \mathbf{w} = \mathbf{u} \bmod q;\\
|
||||
\mathbf{c}_1 = \mathbf{B}^T\cdot \mathbf{e}_0 + \mathbf{e}_1 \bmod q; \hspace*{5pt} \mathbf{c}_2 = \mathbf{G}_0^T\cdot \mathbf{e}_0 + \mathbf{e}_2 + \lfloor q/2\rfloor\cdot \mathbf{y} \bmod q.
|
||||
\end{cases}
|
||||
\]
|
||||
|
||||
Using the same strategy as in Sections~\ref{subsection:zk-for-commitments} and~\ref{subsection:zk-for-signature}, we can derive a statistical \textsf{ZKAoK} for the above relation from the protocol in Section~\ref{sse:stern-abstraction}. As the transformations are similar to those in Section~\ref{subsection:zk-for-signature}, we only sketch main points.
|
||||
|
||||
In the first step, we combine the given equations to an equation of the form:
|
||||
\[\vspace*{-3.5pt}
|
||||
\mathbf{M}\cdot \left(
|
||||
\begin{array}{c}
|
||||
\mathbf{d}_1 \\
|
||||
\mathbf{s} \\
|
||||
\mathbf{z} \\
|
||||
\end{array}
|
||||
\right) + \mathbf{M}_0 \cdot \mathbf{d}_2 + \sum_{j=1}^\ell \mathbf{M}_j(\mathrm{id}[j]\mathbf{d}_2) + \mathbf{M}' \cdot \left(
|
||||
\begin{array}{c}
|
||||
\mathbf{w} \\
|
||||
\mathbf{y} \\
|
||||
\end{array}
|
||||
\right) + \mathbf{M}'' \cdot \left(
|
||||
\begin{array}{c}
|
||||
\mathbf{e}_0 \\
|
||||
\mathbf{e}_1 \\
|
||||
\mathbf{e}_2 \\
|
||||
\end{array}
|
||||
\right) = \mathbf{v} \bmod q,
|
||||
\]
|
||||
where matrices $\mathbf{M}, \mathbf{M}_0, \ldots, \mathbf{M}_\ell, \mathbf{M}', \mathbf{M}''$ and vector $\mathbf{v}$ are built from the input.
|
||||
|
||||
We then apply the techniques of \cref{sse:stern-abstraction} for %the vectors
|
||||
$\mathbf{x}_0 = (\mathbf{d}_1^T \| \mathbf{s}^T \| \mathbf{z}^T)^T \in [-\beta, \beta]^{7m}$, $\mathbf{d}_2 \in [-\beta,\beta]^m$; $\mathbf{x}_1 = (\mathbf{w}^T \| \mathbf{y}^T)^T\in \{0,1\}^{3m}$; and $\mathbf{x}_2 = (\mathbf{e}_0^T \| \mathbf{e}_1^T \| \mathbf{e}_2^T)^T \in [-B,B]^{n + 3m}$. This allows us to obtain a unified equation $\mathbf{P}\cdot \mathbf{x} = \mathbf{v} \bmod q$, and to define the sets $\mathsf{VALID}$, $\mathcal{S}$, and permutations $\{T_\pi: \pi \in \mathcal{S}\}$ so that the conditions in~(\ref{eq:zk-equivalence}) hold, in a similar manner as in Section~\ref{subsection:zk-for-signature}.
|
||||
|
||||
@ -24,6 +24,7 @@
|
||||
\newcommand{\Keygen}{\ensuremath{\mathsf{Keygen}}\xspace}
|
||||
\newcommand{\param}{\ensuremath{\mathsf{par}}\xspace}
|
||||
\newcommand{\pk}{\ensuremath{\mathsf{pk}}\xspace}
|
||||
\newcommand{\vk}{\ensuremath{\mathsf{vk}}\xspace}
|
||||
\newcommand{\sk}{\ensuremath{\mathsf{sk}}\xspace}
|
||||
%% ZK
|
||||
\newcommand{\trans}{\textsf{trans}\xspace}
|
||||
|
||||
15
these.bib
15
these.bib
@ -84,17 +84,6 @@
|
||||
year = {2015},
|
||||
}
|
||||
|
||||
@InCollection{SSE+12,
|
||||
author = {Sakai, Y. and Schuldt, J. and Emura, K. and Hanaoka, G. and Ohta, K.},
|
||||
title = {On the Security of Dynamic Group Signatures: Preventing Signature Hijacking},
|
||||
booktitle = {{PKC}},
|
||||
publisher = {Springer},
|
||||
year = {2012},
|
||||
volume = {7293},
|
||||
series = {LNCS},
|
||||
pages = {715--732},
|
||||
}
|
||||
|
||||
@InProceedings{ACDN13,
|
||||
author = {Abe, Masayuki and Camenisch, Jan and Dubovitskaya, Maria and Nishimaki, Ryo},
|
||||
title = {Universally composable adaptive oblivious transfer (with access control) from standard assumptions},
|
||||
@ -1130,7 +1119,7 @@
|
||||
pages = {457--473},
|
||||
}
|
||||
|
||||
@InProceedings{SSE+12a,
|
||||
@InProceedings{SSE+12,
|
||||
author = {Sakai, Y. and Schuldt, J. and Emura, K. and Hanaoka, G. and Ohta, K.},
|
||||
title = {On the Security of Dynamic Group Signatures: Preventing Signature Hijacking},
|
||||
booktitle = {PKC},
|
||||
@ -2818,7 +2807,7 @@
|
||||
booktitle = {Asiacrypt},
|
||||
year = {2017},
|
||||
series = {LNCS},
|
||||
pages = {347--374},
|
||||
pages = {347--374o},
|
||||
publisher = {Springer},
|
||||
}
|
||||
|
||||
|
||||
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Reference in New Issue
Block a user