In this Chapter, we aim at lifting the \textit{signature with efficient protocols} from~\cite{LPY15} into the random oracle model in order to get an efficient construction~\cite{BR93}.
Signatures with efficient protocols in the Camenish and Lysyanskaya fashion~\cite{CL04a} are digital signatures which come with companion zero-knowledge proofs that allow a signature holder to prove knowledge of the signature of a commited message as well as proving possession of a hidden message-signature pair in a zero-knowledge manner.
This building block proved useful in the design of many efficient anonymity-related protocols such as anonymous credentials~\cite{CL01}, which are similar to group signatures except that anonymity is irrevocable (meaning that there is no opening authority).
As explained in \cref{ch:proofs}, the \textit{quality} of a scheme depends on both its efficiency and the simplicity of the assumptions it relies on.
Before the works described in this Chapter, most signature schemes rely on groups of hidden order~\cite{CL04a} or non-standard assumptions in groups with bilinear maps~\cite{CL04, Oka06}.
To illustrate this multi-criteria quality evaluation, we can see that Camenisch and Lysyanskaya proposed a signature scheme that is secure in pairing-friendly groups but relies on the non-interactive LRSW assumption~\cite{LRSW99}; but this signature scheme requires $\mathcal{O}(n)$ group elements to encode an $\ell$-block message.
Pointcheval and Sanders improved this signature to go down to $\bigO(1)$ group elements for an $\ell$-block message, but which is only proven secure in the generic group model (a model where group accesses are handled by an oracle that performs the group operations).
We note that beside the scheme presented in this section, we are only aware of two schemes based on fixed-size assumptions: (1) a variant of the Camenisch and Lysyanskaya scheme~\cite{CL04} due to Gerbush, Lewko, O'Neill and Waters~\cite{GLOW12} in composite order groups.
Due to this assumption, the groups that are used are inherently bigger and leads to less efficient representations than in prime order groups: for equivalent security level, Freeman~\cite{Fre10} estimates that computing a pairing over a group $N = pq$ is at least $50$ times slower than the same pairing in the prime order group setting.
(2) A construction by Yuen, Chow, Zhang and Yu~\cite{YCZY14} under the decision linear assumption~\cite{BBS04} which unfortunately does not support ``randomizable signature'', which is an important property in privacy-enhancing cryptography. An application of this property is, in the context of group signatures, the re-randomization of credentials accross distinct privacy-preserving authentication.
This signature length is made possible by using efficient $\QANIZK$ arguments -- as presented in~\cref{sse:zk-nizk} and formally defined in~\cite{JR13} -- to prove the belonging to some linear subspace spanned by the rows of a matrix.
For this purpose, it was shown that for this specific task, the size of the argument may be independent of the dimension of the considered subspace~\cite{JR14,LPJY14,KW15}.
The signature scheme described in this chapter (\cref{scal-sig}) crucially takes advantage of this observation as $\ell$-block messages are certified using a $\QANIZK$ argument for a subspace of dimension $\bigO(\ell)$.
This construction natively supports efficient protocols to enhance privacy as described in \cref{new-proto}. Hence, our signature scheme enables the design of an efficient anonymous credentials system based on the sole \SXDH assumption.
As another showcase for this signature, we also design another primitives.
Namely, a dynamic group signature scheme, as described in \cref{ch:gs-background}, which is practical and relies on simple assumptions (namely \SXDH and \SDL).
This construction is competitive both in term of signature size and computation time with the best solutions based on non-interactive assumptions~\cite{BBS04,DP06} (in these cases, the Strong Diffie-Hellman assumption~\cite{BB04}).
Concretely, at the 128-bits security, each signature fits within $320$ bytes while providing the strongest sense of anonymity (meaning the definition in \cref{sec:RGSdefsecAnon}).
In this Chapter, we will first recall the useful building blocks that are used to design and prove our signature scheme that supports efficient protocols in the~\cite{CL02a} fashion. Then we describe this scheme and we next give the construction and the proof for the group signature scheme for dynamically growing groups. Finally, we show the experimental results we obtain for this group signature scheme.
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.
In \cite{LPY15}, Libert \textit{et al.} described an F-unforgeable signature%
\footnote{In F-unforgeability, the adversary only has to output a forgery for a message $M$ without outputting the message, but the image $F(M)$ for an injective function $F$ that is not necessarily efficiently invertible instead~\cite{BCKL08}. In~\cite{LPY15}, the function $F$ is $M \mapsto\hat{g}^M$.}
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 \QANIZK 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.
The signature on $\ell$ scalars thus only consists of $4$ elements in $\GG$
while the verification equation only involves a computation of $5$ pairings\footnote{Actually only $4$ pairing computations are necessary, as $e(\Omega, \hat{g}_{2\ell+4})$ is independent of the inputs $\pi$ and $\vec{m}$, and can hence be precomputed.}.
The above signature scheme is existentially unforgeable under chosen-message attacks (\textsf{eu-cma}) if the $\SXDH$ assumption holds in $(\GG, \Gh, \GT)$.
\item[Type B signatures:] These use a random value $\omega' \in_R \Zp$ instead of the secret key $\omega$. We pick random $\omega', s, s_1\sample\U(\Zp)$ and
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 $\omega\in\Zp$.
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
\{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
In this section, we give $\Sigma$-protocols (\cref{sse:sigma-protocols}) for issuing a signature on a committed multi-block message and for proving knowledge of a valid message-signature pair.
We give $\Sigma$-protocols for proving the knowledge of a signature-message pair $({\sigma},\vec{m})$ satisfying the verification equation~\eqref{sig-ver-1} of the scheme of Section~\ref{scal-sig}
where ${\sigma}=(\sigma_1,\sigma_2,\sigma_3,\pi)$ and $\vec{m}=(m_1,\ldots,m_\ell)$.
We note that, as shown in the proof of Theorem \ref{th:eu-cma-1}, a candidate signature $(\sigma_1,\sigma_2,\sigma_3,\pi)$ may satisfy the verification equation
although $\log_g(\sigma_2)\neq\log_h(\sigma_3)$. In applications to anonymous credentials, a malicious credential issuer could take advantage of this fact in attempts to
break the anonymity of the scheme (e.g., by linking two authentications involving the same credential). For this reason, we consider a protocol for proving possession
We thus consider the case of arbitrary valid signatures that may have been maliciously computed by a signer who, {e.g.}, aims at tracing provers across different authentications. In this setting, we can still obtain a perfect SHVZK $\Sigma$-protocol to hedge against such attacks.
A first attempt to efficiently build such a protocol is to ``linearize'' the verification equation (\ref{eq-mult-sig}) by making sure that two witnesses are never paired together. However, we will still have to deal with (parallelizable) intermediate $\Sigma$-protocols for quadratic scalar relations.
Even though a quadratic pairing-product equation $e(x_1,\hat{a})\cdot e(x_2,\hat{y})$ -- for variables $x_1,x_2,\hat{y}$ and constant $\hat{a}$ -- can be linearized by partially randomizing the variables so as to get the equation $e(x_1\cdot x_2^{r},\hat{a})\cdot e(x_2,\hat{y}\cdot\hat{a}^{-r})$ (which allows $\hat{y}'=\hat{y}\cdot\hat{a}^{-r}$ to appear in the
clear), proving knowledge of a valid signature still requires proving a statement about some representation of $\hat{y}$ which now appears in committed form. Somehow, going through the randomizing factor $\hat{a}^{-r}$ involves a quadratic relation between some known exponents to get special-soundness. To ease the entire proof we rather directly commit to the variables in $\GG$ and $\hat{\GG}$ using their available generator $g$ and $\hat{g}$ which are not among the constants of the verification equation of the signature. We additionally need an extra generator $f$ of $\GG$ whose discrete logarithm is unknown.
$r_1,r_2\sample\U(\Zp)$ and compute $\hat{D}_1=d_1\cdot\hat{g}^{r_1}$ and $\hat{D}_2=d_2\cdot\hat{g}^{r_2}$.
\item In order to prove knowledge of an opening of commitments $\hat{D}_1,\hat{D}_2\in\Gh$ to the same message $\vec{m}=(m_1,\ldots,m_\ell)\in\Zp^\ell$,
\item[\textsf{Challenge}] Given $\mathsf{com}$ as per (\ref{eq-comm-2}), pick $\rho\sample\U(\Zp)$ uniformly at random and return $\mathsf{chall}=\rho$.
\item[\textsf{Response}] On inputs $\mathsf{com}$, $\mathsf{aux}$ and $\mathsf{chall}=\rho$, compute: % the following elements over $\Zp$:
\item[\textsf{Verify}] Given $(\mathsf{com};\mathsf{chall};\mathsf{resp})$ return $0$ if it does not parse correctly or if the following relations do not hold:
It is worth noticing that no pairing evaluation is required until the final step of $\mathsf{Verify}$, which is almost as efficient as the verification of
Moreover, the prover's first message $\mathsf{com}$ is of constant-size and the communication complexity of the protocol exceeds the length of the witness by
Moreover, together with previously extracted $(r_1,r_2)$, step 2 of $\mathsf{Verify}$ also guarantees that $t_4$ satisfies $C_0=C_2^{r_1}\cdot C_3^{r_2}\cdot g^{t_4}$.
$\sigma_0\cdot g^{t_0}=\sigma_2^{r_1}\cdot\sigma_3^{r_2}\cdot g^{t_2\cdot r_1+t_3\cdot r_2+t_4}$. Hence, we are done if we can show that $t_0=t_2r_1+t_3r_2+t_4$. But this exactly what step 3 of $\mathsf{Verify}$ and the
special soundness of the sub-protocol involving $(T_0,T_2,T_3,T_4)$ tells us. First, we have a representation of these
$T_i$'s {w.r.t.} the basis $(g,f)\in\GG^2$ which guarantees that we are working on the already extracted $(t_0,t_2,t_3,t_4)$ involved in the expressions of $\hat{D}_0$ and
Second, the verification equation (\ref{last-ver-sig}) ensures that $T_0=T_2^{r_1}\cdot T_3^{r_2}\cdot T_4$ and the final result follows by replacing them by their
At a high level, the protocol involves a committer who wants to get a signature on ${\mathbf{m}}=(m_1,\ldots,m_\ell)$ and first computes a commitment of the form $c_v=v_1^{m_1}\cdots v_\ell^{m_\ell}\cdot u^{r}$, where $u$ is the extra public parameter (with unknown discrete log). The signer gives back elements of the form $\tau_1=g^\omega c_v^s$, $\tau_2=g^s$, $\tau_3=h^s$ which is almost the desired signature. To get the component $\sigma_1$ of the right form relatively to $\tau_2,\tau_3$ the committer has to remove the factor $u^{rs}$ from $\tau_1$. Then, the signer also sends $\tau_0=u^s$ to enable removing $\tau_0^r$.
In the protocol some randomizing steps are included as well as other additional components allowing the committer to extract $\pi$, the \QANIZK part of the signature. In the security proof of the protocol we thus have to show that the additional value $\tau_0=u^s$ does not affect the unforgeability of the signature. \smallskip
At the beginning of a new run of the protocol, the committer has a vector ${\mathbf{m}}=(m_1,\ldots,m_\ell)$, the public-key of the signature scheme and the extra generator $u\in\GG$ (which can be a hashed point), the signer also has the secret key of the signature scheme but not ${\mathbf{m}}$.
To get a signature on ${\mathbf{m}}$, the committer picks $r\sample\U(\Zp)$ and computes a perfectly hiding commitment $c_v=v_1^{m_1}\cdots v_\ell^{m_\ell}\cdot u^{r}\in\GG$.
The signer receives these commitments and they both engage in an interactive proof of knowledge of an equal representation of $c_v$ relatively to the basis $(v_1,\ldots,v_\ell;u)$ and $c_z$ relatively to the basis $(z_2,\ldots,z_{\ell+1};u)$,
We notice that the number of transmitted group elements is constant and no pairing is needed before the signature verification phase.
In comparison, the construction of \cite{CL02a} requires groups of larger hidden order and their protocol for signing committed message blocks requires a linear number of range proofs. \smallskip
We briefly sketch the proof of the above protocol in front of malicious entities since classical arguments can be applied. Assuming that the committer uses secure ZKPK and does not output $\bot$, a malicious signer which receives perfectly hiding commitments $c_v,c_z$ cannot tell apart an honest proof from a simulated proof. Consequently the signer learns nothing from ${\mathbf{m}}$ during the execution of the protocol.
In the other case, we have to show that a corrupted committer remains unable to produce valid signature on a new vector ${\mathbf{m}^\star}$. First, since the generation of $u$ is not under the controlled of the committer but of the random oracle, $u$ can be made independent of rest of $\mathsf{pk}$. Then, we only need to show that the signature remains unforgeable when $\tau_0$ is given in the signature. Since ${\mathbf{m}}$ and $s$ can be extracted from the proof of knowledge the reduction can output a signature on ${\mathbf{m}}$. Moreover it is easy to see from the security proof (in \cref{sse:sigmasig-qa-nizk}) of the signature how this additional element can be simulated. Actually the only place in the reduction where $\tau_0$ could not be computed directly as $u^s$ for a known $s$ is when the challenger $\bdv$ has to embed an $\SXDH$ challenge in a simulated signature. Given ($g,h,g^b,h^{b+c}$), $\bdv$ can compute $u=g^{a_u}h^{b_u}$ from random $a_u,b_u\gets\Zp$ and program the random oracle to output this element $u$ as the specification of the public-key would do. Then to simulate $\tau_0$$\bdv$ simply has to compute $\tau_0=(g^b)^{a_u}(h^{b+c})^{b_v}=u^bh^{c\cdot b_v}$ which is $u^b$ or random. The rest of the reduction remains unchanged since the value $a_u,b_u$ are completely independent of those already described in the sketch of proof in \cref{sse:sigmasig-qa-nizk}. \smallskip
Since a malicious signer may know the simulation trapdoor $\mathsf{tk}=\{\chi_i\}_{i=1}^{2\ell+4}$ of the underlying \QANIZK argument, he could produce valid signature so that $\log_g \sigma_2\neq\log_h \sigma_3$. Then, if the committer later needs to proof knowledge of the received signature it then has to use the sigma protocol of Section~\ref{scal-sig} where both $\sigma_2$ and $\sigma_3$ only appear in committed form.
\indent At a high level, each group member obtains a membership certificate consisting of a signature $(\sigma_1,\sigma_2,\sigma_3,\pi)$ on
a message $\ID\in\Zp$ which is only known to the group member. During the joining protocol, each group member thus obtains a signature
on a committed message $\ID\in\Zp$. Here, we use a deterministic commitment to $\ID$, which suffices to ensure security against framing attacks and allows for a better
efficiency. When signing a message, each group member verifiably encrypts the components $(\sigma_1,\pi)$ of his membership certificate that depend on $\ID$ (and not $\sigma_2,\sigma_3$ which can be assumed to be honestly computed here, unlike in the previous section).
For the sake of efficiency, we use a randomness re-using \cite{BBKS07} variant of the Cramer-Shoup encryption scheme \cite{CS98} whereby $\sigma_1$ and $\pi$ are both encrypted using
the same encryption exponent $\theta\in\Zp$. For public verifiability purposes, the validity of Cramer-Shoup ciphertexts is demonstrated using
$\Sigma$-protocols and the Fiat-Shamir heuristic \cite{FS86} (somewhat in the fashion of \cite{SG98}) rather than designated verifier $\NIZK$ proofs \cite{CS98}. \\
In the join protocol, the user proves knowledge of his membership secret $\ID\in\Zp$ in a zero-knowledge manner, which restricts the group manager to sequentially interact
with prospective users. However, this limitation can be removed using an extractable commitment as in \cite{DP06}.
%\item Parse the signature $\Sigma$ as per~\eqref{gsig-sigma} and $\mathcal{S}_\OA$ as $ \bigl(x_z, y_z, x_r, y_r, x_\sigma, y_\sigma, x_\ID, y_\ID \bigr)$.
It is possible to spare one group element in the signature by eliminating the encryption $C_{\ID}$ of $v^\ID$ which is only used to open signatures in constant time.
This results in a modified opening algorithm which takes $O(N)$ in the worst-case. In applications where signature openings are infrequent, this is acceptable.
The security of the above dynamic group signature scheme, namely full anonymity, security against misidentifications and security against framing attacks that are defined in \cref{sse:gs-sec-notions} are expressed in \cref{th:sgsig-anonymity}, \cref{th:sgsig-mis-identification} and~\cref{th:sgsig-non-frameability} respectively.
The security relies on the \SXDH assumption for anonymity and misidentification, and on the \SDL assumption for non-frameability.
At the first transition, we need to rely on the security of the computational soundness of the \QANIZK argument of Section~\ref{sse:sigmasig-qa-nizk} which relies on the $\SXDH$ assumption, since $\tilde\sigma_2$ and
In the challenge phase, the adversary outputs two valid membership certificates and membership secrets $(\crt_0^\star,\scr_0^\star),(\crt_1^\star,\scr_1^\star)$ and obtains a challenge signature which the challenger computes using $(\crt_d^\star,\scr_d^\star)$, where $d \sample\U(\bit)$. We define $W_0$ to be the event that
$\theta, \ID$. Namely, we pick $c,~ s_\theta,~ s_\ID\sample\U(\Zp)$ at random and set $ R_1= g^{s_\theta}\cdot C_1^{-c}$, $R_2= h^{s_\theta}\cdot C_2^{-c},$
\item[Game 4:] Here, we modify the distribution of the challenge signature and replace $C_2= h^\theta$ by $C_2= h^{\theta+\theta'}$, for a randomly chosen $\theta'
the challenger chooses a random $\alpha\sample\U(\Zp)$ at the beginning of the game, sets $h = g^\alpha$ and retains the information $\alpha=\log_g(h)$ (note that
that $C_2\neq C_1^\alpha$. Game $5$ remains the same as Game $4$. until the event $E_5$ that $\adv$ queries the opening of a signature
that properly verifies although $C_2\neq C_1^\alpha$. Lemma~\ref{le-gsig-4} states that $\Pr[E_5]\leq q_O \cdot q_H/ p$, where $q_O$ is the number of opening
decide with non-negligible probability $\varepsilon$ whether $c =0$ or $c \in_R \Zp$. To achieve this, $\bdv$ sets $h = g^a$ and computes the challenge signature as $ C_1= g^b$ and $ C_2=\eta$.
The rest of the game continues like in Game $3$ (which is also the same as in Game $2$).
If $\adv$ wins and correctly guesses $d'=d \in\bit$, $\bdv$ outputs $1$, meaning that $C_2= h^{b }= g^{ab}$. Otherwise, $\bdv$ returns $0$ meaning that $(g^a, g^b, \eta)\in_R \GG^3$.
which can be interpreted as a linear system with unknowns $(c,s_\theta)\in\Zp^2$
\begin{equation}\label{gsig-proof-sys}
\begin{cases}
\log_g(R_1) = s_\theta - \log_g(C_1) \cdot c &\bmod p, \\
\log_h(R_2) = s_\theta - \log_h(C_2) \cdot c &\bmod p.
\end{cases}
\end{equation}
We can assume w.l.o.g. that each opening query is preceded by the corresponding random oracle query (otherwise, the reduction can simply make the hash query for itself).
The input of each hash query contains a pair $(R_1, R_2)$ determining the non-homogeneous terms of the linear
system~\eqref{gsig-proof-sys}. Since $\log_g(C_1)\neq\log_h(C_2)$, the system is full-rank, so that for each $(R_1,R_2)$, there is exactly
one pair $(c, s_\theta)\in\Zp^2$ that satisfies (\ref{gsig-proof-sys}). The probability that, in response to a random oracle query, the reduction returns
the proof of knowledge of $\ID$ at each execution of the join protocol with the adversary attempting to escape traceability.
For this
reason, we need to assume that users join the system sequentially, rather than concurrently.
However, this problem can be solved as in \cite{DP06} by having the user send an extractable commitment to $\ID$ and non-interactively prove (via the Fiat-Shamir heuristic) that he did so correctly.
This allows the reduction to
extract $\ID$ without rewinding the user at each execution of $\mathsf{Join}$. Then, the proof of security against framing attacks must be modified by having the reduction
which consists in implicitly rewinding the zero-knowledge proof by running the adversary twice and changing the outputs of the random oracle after the hash query that involves
the forgery message. The Forking Lemma~\cite{PS00} -- more precisely, its generalization given by Bellare and Neven~\cite{BN06} -- ensures that, after two runs of the adversary,
\textsf{Join} query, the reduction $\bdv$ rewinds the adversary $\adv$ in order to extract the witness $\ID=\log_v(V_{\ID})$ of which $\adv$ demonstrates knowledge at step 3 of the
join protocol. Having extracted $\ID\in\Zp$, $\bdv$ invokes its own
signing oracle on the message $\ID$ to obtain $(\sigma_1, \sigma_2, \sigma_3, z, r)$. Then, $\bdv$ returns $\crt_i=(i,V_{\ID},\sigma_1,\sigma_2,\sigma_3,z,r)$ as in a normal execution
which the opening algorithm does not reveal a properly registered identity. With all but negligible probability, $\adv$ must have queried the random oracle value
Thus, $\bdv$ replays the adversary $\adv$ with the \emph{same} input and random tape as in the first run. In the second run, the random oracle is also the same until the hash query
two suitably related forgeries with non-negligible probability $\varepsilon\cdot(\varepsilon/ q_H -1/p)$. Namely, $\bdv$ will obtain two matching transcripts
$s_\ID^\dag$ (that necessarily involve the same identifier $\ID^\star$ which is uniquely determined by $C_{\mathsf{CS}}^\star=(C_1^\star, C_2^\star, C_z^\star, C_\sigma^\star, C_\ID^\star)$), $\bdv$ runs the knowledge extractor of to obtain
\begin{proof} Let us assume that a PPT adversary $\adv$ can create, with advantage $\varepsilon$, a forgery $(M^\star,\sigma^\star)$ that opens to some honest user $i\in U^b$ who did not sign $M^\star$. We give a reduction $\bdv$ that uses $\adv$ to break SDL. \\
\indent Algorithm $\bdv$ takes as input an SDL instance $(g,\hat{g},g^a,\hat{g}^a)$ and uses its interaction with the adversary $\adv$ to compute $a \in\Zp$.
At step 1, $\bdv$ defines the generators $g,\hat{g}$ in $\pk_s$ to be those of its input and computes $h=g^{\alpha_h}$, $v=g^{\alpha_v}$, $w=g^{\alpha_w}$, $\hat{g}_z=\hat{g}^{\alpha_z}$ for randomly chosen scalars $\alpha_h,\alpha_v,\alpha_w,\alpha_z \sample\U(\Zp)$.
In order to compute $\{z_j\}_{j=1}^3$ of $\mathsf{crs}$ contained in $\pk_s$, $\bdv$ chooses $\mathsf{tk}=\{\chi_j\}_{j=1}^6$ of step 4 of the key generation algorithm of the signature scheme of Section \ref{scal-sig} with $\ell=1$. (Note that when $\ell=1$, $n=6$ and that $\{z_j\}_{j=1}^3$ are \QANIZK argument for the vectors $(g,1,1,1,1,h)$, $(v,g,1,h,1,1)$ and $(w,1,g,1,h,1)$. Moreover $\{\hat{g}_i=\hat{g}_z^{\chi_i}\}_{i=1}^6$ are the verifying key.)
As a result of this setup phase, $\bdv$ knows $\mathcal{S}_{\GM}=\sk_s=\omega$, $\mathcal{S}_{\OA}=\bigl(x_z, y_z, x_\sigma, y_\sigma, x_\ID, y_\ID\bigr)$ and even $\mathsf{tk}$. The adversary $\adv$ is run on input of the group public key $\gspk:=(\pk_s,(X_z,X_\sigma,X_{\ID}),H)$, which has the same distribution as in the real attack game. \medskip
Should $\adv$ decide to corrupt the group manager or the opening authority during the game, $\bdv$ is able to reveal $\mathcal{S}_{\GM}=\sk_s$ and $\mathcal{S}_{\OA}$ when requested.
\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. Then, $\bdv$ runs $\mathsf{J}_{\mathsf{user}}$ on behalf of the honest user in an execution of $\mathsf{Join}$. %The actions taken by $\bdv$ are dictated by the index $j \in \{1,\ldots,q_b\}$ of the $Q_{\bjoin}$-query. \\
the membership certificate $\crt_{i}=(i, V_i, \sigma_1, \sigma_2, \sigma_3, \pi)$ to compute the ciphertext $C_{\mathsf{CS}}$ at steps 1-2 of the signing algorithm.
While $\bdv$ does not know the witness $\ID_i=a\cdot\delta_i\in\Zp$ to generate a proof at step 3, $\bdv$ is able to simulate the
programs $H$ to return $c$ on inputs $(M, C_{\mathsf{CS}},\tilde\sigma_2, \tilde\sigma_3, R_1, R_2, R_3, R_4)$. In the event that $H$ is already defined at that point,
$\bdv$ aborts.
The probability to fail at one signing query is $\leq q_s/p^3$, where
for some message $M^\star$, that opens to ${i^\star}\in U^b$ although user $i^\star$ did not sign $M^\star$. With high probability, $\adv$ must have queried the hash value
$H(M^\star, C_{\mathsf{CS}}^\star, \tilde\sigma_2^\star,\tilde\sigma_3^\star,R_1^\star, R_2^\star, R_3^\star, R_4^\star)$, which would be unpredictable otherwise.
Hence, $\bdv$ can run $\adv$ a second time with the same input and random tape.
At the moment when $\adv$ queries $H(M^\star, C_{\mathsf{CS}}^\star, \tilde\sigma_2^\star,\tilde\sigma_3^\star,R_1^\star, R_2^\star, R_3^\star, R_4^\star)$ in the second run, $\bdv$ starts responding with different random oracle values which depart from those of the initial run.
This observation tells us that, if $\adv$ has advantage $\varepsilon$ as a framing adversary making $q_H$ random oracle queries, then $\bdv$ implies an algorithm solving the SDL problem with probability $\varepsilon(\varepsilon/ q_H -1/p)$.
%On the other hand, the comparison of computational cost is not straightforward, as it is not clear if the computation of $e(x^\alpha, \hat x) \cdot e(y^\beta, \hat y)$ is easier than the computation
An implementation of the aforementioned group signature scheme has been made in \texttt{C} using the \textit{Relic toolkit} for pairing-based cryptography~\cite{AG} and is available at the following address:~\url{https://gforge.inria.fr/projects/sigmasig-c/}.
The benchmarking was made on a single-core of an \textit{Intel\textregistered{} Core\texttrademark{} i5-7500 CPU @ 3.40GHz} (Kaby Lake architecture) with 6MB of cache.
To implement pairings, the relic library implements the Barreto-Naehrig~\cite{BN06} curve over a 256 bits curve.
Figures are available in Table~\ref{ta:sigmasig-figures}.
