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{Cha85,CL01}, which are similar to group signatures except that anonymity is irrevocable (meaning that there is no opening authority).
In other words, an anonymous credential scheme involves one (or more) credential issuer(s) and a set of users who have a long term secret key which can be seen as their digital identity, and pseudonyms that can be seen as commitments to their secret key.
Users can dynamically obtain credentials from an issuer that only knows users' pseudonyms and obliviously sign users' secret keys as well as a set of attributes.
Later on, users can make themselves known to verifiers under a different pseudonym and demonstrate possession of the issuer's certificate on their secret key withour revealing neither the signature nor the key.
In this context, signature with efficient protocols can typically be used as follows:
the user obtains the issuer's signature on a committed message via an interactive protocol, and uses a protocol for proving that two commitments open to the same value (which allows proving that the same secret underlies two distinct pseudonyms) and finally a protocol for proving possession of a secret message-signature pair.
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
