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%\chapter{Lattice-Based Oblivious Transfer with Access Control} \label{ch:ac-ot}
%\addcontentsline{tof}{chapter}{\protect\numberline{\thechapter} Transfert inconscient adaptatif avec contrôle d'accès à base de réseaux euclidiens}
%\label{ch:ot-lwe}
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\begin{comment}
\section{Introduction}
\end{comment}

Kiayias, Tsiounis and Yung~\cite{KTY07} presented group encryption ($\mathsf{GE}$) as the encryption analogue of group signatures~\cite{CVH91}, which allow users  to anonymously    sign messages on behalf of an entire group they belong to.
While group signatures aim at hiding the source of some message within a crowd administered by some group manager, group encryption rather seeks to hide its destination within a group of legitimate receivers.
In both cases, a verifier should be convinced that the anonymous signer/receiver  indeed belongs to a purported population.
In order to keep users accountable for their actions, an opening authority ($\mathsf{OA}$)  is further empowered with some information allowing it to un-anonymize signatures/ciphertexts.  

Kiayias, Tsiounis and Yung~\cite{KTY07} formalized  $\mathsf{GE}$ schemes as a primitive   allowing the sender to generate publicly verifiable guarantees that:
(1) The ciphertext is well-formed and intended for some registered group member who will be able to decrypt;
(2) the opening authority will be able identify the receiver if necessary; (3) The plaintext satisfies certain properties such as being a witness for some
public relation or the private key that underlies a given public key. In the model of Kiayias \textit{et al.}~\cite{KTY07}, the message secrecy and anonymity
properties are required to withstand active adversaries, which are granted access to decryption oracles in all security experiments.

As a natural application, group encryption allows a firewall to filter all incoming encrypted  emails  except those intended for some certified organization
member   and the content of  which is additionally guaranteed to satisfy certain requirements, like the absence of malware.

 $\mathsf{GE}$~schemes are also motivated by natural privacy applications such as anonymous trusted third parties, key recovery mechanisms or oblivious retriever
 storage systems. In optimistic protocols, $\mathsf{GE}$ allows verifiably encrypting messages to \emph{anonymous} trusted third parties which mostly remain off-line
 and  only come  into play to sort out conflicts.  In order to  protect privacy-sensitive information such as users' citizenship, group encryption
 makes it possible to hide the identity of users' preferred trusted third parties within a set of properly certified trustees.

In cloud storage services, $\mathsf{GE}$ enables privacy-preserving asynchronous transfers of encrypted datasets. Namely, it allows users to archive  encrypted datasets
on  remote  servers while convincing those servers that the data is indeed intended for some anonymous certified client who paid a subscription to the storage
provider.  Moreover,  a judge should be able to identify the archive's recipient in  case a misbehaving server is found guilty of hosting   suspicious
transaction records or any other illegal content.

As pointed out by Kiayias \textit{et al.}~\cite{KTY07}, group encryption also implies a form of hierarchical group signatures~\cite{TW05}, where signatures can only be opened by a set of eligible trustees   operating in a very specific manner determiner by the signer.

The design of numerous privacy-preserving cryptographic protocols crucially relies on  zero-knowledge proofs~\cite{GMR85} to prove properties about encrypted or committed values so as to enforce honest behavior on behalf of participants or protect the privacy of users.
In the lattice settings, efficient zero-knowledge proofs are non-trivial to construct due to the limited amount of algebraic structure.
While natural methods of proving knowledge of secret keys \cite{MV03,Lyu08,KTX08,LNSW13} are available, they are only known to work for specific languages.
When it comes to proving circuit satisfiability, the best known methods are designed for the $\mathsf{LPN}$ setting~\cite{JKPT12} or take advantage of the extra structure available in the ring $\LWE$
setting~\cite{XXW13,BKLP15}.
Hence, these methods are not known to readily carry over to standard (i.e., non-ideal) lattices.
In the standard model, the problem
is even trickier as we do not have a lattice-based counterpart of Groth-Sahai proofs~\cite{GS08} and efficient non-interactive proof systems are only available
for specific problems~\cite{PV08}.

The difficulty of designing efficient  zero-knowledge proofs for lattice-related languages makes it highly non-trivial to adapt privacy-preserving cryptographic
primitives  in the lattice setting. In spite of these technical hurdles, a recent body of work successfully designed anonymity-enabling mechanisms like ring
signatures \cite{KTX08,ABB+13}, blind signatures \cite{Ruec10},  group signatures \cite{GKV10,LLLS13,LLNW14,BCK+14,NZZ15,LNW15,LLNW16}
or, more recently, signature schemes with companion zero-knowledge protocols~\cite{LLM+16}.  A common feature of all these works is that the zero-knowledge
layer of the proposed protocols only deals with linear equations, where witnesses are only multiplied by public values.

In this chapter, motivated by the design of advanced privacy-preserving protocols in the lattice setting, we construct zero-knowledge arguments for  non-linear
statements among witnesses consisting of vectors and matrices.  For suitable parameters $q,n,m \in \ZZ$, we consider zero-knowledge argument systems whereby a
prover can demonstrate knowledge of secret matrices $\mathbf{X} \in \ZZ_q^{m \times n}$ and vectors $\mathbf{s} \in \ZZ_q^n$, $\mathbf{e} \in \ZZ^m$ such that:
(i) $\mathbf{e} \in \ZZ^m$ has small norm;
(ii) A public vector $\mathbf{b} \in \ZZ_q^n$ equals  $\mathbf{b} = \mathbf{X}\cdot \mathbf{s} + \mathbf{e} \bmod q$;
(iii) The underlying pair $(\mathbf{X},\mathbf{s})$ satisfies additional algebraic relations: for instance, it should be possible to prove possession of a signature on some representation of the matrix $\mathbf{X}$.
In particular, our zero-knowledge argument  makes it possible to prove that a given ciphertext is a well-formed $\LWE$-based encryption with respect to some
hidden, but certified public key. This protocol comes in handy in the design of \textit{group encryption} schemes~\cite{KTY07}, where such languages naturally
arise.
Using these advances, we thus construct, in this chapter, the first construction of group encryption under lattice assumptions.

\paragraph{Related work.}
Kiayias, Tsiounis and Yung (KTY) \cite{KTY07} formalized the notion of group encryption and provided a modular design using
zero-knowledge proofs, digital signatures, anonymous CCA-secure public-key encryption and commitment schemes. They also gave an efficient instantiation using
Paillier's cryptosystem~\cite{Pail99} and Camenisch-Lysyanskaya signatures \cite{CL02}.

Cathalo, Libert and Yung \cite{CLY09}
designed a non-interactive system in the standard model under non-interactive pairing-related assumptions.  El~Aimani and Joye \cite{EJ13} suggested various
efficiency improvements with both interactive and non-interactive proofs.

Libert \textit{et al.}~\cite{LYJP14} empowered the $\GE$ primitive with a refined traceability mechanism akin to that of traceable signatures~\cite{KTY04}. Namely,
by releasing a user-specific trapdoor, the opening authority can allow anyone to publicly trace ciphertexts encrypted for this specific group member  without
affecting the privacy of other users.  Back in 2010, Izabachène, Pointcheval and Vergnaud~\cite{IPV10} considered the problem of eliminating subliminal
channels in a different form of traceable group encryption.

As a matter of fact,  all existing realizations of group encryption or similar primitives rely on traditional number theoretic assumptions like the hardness of
factoring or computing discrete logarithms.  In particular, all of them are vulnerable to quantum attacks.  For the sake of not putting all one's eggs in the
same basket, it is highly desirable to have instantiations based on alternative, quantum-resistant foundations.
\bigskip

In the next sections, we first present the definitions of a group encryption schemes and the required building block.
Then, we describe the zero-knowledge protocol we use to handle these quadratic relations before finally describing our scheme.

\section{Syntax and Definitions of Group Encryption} \label{GE-model}
\index{Group Encryption}

We use the syntax and the security model of Kiayias, Tsiounis and Yung \cite{KTY07}. 
The group encryption (\textsf{GE}) primitive involves a sender, a verifier, a group manager~(\textsf{GM}) that manages the group of receivers and an opening
authority~(\textsf{OA}) which is capable of identifying  ciphertexts' recipients.

In the syntax of \cite{KTY07}, a $\mathsf{GE}$ scheme is specified by the description of a
relation $R$ as well as a tuple
$\mathsf{GE}=\bigl(\mathsf{SETUP},\mathsf{JOIN},\langle
  \mathcal{G}_r,R,\mathsf{sample}_{R}
  \rangle,\mathsf{ENC},\mathsf{DEC},\mathsf{OPEN},\langle
\mathcal{P},\mathcal{V} \rangle \bigr)$ of algorithms or protocols.
In details, $\mathsf{SETUP}$ is a set of initialization procedures that all take (implicitly or explicitly) a security parameter $1^\lambda$ as input. We call them
$\mathsf{SETUP}_{\mathsf{init}}(1^\lambda)$,
$\mathsf{SETUP}_{\mathsf{GM}}(\param)$ and
$\mathsf{SETUP}_{\mathsf{OA}}(\param)$. The first one of these procedures
generates a set of public parameters $\param$ (like the KTY construction \cite{KTY07}, we rely on a common reference string even when using interaction between
provers and verifiers). The latter two procedures are used to produce key pairs
$(\pk_{\GM},\sk_{\GM})$, $(\pk_{\OA},\sk_{\OA})$ for the $\GM$ and the
$\OA$. In the following,   $\param$ is incorporated  in the inputs of all algorithms although we sometimes omit to explicitly write it.

$\mathsf{JOIN}=(\mathsf{J}_{\mathsf{user}},\mathsf{J}_{\GM})$ is an interactive protocol between the $\GM$ and the prospective user.
After the execution of $\mathsf{JOIN}$, the $\GM$ stores the public key $\pk$ and its certificate $\crt_{\pk}$ in a public directory
$\mathsf{database}$. 
 As in \cite{KY05}, we will restrict this
protocol to have minimal interaction and consist of only two messages: the first one is the user's public key $\pk$ sent by $\mathsf{J}_{\mathsf{user}}$ to $\mathsf{J}_{\GM}$
and the latter's response is a certificate $\crt_{\pk}$ for $\pk$ that makes the user's group membership effective. We do not require the user to prove
knowledge of his private key $\sk$ or anything else about it. In our construction, valid   keys will be publicly recognizable and users will not have to prove
their validity. By avoiding proofs of knowledge of private keys, the security proof never has to
rewind the adversary to extract those private keys, which  allows supporting concurrent joins as
advocated by Kiayias and Yung   \cite{KY05}. If applications demand it, it is possible to add
proofs of knowledge of private keys in a modular way but our security proofs do not require
rewinding the adversary in executions of $\mathsf{JOIN}$.

Algorithm $\mathsf{sample}_{R}$ allows sampling pairs $(x,w)\in R$ (made of a public value $x$ and a witness $w$) using keys $(\pk_{R},\sk_{R})$ produced by
$\mathcal{G}_r(1^\lambda)$ which samples public/secret parameters for the relation $R$. Depending on the relation, $\sk_{R}$ may be the empty string (as in the scheme \cite{KTY07} and ours  which both involve publicly samplable relations). The testing procedure   $R(x,w)$ uses $\pk_{R}$ to 
return  $1$ whenever $(x,w)\in R$. To encrypt a witness $w$ such that $(x,w) \in R$ for some public $x$, the sender fetches the pair $(\pk,\crt_{\pk})$
from $\mathsf{database}$ and runs the randomized  encryption algorithm. The latter takes as input $w$, a label $L$, the receiver's   pair $(\pk,\crt_{\pk})$ as
well as public keys $\pk_{\GM}$ and $\pk_{\OA}$. Its output is a ciphertext
$\Psi \gets \mathsf{ENC}(\pk_{\GM},\pk_{\OA},\pk,\crt_{\pk},w,L)$.
On input of the same elements, the certificate $\crt_{\pk}$, the ciphertext $\Psi$ and the random coins $coins_{\Psi}$ that were used to produce  $\Psi$, the
non-interactive algorithm $\mathsf{PP}$ generates a proof $\pi_{\Psi}$ that there exists a certified receiver whose public key was registered in $\mathsf{database}$ and
 who is able to decrypt $\Psi$ and obtain a witness $w$ such that $(x,w) \in R$. The verification algorithm $\mathcal{V}$ takes as input $\Psi$, $\pk_{\GM}$,
$\pk_{\OA}$, $\pi_{\Psi}$  and the description of $R$  and outputs $0$ or $1$. Given $\Psi$, $L$ and the receiver's private key $\sk$, the output of
$\mathsf{DEC}$ is either a witness $w$ such that $(x,w) \in R$ or a rejection symbol $\bot$. Finally,
$\mathsf{OPEN}$ takes as input   a ciphertext/label pair $(\Psi,L)$ and the OA's secret key $\sk_{\OA}$ and returns   a receiver's public key $\pk$.

The  model of \cite{KTY07} considers four properties termed correctness, message security, anonymity and soundness. 
In the security definitions, stateful oracles capture the adversary's
interaction with the system.  In the soundness game, the KTY model requires
that pk belongs to the language of valid public keys. Here, we are implicitly assuming that the space
of valid public keys is dense (all matrices are valid keys, as is the case in our scheme).

In the upcoming definitions,  we sometimes use the notation
\[ \langle \mathsf{output}_A |\mathsf{output}_B \rangle \allowbreak \gets \langle A(\mathsf{input}_A),B(\mathsf{input}_B) \rangle (\mathsf{common\textrm{-}input}) \]
to denote the execution of a protocol between $A$ and $B$ obtaining their own outputs from their respective inputs.
\medskip

\paragraph{Correctness.}
 The correctness property
requires that the following experiment returns $1$ with overwhelming
probability.

\begin{center}
  \procedure{Experiment $\Expt^{\mathrm{correctness}}(\lambda)$}{
    \mathsf{param} \gets
    \mathsf{SETUP}_{\mathsf{init}}(1^\lambda); (\pk_{R},\sk_{R})
    \gets \mathcal{G}_r (\lambda); (x,w) \leftarrow \mathsf{sample}_{R}
    (\pk_{R},\sk_{R}); \\
    (\pk_{\GM},\sk_{\GM}) \leftarrow \mathsf{SETUP}_{\GM}(\mathsf{param}); (\pk_{\OA},\sk_{\OA}) \leftarrow \mathsf{SETUP}_{\OA}(\mathsf{param});  \\
    \langle \pk,\sk,\crt_{\pk} | \pk,\crt_{\pk} \rangle \leftarrow \langle  \mathsf{J}_{\mathsf{user}},\mathsf{J}_{\GM}(\sk_{\GM})  \rangle (\pk_{\GM}); \\
    \Psi \leftarrow
    \mathsf{ENC}(\pk_{\GM},\pk_{\OA},\pk,\crt_{\pk},w,L);\\
    \pi_{\Psi} \leftarrow \mathcal{P}(\pk_{\GM},\pk_{\OA},\pk,\crt,w,L,\Psi,coins_{\Psi}); \\
    \pcif \bigl( (w \neq \mathsf{DEC}(\sk,\Psi,L)) \vee ( \pk \neq
      \mathsf{OPEN}(\sk_{\OA},\Psi,L )) \\ \quad \qquad \vee (\mathcal{V}(\Psi,L,\pi_{\Psi},\pk_{\GM},\pk_{\OA})=0)
    \bigr) \pcthen\\
    \pcind \pcreturn 0\\
    \pcelse\\
    \pcind \pcreturn 1; 
 }
\end{center}

\paragraph{Message Secrecy.}
The message secrecy property is defined by an experiment where the adversary has access to oracles
that may be stateful (and   maintain a state across queries) or
stateless:
%These oracles are the following:
\begin{itemize}
\item[-] $\mathsf{DEC}(\sk)$: is a stateless  oracle for the user decryption function
$\mathsf{DEC}$. When this oracle is restricted not to decrypt a
 ciphertext-label pair $(\Psi,L)$, we denote it by
$\mathsf{DEC}^{\neg \langle \Psi, L \rangle}$.
\item[-] $\mathsf{CH}_{\mathsf{ror}}^b(\lambda,\pk,w,L)$: is a
  real-or-random challenge oracle which is called \textit{once}. It
returns  $(\Psi,coins_{\Psi})$ such that $\Psi \leftarrow
\mathsf{ENC}(\pk_{\GM},\pk_{\OA},\pk,\crt_{\pk},w,L)$ if $b=1$
whereas, if $b=0$, $\Psi \leftarrow
\mathsf{ENC}(\pk_{\GM},\pk_{\OA},\pk,\crt_{\pk},w',L)$ encrypts a
random plaintext   of
length $O(\lambda)$  uniformly sampled in the plaintext space. In both cases, $coins_{\Psi}$ denote the random
coins used to generate $\Psi$.
\item[-]
$\mathsf{PROVE}_{\mathsf{PP},\mathsf{PP}'}^b(\pk_{\GM},\pk_{\OA},\pk,\crt_{\pk},\pk_{R},x,w,\Psi,L,coins_{\Psi})$:
is a stateful oracle that  can be invoked  a polynomial number times. If $b=1$, it  replies by running the real prover $\mathsf{PP}$ on the inputs to
create  an actual proof $\pi_{\Psi}$. If $b=0$, the oracle runs a
simulator $\mathsf{PP}'$ that uses the same inputs as $\mathsf{PP}$  except  witness
$w,coins_{\Psi}$ and generates a simulated proof.
\end{itemize}
These oracles are used in an experiment where the adversary   controls
the $\GM$, the $\OA$ and all members except the honest receiver. The
adversary $\adv$ embodies the dishonest $\GM$ that certifies the honest
receiver in an execution of $\mathsf{JOIN}$. It is granted access to an oracle  $\mathsf{DEC}$ which decrypts on behalf of that receiver. In the
challenge phase, it transmits a state information   $\mathsf{aux}$  to itself and invokes the challenge oracle for a label and a
pair $(x,w) \in R$ of its choice. After the challenge phase, it
can also query the $\mathsf{PROVE}$ oracle  many times
and finally attempts to guess the challenger's bit $b$.

As pointed out in \cite{KTY07,CLY09}, designing an efficient
simulator $\mathsf{PP}'$ (for executing $\mathsf{PROVE}_{\mathsf{PP},\mathsf{PP}'}^b(.)$
when $b=0$) is part of the security proof.
\begin{definition} \label{security-def}
  A $\mathsf{GE}$ scheme satisfies \textit{message security}
if, for any PPT adversary $\adv$, the experiment below returns $1$
with probability at most  $1/2 + \mathsf{negl}(\lambda)$.

\begin{center}
\procedure{Experiment $\Expt_{\adv}^{\mathrm{sec}}(\lambda)$}{
   \param \leftarrow \mathsf{SETUP}_{\mathsf{init}}(1^\lambda);  (\mathsf{aux},\pk_{\GM},\pk_{\OA}) \leftarrow \adv(\param); \\
   \langle \pk,\sk,\crt_{\pk} | \mathsf{aux} \rangle
  \leftarrow \langle \mathsf{J}_{\mathsf{user}},\adv(\mathsf{aux}) \rangle
  (\pk_{\GM}); \\
   (\mathsf{aux},x,w,L,\pk_{R}) \leftarrow
  \adv^{\mathsf{DEC}(\sk,.)}(\mathsf{aux});\\
  \pcif (x,w) \not\in R \pcthen\\
  \pcind\pcreturn 0;\\
   b\sample \{0,1\}; ( \Psi,coins_{\Psi} ) \leftarrow \mathsf{CH}_{\mathsf{ror}}^b(\lambda,\pk,w,L) ; \\
   b' \leftarrow
  \adv^{\mathsf{PROVE}_{\mathsf{PP},\mathsf{PP}'}^b(\pk_{\GM},\pk_{\OA},\pk,\crt_{\pk},\pk_{R},x,w,\Psi,L,coins_{\Psi}),\mathsf{DEC}^{\neg
  \langle \Psi, L \rangle}(\sk,.)}(\mathsf{aux},\Psi) ; \\
   \pcif b=b' \pcthen\\ \pcind\pcreturn 1 \\\pcelse\\ \pcind \pcreturn 0;
}
\end{center}
\end{definition}

\paragraph{Anonymity.}
In the experiment modeling the anonymity property, the adversary
controls the entire system except the opening authority and two well-behaved users.
 The challenger thus introduces two honest users' public keys $\pk_0,\pk_1$ in $\mathsf{database}$ and thus obtains certificate for both $\pk_0,\pk_1$ from the adversarially-controlled $\GM$.
 For a pair $(x,w) \in R$ of its choice, the adversary obtains an encryption of $w$ under $\pk_b$ for some $b\in \bit$ chosen by the challenger.
 The adversary  is provided with decryption oracles w.r.t. both keys $\pk_0,\pk_1$. In addition, it has  the following oracles at disposal:
\begin{itemize}
\item[-] $\mathsf{CH}_{\mathsf{anon}}^b(\pk_{\GM},\pk_{\OA},\pk_0,\pk_1,w,L)$: is a
  challenge oracle that is only queried once by the adversary. It
returns a pair   $(\Psi,coins_{\Psi})$ consisting of a ciphertext
$\Psi \leftarrow
\mathsf{ENC}(\pk_{\GM},\pk_{\OA},\pk_b,\crt_{\pk_b},w,L)$ and the
coin tosses  $coins_{\Psi}$ that were used to generate $\Psi$.
\item[-]
$\mathsf{USER}(\pk_{\GM})$: is a stateful oracle  that obtains  certificates from the adversary by simulating two
executions of $\mathsf{J}_{\mathsf{user}}$   to introduce two honest users
in the group. It uses a string $\mathsf{keys}$ where the outputs $(\pk_0,\sk_0,\crt_{\pk_0})$, $(\pk_1,\sk_1,\crt_{\pk_1})$  of honest users
 are written  as long as the adversarially-supplied certificates $\{\crt_{\pk_d}\}_{d=0}^1$ are valid w.r.t. $\pk_{\GM}$ (i.e., invalid certificates are ignored
by the oracle and no entry is introduced in $\mathsf{keys}$ for them).
\item[-]
$\mathsf{OPEN}(\sk_{\OA},.)$: is a stateless oracle that simulates
the opening algorithm   and, on input of a $\mathsf{GE}$
ciphertext, returns the receiver's public key.
\end{itemize}

   The reason why
the $\mathsf{USER}$ oracle is needed is that both honest users' public keys $\pk_0, \pk_1$ must have been properly
certified by the adversarially-controlled $\mathsf{GM}$ before the challenge phase  because  the adversary subsequently obtains
proofs generated using $(\pk_b,\crt_{\pk_b})$.

\begin{definition} \label{anonymity-def}
  A $\mathsf{GE}$ scheme satisfies \textit{anonymity} if, for any PPT adversary $\adv$, the experiment below returns $1$
with a probability not exceeding  $1/2 + \mathsf{negl}(\lambda)$.
\begin{center}
  \procedure{Experiment $\Expt_{\adv}^{\mathrm{anon}}(\lambda)$}{
    \param \leftarrow
    \mathsf{SETUP}_{\mathsf{init}}(1^\lambda); (\pk_{\OA},\sk_{\OA})
    \leftarrow \mathsf{SETUP}_{\mathsf{OA}}( \param); \\ 
    (\mathsf{aux},\pk_{\GM}) \leftarrow \adv(\param,\pk_{\OA});  
    \mathsf{aux}  \leftarrow
    \adv^{\mathsf{USER}(\pk_{\GM}),\mathsf{OPEN}(\sk_{\OA},.)}
    (\mathsf{aux})  ; \\ 
    \pcif \mathsf{keys} \neq (\pk_0,\sk_0,\crt_{\pk_0},\pk_1,\sk_1,\crt_{\pk_1})(\mathsf{aux})  \pcthen\\
    \pcreturn 0;\\
    (\mathsf{aux},x,w,L,\pk_{R}) \leftarrow
    \adv^{\mathsf{OPEN}(\sk_{\OA},.),
    \mathsf{DEC}(\sk_0,.),\mathsf{DEC}(\sk_1,.)}(\mathsf{aux}); \\
    \pcif (x,w) \not\in R \pcthen\\
    \pcind \pcreturn 0; \\
    b\sample \{0,1\}; ( \Psi,coins_{\Psi} ) \leftarrow \mathsf{CH}_{\mathsf{anon}}^b(\pk_{\GM},\pk_{\OA},\pk_0,\pk_1,w,L) ; \\
    b' \leftarrow
    \adv^{\mathcal{P}(\pk_{\GM},\pk_{\OA},\pk_b,\crt_{\pk_b},x,w,\Psi,L,coins_{\Psi},}
      \\       ^{\mathsf{OPEN}^{\neg \langle \Psi,L \rangle
        }(\sk_{\OA},.),\mathsf{DEC}^{\neg \langle \Psi, L
        \rangle}(\sk_0,.),\mathsf{DEC}^{\neg
    \langle \Psi, L \rangle}(\sk_1,.))}(\mathsf{aux},\Psi) ; \\
    \pcif b=b' \pcthen\\
    \pcind \pcreturn 1\\
    \pcelse\\
    \pcind \pcreturn 0;
  }
\end{center}
\end{definition}

\paragraph{Soundness.}
Here, the adversary creates the group of receivers by interacting with the honest GM.
Its goal is to produce a ciphertext $\Psi$ and a convincing proof
that   $\Psi$ is valid w.r.t. a relation $R$ of its choice but
either: (1) The opening of $\Psi$ reveals a receiver's public key $\pk$ that
does not belong to any group member; (2) The output $\pk$ of
$\mathsf{OPEN}$ is not a valid public key (\textit{i.e.}, $\pk \not\in
\mathcal{PK}$, where $\mathcal{PK}$ is the language of valid public keys); (3) The ciphertext $C$ is   not in the space
$\mathcal{C}^{x,L,\pk_{R},\pk_{\GM},\pk_{\OA},\pk}$ of  valid
ciphertexts. This notion is formalized by a game where the adversary
is given access to a user registration oracle
$\mathsf{REG}(\sk_{\GM},.)$ that simulates $\mathsf{J}_{\GM}$. This oracle
maintains a list $\mathsf{database}$ where registered public keys and
their certificates are stored.

\begin{definition} \label{soundness-def}
  A $\mathsf{GE}$ scheme is \textit{sound} if, for any PPT adversary $\adv$, the experiment below returns $1$
  with negligible probability.
  \begin{center}
    \procedure{Experiment $\Expt_{\adv}^{\mathrm{soundness}}(\lambda)$}{
       \param \leftarrow
      \mathsf{SETUP}_{\mathsf{init}}(1^\lambda);   (\pk_{\OA},\sk_{\OA})
      \leftarrow \mathsf{SETUP}_{\OA}( \param); \\ 
      (\pk_{\GM},\sk_{\GM}) \leftarrow \mathsf{SETUP}_{\GM}( \param);  \\
       (\pk_{R},x,\Psi,\pi_{\Psi},L,\mathsf{aux}) \leftarrow
      \adv^{\mathsf{REG}(\sk_{\GM},.)}(\param,\pk_{\GM},\pk_{\OA},\sk_{\OA});
      \\
       \pcif \mathcal{V}( \Psi,L,\pi_{\Psi},\pk_{\GM},\pk_{\OA})=0 \pcthen\\
       \pcind \pcreturn 0; \\
       \pk \leftarrow \mathsf{OPEN}(\sk_{\OA},\Psi,L); \\
       \pcif \big( (\pk \not\in \mathsf{database} ) \vee (\pk \not \in \mathcal{PK})
        \vee (\Psi \not \in
      \mathcal{C}^{x,L,\pk_{R},\pk_{\GM},\pk_{\OA},\pk}) \big)\pcthen\\
      \pcind \pcreturn 1\\
      \pcelse \\
      \pcind \pcreturn 0;
    }
  \end{center}
\end{definition}

The model of Kiayias \textit{et al.} \cite{KTY07} requires
that $\pk$ belongs to the language of valid public keys, so that the adversary is considered to defeat the soundness property when
$(\Psi,L)$ opens to a key outside the language (i.e.,
$\pk \not \in \mathcal{PK}$). In our scheme, we will assume that the space
of valid public keys is dense in that all matrices of a given dimension are valid public keys, which have an underlying private key.
 We nevertheless use the same definition as \cite{KTY07} in order to emphasize that we are not relaxing the model in any way.











\section{Building Blocks}




\subsection{The Agrawal-Boneh-Boyen IBE Scheme} \label{ap:ABB-IBE}
\index{Identity-Based Encryption!Agrawal-Boneh-Boyen}

\subsubsection{Identity-Based Encryption.} \label{ap:IBE}

An IBE scheme is a tuple of efficient algorithms $(\mathsf{Setup}, \mathsf{Extract}_\mathsf{PP}, \mathsf{Encrypt}_\mathsf{PP},$ $\mathsf{Decrypt}_\mathsf{PP})$ such that
\begin{description}
  \item[\textsf{Setup}$(1^\lambda)$:] On security parameter $\lambda$, this algorithm outputs public parameters $\mathsf{PP}$ and a master secret key $\textsf{msk}$.
  \item[\textsf{Extract}$_\mathsf{PP}(\textsf{msk}, \ID)$:] Takes as input a master secret key $\textsf{msk}$ and an identity $\ID$ and outputs a secret key $\sk_\ID$.
  \item[\textsf{Encrypt}$_\mathsf{PP}(\ID, M)$:] Given an identity  $\ID$ and a message $M$, it outputs a ciphertext $C$.
  \item[\textsf{Decrypt}$_\mathsf{PP}(\sk_\ID, C)$:] Given a secret key $\sk_\ID$ and a ciphertext $C$, outputs either a decryption error symbol $\bot$, or a message $M$.
\end{description}

Correctness   requires that, for any   pair $(\mathsf{PP}, \textsf{msk}) \gets \Setup(1^\lambda)$, any   $\ID$ and any message $M$,  we have
$\mathsf{Decrypt}_\mathsf{PP}\bigl(\textsf{Extract}_\mathsf{PP}(\textsf{msk}, \ID), \mathsf{Encrypt}_\mathsf{PP}(\ID, M)\bigr) = M.$
Our proofs rely on the semantic security of the scheme against selective adversaries (\textsf{IND-sID-CPA})
but also on the stronger property of ciphertext pseudo-randomness.  %in Lemma~\ref{ABB-deux}.
Informally, this notions demands that the adversary be unable to distinguish an
encryption of a message of its choice from a random element of the ciphertext space $\mathcal{C}$.  Notice that this property implies \textsf{IND-sID-CPA} security.

\begin{definition}
  \label{de:pseudorand-cipher}
  An IBE scheme has pseudo-random-ciphertexts if no PPT adversary $\adv$ with   access to private key extraction oracle  \textsf{Extract$_\mathsf{PP}(\textsf{msk}, \cdot)$} has non-negligible advantage
  $ \advantage{\mathrm{ROR}}{\adv}{\lambda} =  | \Pr\bigl[ \mathbf{Expt}_{\adv}^\mathrm{ROR} = 1 \bigr] -  \frac 1 2  | $ in the game described in Figure~\ref{fig:expt-ror}

  \begin{figure}
    \centering
    \procedure{Experiment $\Expt^{\mathrm{ROR}}_{\adv}(\lambda)$}{
      \ID^\star \gets \adv(\textsf{id}, \lambda); (\mathsf{PP}, \textsf{msk}) \gets \mathsf{Setup}(1^\lambda);~\\
      M \gets \adv^{\mathsf{Extract}_\mathsf{PP}(\textsf{msk}, \cdot)}_\textsf{Ch}(\mathsf{PP});\\
      b \sample \U(\bit);\\
      \pcif b = 1 \pcthen\\
      \pcind C^\star \gets \mathsf{Encrypt}_\mathsf{PP}(M, \ID^\star) \\
      \pcelse\\
      \pcind C^\star \sample \U(\mathcal{C});\\
      b' \gets \adv^{\mathsf{Extract}_\mathsf{PP}(\textsf{msk}, \cdot)}(\textsf{guess},C^\star);\\
      \pcif b = b' \pcthen\\
      \pcind \pcreturn 1\\
      \pcelse \\
      \pcind \pcreturn 0
    }
    \caption{Security experiment for the pseudo-random-ciphertext property for an IBE}
    \label{fig:expt-ror}
  \end{figure}
\end{definition}


\subsubsection{The ABB System.} \label{ap:ABB-desc}

  Agrawal, Boneh and Boyen described~\cite{ABB10}  a compact IBE scheme in the standard  model which allows encrypting multi-bit messages. 
%In
%the description   hereunder,   algorithms \textsf{Extract}, \textsf{Encrypt} and \textsf{Decrypt} have implicit access to the public param
%eters $\mathsf{PP}$ defined in the \textsf{Setup} algorithm.

\begin{description}
  \item[\textsf{Setup}$(1^\lambda)$:] Given a security parameter $\lambda$, choose parameters
	$q, n, \sigma, \alpha$ and define $k =\lfloor \log q \rfloor$,  $\bar{m}= nk$, $m = 2 \bar{m}$  and choose a  noise distribution $\chi$ for $\LWE$.
    \begin{enumerate}
      \item Compute $(\bar{\mathbf A}, \mathbf{T}_{\bar{\mathbf{A}}}) \gets \TrapGen(1^n, 1^m, q)$.
      \item Define $\mathbf{G} = \mathbf{I}_n \otimes [1|2|\ldots |2^{k-1}] \in \ZZ_q^{n \times \bar{m}}$.  Sample matrices $\mathbf B \sample \U(\ZZ_q^{ n \times \bar{m}}) $,
			$ \mathbf U \sample \U(\Zq^{n \times m})$.
      \item Let $\mathsf{FRD}: \Zq^n \to \Zq^{n \times n}$ be the full-rank difference mapping from~\cite{ABB10}.
        \end{enumerate} Output
         $
          \mathsf{PP}= \bigl(\bar{\mathbf A}, \mathbf B,  \mathbf U \bigr)$ and $\textsf{msk} = \mathbf{T}_{\bar{\mathbf A}}$.

  \item[\textsf{Extract}$_\mathsf{PP}(\textsf{msk}, \ID)$:] Given $\textsf{msk} = \mathbf{T}_{\bar{\mathbf A}}$ and an identity $\ID \in \Zq^n$, do as follows: \smallskip
    \begin{enumerate}
      \item Define the matrix $\mathbf B_\ID = \mathbf B  + \mathsf{FRD}(\ID) \cdot \mathbf G \in \Zq^{n \times \bar{m}}$.
%\item Use $\mathbf T_A$ to compute a delegated basis $\mathbf T_\ID$ for the dual lattice of the matrix $\mathbf B_{\mathbf A, \ID} = \left[ \mathbf A \mid \mathbf B_\ID \right]$.
      \item Let $\mathbf B_{\mathbf A, \ID} = \left[ \mathbf A \mid \mathbf B_\ID \right] \in \ZZ_q^{n \times (m + \bar{m})}$, use $\mathbf T_A$ to compute a delegated basis $\mathbf T_\ID$ for the lattice $\Lambda^\perp(\mathbf B_{\mathbf A, \ID})$.
      \item Use $\mathbf T_\ID$ to sample a small-norm matrix $\mathbf E_\ID \in \ZZ^{(m+ \bar{m}) \times m}$ satisfying the equality $\mathbf B_{\mathbf A, \ID} \cdot \mathbf E_\ID = \mathbf U \bmod q$.
      \item Output $\sk_\ID = \mathbf E_\ID  \in \ZZ^{(m+ \bar{m}) \times m}$. \smallskip \smallskip
    \end{enumerate}
  \item[\textsf{Encrypt}$_\mathsf{PP}(\ID,\mathbf m)$:] Given an identity $\ID$ and a message $\mathbf m \in \bit^m$,   \smallskip
    \begin{enumerate}
      \item Compute the matrix $\mathbf B_\ID = \mathbf B + \mathsf{FRD}(\ID) \cdot \mathbf G \in \Zq^{n \times \bar{m}}$.
      Sample vectors $\mathbf s \sample \U(\Zq^n), \mathbf x, \mathbf y \sample \chi^m$, $\mathbf R \sample D_{\ZZ,\sigma}^{m \times \bar{m}}$ and compute
			$\mathbf z = \mathbf R^T \cdot \mathbf y \in \ZZ^m$.
      \item Compute
        \begin{equation} \label{eq:ABB-c}
          \begin{cases}
            \mathbf c^{(1)} = \bar{\mathbf A}^T \cdot \mathbf s + \mathbf y \bmod q,\\
            \mathbf c^{(2)} = \mathbf B_\ID^T \cdot \mathbf s + \mathbf z \bmod q,\\
            \mathbf c^{(3)} = \mathbf U^T \cdot \mathbf s + \mathbf x + \mathbf m \cdot \left\lfloor \dfrac{q}{2} \right\rfloor.
          \end{cases}
        \end{equation}
      \item Output $\mathbf c = \bigl(\mathbf c^{(1)},\mathbf c^{(2)},\mathbf c^{(3)}\bigr) \in \ZZ_q^m \times \ZZ_q^{\bar{m}} \times \ZZ_q^m$. \smallskip \smallskip
    \end{enumerate}
  \item[\textsf{Decrypt}$_\mathsf{PP}(\sk_\ID, \mathbf c)$:] Given $\sk_\ID = \mathbf E_\ID$ and  $\mathbf c=\bigl(\mathbf c^{(1)},\mathbf c^{(2)},\mathbf c^{(3)}\bigr)  \in \ZZ_q^m \times \ZZ_q^{\bar{m}} \times \ZZ_q^m$, compute and output
       % \begin{equation} \label{eq:ABB-dec}
       $   \mathbf m' = \left\lfloor \left( \mathbf c^{(3)} - \mathbf E_\ID \cdot \begin{bmatrix} \mathbf c^{(1)} \\ \mathbf c^{(2)} \end{bmatrix}  \right) \cdot \left\lfloor \dfrac{q}{2} \right\rfloor^{-1}  \right\rceil \in \bit^m .$
      %  \end{equation}
\end{description}
 \smallskip



\begin{theorem}[{\cite[Th. 23]{ABB10}}] \label{ABB-pseudorand-prop}
  The ABB IBE scheme has pseudo-random ciphertexts if the $\LWE_{n,q,\chi}$ assumption holds.
\end{theorem}

\section{Warm-up: Decompositions, Extensions, Permutations}
\label{se:decomposition-extensions-permutations}

This section introduces the notations and techniques that will be used throughout the chapter. It details Stern-like protocols that have been introduced in~\cref{sse:stern}. The techniques that will be employed for handling quadratic relations (double-bit extension $\mathsf{ext}(\cdot, \cdot)$, expansion $\expandtimes(\cdot, \cdot)$ of matrix-vector product and the associated permuting mechanisms) are novel contributions.
\subsection{Decompositions}\label{subsection:decomposition}
For any $B \in \ZZ_+$, define the number $\delta_B:=\lfloor \log_2 B\rfloor +1 = \lceil \log_2(B+1)\rceil$ and the sequence $B_1, \ldots, B_{\delta_B}$, where $B_j = \lfloor\frac{B + 2^{j-1}}{2^j} \rfloor$, $\forall j \in [1,\delta_B]$. As observed in~\cite{LNSW13}, the sequence satisfies $\sum_{j=1}^{\delta_B} B_j = B$ and
 any integer $v \in [0, B]$ can be decomposed into a binary vector $\mathsf{idec}_B(v) \hspace*{-1pt}= \hspace*{-1pt}(v^{(1)}, \ldots, v^{(\delta_B)})^T \hspace*{-2pt}\in \hspace*{-1pt}\{0,1\}^{\delta_B}$ such that $\sum_{j=1}^{\delta_B}B_j \cdot v^{(j)} \hspace*{-1pt}=\hspace*{-1pt} v$. We describe this decomposition procedure in a deterministic manner:
\begin{enumerate}
\item $v': = v$
\item For $j=1$ to $\delta_B$ do:
    \begin{enumerate}[(i)]
    \item If $v' \geq B_j$ then $v^{(j)}: = 1$, else $v^{(j)}: = 0$;
    \item $v': = v' - B_j\cdot v^{(j)}$.
    \end{enumerate}
\item Output $\mathsf{idec}_B(v) = (v^{(1)}, \ldots, v^{(\delta_B)})^T$.
\end{enumerate}

Next, for any positive integers $\mathfrak{m}, B$, we define the decomposition matrix:
\begin{eqnarray}
\mathbf{H}_{\mathfrak{m},B}: = \begin{bmatrix} B_1 \ldots  B_{\delta_B} &  & & & \\
  &  B_1 \ldots  B_{\delta_B} &   &  & \\
	  &   &  &  \ddots  &  \\
			  &   &  &    & B_1 \ldots  B_{\delta_B}  \\
\end{bmatrix} \in \mathbb{Z}^{\mathfrak{m} \times \mathfrak{m}\delta_B},
\end{eqnarray}
and the following injective functions:
\begin{enumerate}[(i)]
\item $\mathsf{vdec}_{\mathfrak{m}, B}: [0,B]^{\mathfrak{m}} \rightarrow \{0,1\}^{\mathfrak{m}\delta_B}$ that maps vector $\mathbf{v} = (v_1, \ldots, v_{\mathfrak{m}})^T$ to vector $\big(\mathsf{idec}_B(v_1)^T \| \ldots \| \mathsf{idec}_B(v_{\mathfrak{m}})^T\big)^T$. Note that $\mathbf{H}_{\mathfrak{m}, B} \cdot \mathsf{vdec}_{\mathfrak{m}, B}(\mathbf{v}) = \mathbf{v}$. \smallskip
\item $\mathsf{vdec}'_{\mathfrak{m}, B}: [-B,B]^{\mathfrak{m}} \rightarrow \{-1,0,1\}^{\mathfrak{m}\delta_B}$ that maps vector
 $\mathbf{w} = (w_1, \ldots, w_{\mathfrak{m}})^T$ to vector
    $\big(\sigma(w_1)\cdot\mathsf{idec}_B(w_1)^T \| \ldots \| \sigma(w_{\mathfrak{m}})\cdot\mathsf{idec}_B(w_{\mathfrak{m}})^T\big)^T$, where for each $i=1, \ldots, \mathfrak{m}$: $\sigma(w_i) = 0$ if $w_i =0$; $\sigma(w_i) = -1$ if $w_i <0$; $\sigma(w_i) = 1$ if $w_i >0$. Note that $\mathbf{H}_{\mathfrak{m}, B} \cdot \mathsf{vdec}'_{\mathfrak{m}, B}(\mathbf{w}) = \mathbf{w}$.
\end{enumerate}

We also define the following matrix decomposition procedure. For positive integers $n,m,q$, define the injective function $\mathsf{mdec}_{n,m,q}: \mathbb{Z}_q^{m \times n} \rightarrow \{0,1\}^{mn\delta_{q-1}}$ that maps matrix $\mathbf{X} = [\mathbf{x}_1 | \ldots | \mathbf{x}_n] \in \mathbb{Z}_q^{m \times n}$, where $\mathbf{x}_1, \ldots, \mathbf{x}_n \in \mathbb{Z}_q^m$, to vector
\begin{align*}
\mathsf{mdec}_{n,m,q}(\mathbf{X}) &= \big(\mathsf{vdec}_{m, q-1}(\mathbf{x}_1)^T \| \ldots \|\ \mathsf{vdec}_{m,q-1}(\mathbf{x}_n)^T\big)^T \\
  &= (x_{1,1}, \ldots, x_{1, mk}, x_{2,1}, \ldots, x_{2,mk}, \ldots, x_{n,1}, \ldots, x_{n,mk})^T \\
  & \hspace{.6\textwidth}\in \{0,1\}^{nm \delta_{q-1}},
\end{align*}
where, for each $(i,j) \in [n] \times [m \delta_{q-1}]$, $x_{i,j} \in \{0,1\}$ denotes the $j$-th bit of the  decomposition of the $i$-th column  of $\mathbf{X}$. \\ \indent Looking ahead,
when proving
knowledge of witnesses $(\mathbf{X},\mathbf{s}) \in \ZZ_q^{m \times n} \times \ZZ_q^{n}$ satisfying $\mathbf{b} = \mathbf{X} \cdot \mathbf{s} + \mathbf{e} \bmod q$,  we will have to consider terms of the form $x_{i,j} \cdot s_{i,t}$, where $\mathbf{s}=(s_1,\ldots,s_n)^T \in \ZZ_q^n$ and
 $(s_{i,1},\ldots,s_{i,\delta_{q-1}})^T=\mathsf{idec}_{q-1}(s_i)$ for each
$i \in [n]$.

\subsection{Extensions and Permutations}\label{subsection:warm-up-ext-perm}
We now introduce the extensions and permutations which will be essential for proving quadratic relations.
\begin{itemize}
    \item For each $c \in \{0,1\}$, denote by $\overline{c}$ the bit $1-c \in \{0,1\}$.
    \item For $c_1,c_2 \in \{0,1\}$, define the vector $$\mathsf{ext}(c_1,c_2) = (\overline{c}_1\cdot \overline{c}_2, \overline{c}_1\cdot {c}_2, {c}_1\cdot \overline{c}_2, c_1\cdot c_2)^T \in \{0,1\}^4.$$
    \item For  $b_1,b_2 \in \{0,1\}$, define the permutation $T_{b_1,b_2}$ that transforms vector $\mathbf{v} = (v_{0,0}, v_{0,1}, v_{1,0}, v_{1,1})^T \in \mathbb{Z}_q^4$ to vector $(v_{{b}_1, {b}_2}, v_{{b}_1, \overline{b}_2}, v_{ \overline{b}_1,{b}_2}, v_{\overline{b}_1, \overline{b}_2})^T$.

        Note that, for all $c_1, c_2, b_1, b_2 \in \{0,1\}$, we have the following:
        \begin{eqnarray}
         \mathbf{z} = \mathsf{ext}(c_1, c_2) \hspace*{5pt}\Longleftrightarrow \hspace*{5pt} T_{b_1, b_2}(\mathbf{z}) = \mathsf{ext}(c_1 \oplus b_1, c_2 \oplus b_2),
        \end{eqnarray}

\end{itemize}
where $\oplus$ denotes the bit-wise addition modulo $2$.

Now, for positive integers $n,m,k$, and for vectors $$\mathbf{x} = (x_{1,1}, \ldots, x_{1, mk}, x_{2,1}, \ldots, x_{2,mk}, \ldots, x_{n,1}, x_{n,mk})^T \in \{0,1\}^{nmk}$$
and $\mathbf{s}_0 = (s_{1,1}, \ldots, s_{1,k}, s_{2,1}, \ldots, s_{2,k}, \ldots, s_{n,1}, \ldots, s_{n,k})^T \in \{0,1\}^{nk}$, we define the vector $ \expandtimes (\mathbf{x}, \mathbf{s}_0) \in \{0,1\}^{4nmk^2}$ as
\begin{align*}
  \expandtimes (\mathbf{x}, \mathbf{s}_0) =
    &\bigl( \mathsf{ext}^T(x_{1,1}, s_{1,1}) \| \mathsf{ext}^T(x_{1,1}, s_{1,2}) \| \ldots \| \mathsf{ext}^T(x_{1,1}, s_{1,k}) \| \\
    &\| \mathsf{ext}^T(x_{1,2}, s_{1,1}) \| \mathsf{ext}^T(x_{1,2}, s_{1,2}) \| \ldots \| \mathsf{ext}^T(x_{1,2}, s_{1,k}) \| \ldots \\
    &\| \mathsf{ext}^T(x_{1,mk}, s_{1,1}) \| \mathsf{ext}^T(x_{1,mk}, s_{1,2}) \| \ldots \| \mathsf{ext}^T(x_{1,mk}, s_{1,k}) \| \\
    &\| \mathsf{ext}^T(x_{2,1}, s_{2,1}) \| \mathsf{ext}^T(x_{2,1}, s_{2,2}) \| \ldots \| \mathsf{ext}^T(x_{2,1}, s_{2,k}) \| \ldots \\
    &\|  \mathsf{ext}^T(x_{2,mk}, s_{2,1}) \| \mathsf{ext}^T(x_{2,mk}, s_{2,2}) \| \ldots \| \mathsf{ext}^T(x_{2,mk}, s_{2,k}) \| \ldots \\
    &\| \mathsf{ext}^T(x_{n,1}, s_{n,1}) \| \mathsf{ext}^T(x_{n,1}, s_{n,2}) \| \ldots \| \mathsf{ext}^T(x_{n,1}, s_{n,k}) \| \ldots \\
    &\|  \mathsf{ext}^T(x_{n,mk}, s_{n,1}) \| \mathsf{ext}^T(x_{n,mk}, s_{n,2}) \| \ldots \| \mathsf{ext}^T(x_{n,mk}, s_{n,k})
  \bigr)^T\hspace*{-2.5pt}.
\end{align*}
That is, $ \expandtimes (\mathbf{x}, \mathbf{s}_0)$ is obtained by  applying $\mathsf{ext}$ to all pairs of the form $(x_{i,j},s_{i,t})$ for $(i,j,t) \in [n] \times [mk] \times [k]$.


Now, for $\mathbf{b} = (b_{1,1}, \ldots, b_{1, mk}, b_{2,1}, \ldots, b_{2,mk}, \ldots, b_{n,1}, b_{n,mk})^T \in \{0,1\}^{nmk}$ and $\mathbf{d} = (d_{1,1}, \ldots, d_{1,k}, d_{2,1}, \ldots, d_{2,k}, \ldots, d_{n,1}, \ldots, d_{n,k})^T \in \{0,1\}^{nk}$, we define the permutation $P_{\mathbf{b}, \mathbf{d}}$ that transforms
vector
\begin{align*}
\mathbf{v} = &\big( (\mathbf{v}_{1,1,1}^T \| \ldots \| \mathbf{v}_{1,1, k}^T ) \| ( \mathbf{v}_{1,2,1}^T \| \ldots \| \mathbf{v}_{1,2,k}^T ) \| \ldots \| ( \mathbf{v}_{1,mk,1}^T \| \ldots \| \mathbf{v}_{1,mk,k}^T ) \| \\
&~  (\mathbf{v}_{2,1,1}^T \| \ldots \| \mathbf{v}_{2,1, k}^T ) \|  (\mathbf{v}_{2,2,1}^T \| \ldots \| \mathbf{v}_{2,2,k}^T ) \| \ldots \| ( \mathbf{v}_{2,mk,1}^T \| \ldots \| \mathbf{v}_{2,mk,k}^T  ) \| \\
&~ \hspace*{-25pt}  (\mathbf{v}_{n,1,1}^T \| \ldots \| \mathbf{v}_{n,1, k}^T ) \| ( \mathbf{v}_{n,2,1}^T \| \ldots \| \mathbf{v}_{n,2,k}^T ) \| \ldots \| ( \mathbf{v}_{n,mk,1}^T \| \ldots \| \mathbf{v}_{n,mk,k}^T )
\big)^T  \hspace*{-3.5pt}\in \hspace*{-1.5pt}\mathbb{Z}^{4nmk^2},
\end{align*}
consisting of $nmk^2$ blocks of length $4$, to the vector $P_{\mathbf{b}, \mathbf{d}}(\mathbf{v})$ of the form
\begin{align*}
\big(~& (\mathbf{w}_{1,1,1}^T \| \ldots \| \mathbf{w}_{1,1, k}^T ) \| ( \mathbf{w}_{1,2,1}^T \| \ldots \| \mathbf{w}_{1,2,k}^T ) \| \ldots \| ( \mathbf{w}_{1,mk,1}^T \| \ldots \| \mathbf{w}_{1,mk,k}^T ) \| \\
& ( \mathbf{w}_{2,1,1}^T \| \ldots \| \mathbf{w}_{2,1, k}^T ) \| ( \mathbf{w}_{2,2,1}^T \| \ldots \| \mathbf{w}_{2,2,k}^T ) \| \ldots \| ( \mathbf{w}_{2,mk,1}^T \| \ldots \| \mathbf{w}_{2,mk,k}^T ) \| \\
& (\mathbf{w}_{n,1,1}^T \| \ldots \| \mathbf{w}_{n,1, k}^T ) \|  (\mathbf{w}_{n,2,1}^T \| \ldots \| \mathbf{w}_{n,2,k}^T ) \| \ldots \|  (\mathbf{w}_{n,mk,1}^T \| \ldots \| \mathbf{w}_{n,mk,k}^T )
~ \big)^T,
\end{align*}
where for each $(i,j,t) \in [n]\times [mk] \times [k]$: \hspace*{2.5pt}$\mathbf{w}_{i,j,t} = T_{b_{i,j}, d_{i,t}}(\mathbf{v}_{i,j,t})$.
\smallskip

Observe that, for all $\mathbf{b} \in \{0,1\}^{nmk}, \mathbf{d} \in \{0,1\}^{nk}$, we have:
\begin{eqnarray}\label{eq:expand-permuting}
\mathbf{z} = \expandtimes (\mathbf{x}, \mathbf{s}_0) \hspace*{5pt}\Longleftrightarrow \hspace*{5pt} P_{\mathbf{b},\mathbf{d}}(\mathbf{z}) = \expandtimes (\mathbf{x} \oplus \mathbf{b}, \mathbf{s}_0 \oplus \mathbf{d}).
\end{eqnarray}
\smallskip

\noindent
Next, we recall the notations, extensions and permutations used in previous Stern-like protocols~\cite{LNSW13,LNW15,ELL+15,LLM+16} for proving linear relations.

For any positive integer $t$, denote by $\permutations_t$ the symmetric group of all permutations of~$t$ elements, by $\mathsf{B}_{2t}$ the set of all vectors in $\{0,1\}^{2t}$ having Hamming weight~$t$, and by $\mathsf{B}_{3t}$ the set of all vectors in $\{-1,0,1\}^{3t}$ having exactly $t$ coordinates equal to $j$, for each $j \in \{-1,0,1\}$.
Note that for any $\phi \in \permutations_{2t}$ and $\psi\in \permutations_{3t}$, we have the following equivalences:
\begin{eqnarray}\label{eq:permuting-B_2t_B_3t}
\mathbf{x} \in \mathsf{B}_{2t} \Longleftrightarrow \phi(\mathbf{x}) \in \mathsf{B}_{2t} \hspace*{7.5pt}\text{  and  }\hspace*{7.5pt} \mathbf{y} \in \mathsf{B}_{3t}  \Longleftrightarrow \psi(\mathbf{y}) \in \mathsf{B}_{3t}.
\end{eqnarray}
The following extending procedures are defined for any positive integers $t$.
\begin{itemize}
  \item $\mathsf{ExtendTwo}_t: \{0,1\}^{t} \rightarrow \mathsf{B}_{2t}$. On input vector $\mathbf{x}$ with Hamming weight $w$, it outputs
    \[\mathbf{x}' = (\mathbf{x}^T \| \mathbf{1}^{t-w} \| \mathbf{0}^{w})^T. \]
  \item $\mathsf{ExtendThree}_t: \{-1,0,1\}^{t} \rightarrow \mathsf{B}_{3t}$. On input vector $\mathbf{y}$ containing $n_j$ coordinates equal to $j$ for $j \in \{-1,0,1\}$, this procedure outputs the vector
    \[\mathbf{y}' = (\mathbf{y}^T \| \mathbf{1}^{t-n_1}
    \| \mathbf{0}^{t-n_0} \| \mathbf{(-1)}^{t-n_{-1}}).\]
\end{itemize}
We also use the following encoding and permutation to achieve fine-grained control over coordinates of binary witness-vectors.
\begin{itemize}
    \item For any positive integer $t$, define the function $\mathsf{encode}_t$ that encodes vector $\mathbf{x} = (x_1, \ldots, x_t)^T\in \{0,1\}^t$ to vector $\mathsf{encode}_t(\mathbf{x}) = (\bar{x}_1, x_1, \ldots, \bar{x}_t, x_t)^T \in \{0,1\}^{2t}$.
    \item For any positive integer $t$ and any vector $\mathbf{c} = (c_1, \ldots, c_t)^T \in \{0,1\}^t$, define the permutation $F_{\mathbf{c}}^{(t)}$ that transforms vector $\mathbf{v} = (v_1^{(0)}, v_1^{(1)}, \ldots, v_t^{(0)}, v_t^{(1)})^T \in \ZZ^{2t}$ into vector $F_{\mathbf{c}}^{(t)}(\mathbf{v}) = (v_1^{(c_1)}, v_1^{(\bar{c}_1)}, \ldots, v_t^{(c_t)}, v_t^{(\bar{c}_t)})^T$.
\end{itemize}
Note that the following equivalence holds for all $t, \mathbf{c}$:
\begin{eqnarray}\label{eq:equivalence-encoding}
\mathbf{y} = \mathsf{encode}_t(\mathbf{x}) \hspace*{7.5pt}\Longleftrightarrow \hspace*{7.5pt} F_{\mathbf{c}}^{(t)}(\mathbf{y}) = \mathsf{encode}_t(\mathbf{x} \oplus \mathbf{c}).
\end{eqnarray}

To close this warm-up section, we remark that the equivalences observed in~\eqref{eq:expand-permuting}, \eqref{eq:permuting-B_2t_B_3t} and~\eqref{eq:equivalence-encoding} will play crucial roles in our zero-knowledge layer.

\section{The Supporting Zero-Knowledge Layer} \label{groupenc-zk-layer}

In this section, we first demonstrate how to prove in zero-knowledge that a given vector $\mathbf{b}$ is a correct \textsf{LWE} evaluation, i.e., $\mathbf{b} = \mathbf{X}\cdot \mathbf{s} + \mathbf{e} \bmod q$, where the hidden matrix $\mathbf{X}$ and vector $\mathbf{s}$ may satisfy additional conditions.

This sub-protocol, which we believe will have other applications, is one of the major challenges in our road towards the design of lattice-based group encryption. We then plug  this building block into the big picture as described in~\cref{sse:stern}, and construct the supporting zero-knowledge argument of knowledge (\textsf{ZKAoK}) for our group encryption scheme (Section~\ref{groupenc-scheme}).

\subsection{Proving the LWE Relation With Hidden Matrices}\label{subsection:quadratic-relation}
Let $n,m,q, \beta$ be positive integers where $\beta \ll q$, and let $k = \delta_{q-1}= \lceil \log_2 q\rceil$. We identify $\Zq$ as the set $\{0,1, \ldots, q-1\}$.
We consider a zero-knowledge argument system allowing prover $\mathcal{P}$ to convince verifier $\mathcal{V}$ on input $\mathbf{b} \in \Zq^m$ that $\mathcal{P}$ knows secret matrix $\mathbf{X} \in \Zq^{m \times n}$, and vectors $\mathbf{s} \in \Zq^n$, $\mathbf{e} \in [-\beta, \beta]^m$ such that:
\begin{eqnarray}\label{eq:quadratic-LWE-original}
\mathbf{b} = \mathbf{X}\cdot \mathbf{s} + \mathbf{e} \bmod q.
\end{eqnarray}
Moreover, the argument system should be readily extended to proving that $\mathbf{X}$ and $\mathbf{s}$ satisfy additional conditions, such as:
\begin{itemize}
    \item The bits representing $\mathbf{X}$ are certified by an authority, and the prover also knows that secret signature-certificate.
    \item The (secret) hash of $\mathbf{X}$ is correctly encrypted to a given ciphertext.
    \item The \textsf{LWE} secret $\mathbf{s}$ is involved in other linear equations.
\end{itemize}

Let $q_1, \ldots, q_k \in \Zq$ be   the sequence of integers obtained by decomposing  $q-1$ using the technique  recalled in
Section \ref{subsection:decomposition}, and define the row vector $\mathbf{g} = (q_1, \ldots, q_k)$.
Let $\mathbf{X} = [\mathbf{x}_1 | \ldots | \mathbf{x}_n] \in \Zq^{m \times n}$ and $\mathbf{s}= (s_1, \ldots, s_n)^T$.
For each index $i \in [n]$, let  us consider $\mathsf{vdec}_{m,q-1}(\mathbf{x}_i) = (x_{i,1}, \ldots, x_{i,mk})^T \in \{0,1\}^{mk}$.
Let
\[ \mathsf{vdec}_{n,q-1}(\mathbf{s})= (s_{1,1}, \ldots, s_{1,k}, s_{2,1}, \ldots, s_{2,k}, \ldots, s_{n,1}, \ldots s_{n,k})^T \in \{0,1\}^{nk} \]
and observe that $s_i = \mathbf{g} \cdot \mathsf{idec}_{q-1}(s_i)= \mathbf{g}\cdot (s_{i,1}, \ldots, s_{i,k})^T$  for each $i \in [n]$.
We   have:
\begin{eqnarray*}
\mathbf{X}\cdot \mathbf{s}  &=& \sum_{i=1}^n \mathbf{x}_i\cdot s_i = \sum_{i=1}^n \mathbf{H}_{m,q-1}\cdot \mathsf{vdec}_{m,q-1}(\mathbf{x}_i)\cdot s_i \\
&=& \mathbf{H}_{m,q-1}\cdot \Big(\sum_{i=1}^n (x_{i,1}\cdot s_i, \ldots, x_{i,mk}\cdot s_i)^T\Big) \bmod q.
\end{eqnarray*}
Observe that, for each $i \in [n]$ and each $j \in [mk]$, we have
\begin{align*}
x_{i,j}\cdot s_i &= x_{i,j}\cdot \mathbf{g} \cdot (s_{i,1}, \ldots, s_{i,k})^T \\
                 &= (q_1, \ldots, q_k) \cdot (x_{i,j}\cdot s_{i,1}, \ldots, x_{i,j}\cdot s_{i,k})^T.
\end{align*}
We now extend vector $(q_1, q_2, \ldots, q_k)$ to $\mathbf{g}' \hspace*{-1.5pt}=\hspace*{-1.5pt} (0,0,0,q_1, 0,0,0, q_2, \ldots, 0,0,0,q_k) \in \Zq^{4k}$.
For all $(i,j) \in [n]\times [mk]$, we have:
$$
x_{i,j}\cdot s_i = \mathbf{g}' \cdot (\mathsf{ext}^T(x_{i,j}, s_{i,1}) \| \ldots \| \mathsf{ext}^T(x_{i,j},s_{i,k}))^T.
$$
Let us define the matrices
\begin{eqnarray} \label{Q0-def}
  \mathbf{Q}_0: = \mathbf{I}_{mk}\otimes \mathbf{g}' =  \begin{bmatrix} \mathbf{g}' &  & & & \\
    &  \mathbf{g}' &   &  & \\
    &   &  &  \ddots  &  \\
    &   &  &    & \mathbf{g}'  \\
  \end{bmatrix} \in \Zq^{mk \times 4mk^2},
\end{eqnarray}
and $\widehat{\mathbf{Q}} = [\overbrace{\mathbf{Q}_0 | \ldots | \mathbf{Q}_0}^{n \text{ times  }}] \in \Zq^{mk \times 4nmk^2}$. For each $i \in [n]$, define
\begin{align*}
  \mathbf{y}_i = \bigl( &\mathsf{ext}^T(x_{i,1}, s_{i,1}) \| \ldots \| \mathsf{ext}^T(x_{i,1},s_{i,k}))^T \| \mathsf{ext}^T(x_{i,2},s_{i,1}) \| \ldots \| \mathsf{ext}^T(x_{i,2}, s_{i,k}) \\
    & \| \ldots \|\mathsf{ext}^T(x_{i,mk},s_{i,1} \| \ldots \| \mathsf{ext}^T(x_{i,mk}, s_{i,k}) \bigr)^T \in \{0,1\}^{4mk^2}.
  \end{align*}
Then, for all $i \in [n]$, we have:
$
(x_{i,1}\cdot s_i, \ldots, x_{i,mk}\cdot s_i)^T = \mathbf{Q}_0 \cdot \mathbf{y}_i.
$
Now, we note that $$(\mathbf{y}_1^T \| \ldots \| \mathbf{y}_n^T)^T = \expandtimes \bigl(\mathsf{mdec}_{n,m,q}(\mathbf{X}), \mathsf{vdec}_{n,q-1}(\mathbf{s})\bigr),$$
and
\begin{multline} \label{almost}
 \sum_{i=1}^n (x_{i,1}\cdot s_i, \ldots, x_{i,mk}\cdot s_i)^T \\ = \sum_{i=1}^n \mathbf{Q}_0 \cdot \mathbf{y}_i  = \widehat{\mathbf{Q}}\cdot \expandtimes \bigl(\mathsf{mdec}_{n,m,q}(\mathbf{X}), \mathsf{vdec}_{n,q-1}(\mathbf{s}) \bigr). \qquad
\end{multline}

Letting $\mathbf{Q}= \mathbf{H}_{m,q-1}\cdot \widehat{\mathbf{Q}}  \in \Zq^{m \times 4nmk^2}$ and left-multiplying~\eqref{almost}  by $ \mathbf{H}_{m,q-1}$, we obtain the equation:
\[
\mathbf{X} \cdot \mathbf{s} = \mathbf{Q}\cdot
 \expandtimes \bigl(\mathsf{mdec}_{n,m,q}(\mathbf{X}), \mathsf{vdec}_{n,q-1}(\mathbf{s})\bigr) \bmod q.
\]
\begin{comment}
\begin{table}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
  \hline
  $x_1$ & $x_2$ & $b_1$ & $b_2$ & ~$\mathsf{ext}(x_1,x_2)$~ & ~$T_{b_1,b_2}(\mathsf{ext}(x_1,x_2))$~ & ~$x_1 \oplus b_1$~& ~$x_2 \oplus b_2$~ &~$\mathsf{ext}(x_1 \oplus b_1, x_2 \oplus b_2)$~\\
  \hline
  \rule{0pt}{3ex}
  $0$ & $0$ & $0$ & $0$ & $(1000)^T$ & $(1000)^T$ & $0$ & $0$ & $(1000)^T$ \\[5pt]

  $0$ & $0$ & $0$ & $1$ & $(1000)^T$ & $(0100)^T$ & $0$ & $1$ & $(0100)^T$ \\[5pt]

  $0$ & $0$ & $1$ & $0$ & $(1000)^T$ & $(0010)^T$ & $1$ & $0$ & $(0010)^T$ \\[5pt]

  \hline
\end{tabular}
\end{table}
\end{comment}

This means that the task of proving knowledge of $(\mathbf{X},\mathbf{s},\mathbf{e}) \in \Zq^{m \times n} \times \Zq^n \times [-\beta,\beta]^m$ such that $\mathbf{b}=\mathbf{X} \cdot \mathbf{s} + \mathbf{e} \bmod q$
boils down to proving knowledge of $\mathbf{z} \in \{0,1\}^{4nmk^2}$, $\mathbf{x} \in \{0,1\}^{nmk}$, $\mathbf{s}_0 \in \{0,1\}^{nk}$  and a short $\mathbf{e} \in \ZZ^m$  such that
\[
\mathbf{b} = \mathbf{Q}\cdot \mathbf{z} + \mathbf{I}_m \cdot \mathbf{e} \bmod q
\qquad \mbox{ and } \qquad
\mathbf{z}  =  \expandtimes (\mathbf{x},\mathbf{s}_0).
\]

As the knowledge of small-norm vector $\mathbf{e}$ can easily be proven with Stern-like protocol (e.g.,~\cite{LNSW13}), the challenging part is to prove in zero-knowledge the constraint ``$\mathbf{z}  =  \expandtimes (\mathbf{x},\mathbf{s}_0)$''.
To this end, we will use the following permuting technique inspired by the equivalence of equation~\eqref{eq:expand-permuting}.
We sample uniformly random $\mathbf{d}_x \in \{0,1\}^{n{m}k}$ and $\mathbf{d}_s \in \{0,1\}^{nk}$, send $\mathbf{x}' = \mathbf{x} \oplus \mathbf{d}_x$ and $\mathbf{s}' = \mathbf{s}_0 \oplus \mathbf{d}_s$ to the verifier, and let the latter check that $P_{\mathbf{d}_x, \mathbf{d}_s}(\mathbf{z}) = \expandtimes(\mathbf{x}', \mathbf{s}')$.
This will be sufficient to convince the verifier that the original vector $\mathbf{z}$ satisfies the required constraint.
The crucial point is that no additional information about $\mathbf{x}$ and $\mathbf{s}_0$ is leaked, since these binary vectors are perfectly hidden under the ``one-time pad'' $\mathbf{d}_x$ and $\mathbf{d}_s$, respectively.

In the framework of Stern's protocol, the idea of using ``one-time-pad'' permutations further allows us to prove that $\mathbf{x}$ and $\mathbf{s}_0$ satisfy additional conditions, i.e., they appear in other equations.
This is done by first setting up an equivalence similar to~\eqref{eq:expand-permuting} in the places where these objects appear, and then, using the same ``one-time pad'' for each of them in all appearances.
We will explain in detail how   this technique can be realized in the next subsection.

%**************************************************
\section{Our Lattice-Based Group Encryption Scheme} \label{groupenc-scheme}

To build a $\mathsf{GE}$ scheme using our zero-knowledge argument system, we need to choose a specific key-private CCA2-secure encryption scheme.
The first idea is to use the CCA2-secure public-key cryptosystem which is implied by the Agrawal-Boneh-Boyen  identity-based encryption (IBE) scheme \cite{ABB10} (which is
recalled in \cref{ap:ABB-IBE}) via the Canetti-Halevi-Katz (CHK) transformation \cite{CHK04}.
The ABB scheme is a natural choice since it has pseudo-random ciphertexts  (which  implies  the key-privacy \cite{BBDP01} when the CHK paradigm
is applied)  and provides
one of the most efficient CCA2 cryptosystem based on the hardness of  $\LWE$ in the standard model.
One difficulty is that  the Kiayias-Tsiounis-Yung model \cite{KTY07} requires
that certified  public keys be   valid public keys (i.e., which have a matching secret key).
When new group members join the system and request a certificate for their
public key $\mathbf{B}_{\mathsf{U}} \in \Zq^{n \times \bar{m}}$,  a direct use of the ABB/CHK technique would incur of proof
of existence of  a  GPV trapdoor  \cite{GPV08} corresponding to $\mathbf{B}_{\mathsf{U}}$ (i.e., a small-norm matrix $\mathbf{T}_{\mathbf{B}_{\mathsf{U}}} \in \ZZ^{\bar{m} \times \bar{m} } $ s.t.
$\mathbf{B} \cdot \mathbf{T}_{\mathbf{B}_{\mathsf{U}}}= \mathbf{0}^n \bmod q$).
While the techniques of Peikert and Vaikuntanathan
 \cite{PV08} would provide a solution to this problem (as they allow proving that  $\mathbf{T}_{\mathbf{B}_{\mathsf{U}}} \in \ZZ^{\bar{m} \times \bar{m}} $  has full-rank),
we found it simpler  to rely on the trapdoor mechanism
of Micciancio and Peikert \cite{MP12}.

If we assume public parameters containing  a random matrix $\bar{\mathbf{A}} \in \Zq^{n \times m}$, each user's public key
can consist of a matrix $\mathbf{B}_{\mathsf{U}} = \bar{\mathbf{A}} \cdot \mathbf{T}_{\mathsf{U}} \in \Zq^{n \times \bar{m}}$, where $\mathbf{T}_{\mathsf{U}} \in \ZZ^{m \times \bar{m}}$ is a
small-norm matrix whose calms are sampled from a discrete Gaussian distribution. Note that, if $\bar{\mathbf{A}} \in \Zq^{n \times m}$ is uniformly distributed, then
\cite[Lemma 5.1]{GPV08} ensures that, with overwhelming probability, any
matrix $\mathbf{B}_{\mathsf{U}} \in \Zq^{n \times \bar{m}}$ has an underlying small-norm matrix satisfying $\mathbf{B}_{\mathsf{U}} = \bar{\mathbf{A}} \cdot \mathbf{T}_{\mathsf{U}}  \bmod q $. This simplifies
the joining procedure by eliminating the need for proofs of public key validity.

In the encryption algorithm,  the sender computes a dual Regev encryption~\cite{GPV08} of the witness $\mathbf{w} \in \{0,1\}^m$  using a matrix
$[\bar{\mathbf{A}} ~|~ \mathbf{B}_\mathsf{U} + \mathsf{FRD}(\vk) \cdot \mathbf{G} ] \in \Zq^{n \times (m + \bar{m})}$ such that: (i) $\vk \in \Zq^n$ is the verification key of a one-time signature; (ii)
$\mathsf{FRD} : \Zq^n \rightarrow \Zq^{n \times n}$ is the full-rank difference\footnote{This means that, for any two distinct one-time verification keys $\vk ,\vk' \in \Zq^n$, the difference
$\mathsf{FRD}(\vk) - \mathsf{FRD}(\vk') \in \Zq^{n \times n}$ is invertible over $\Zq$.}   function of \cite{ABB10};
 (iii) $\mathbf{G} = \mathbf{I}_n \otimes [1|2| \ldots |2^{k-1}] \in \Zq^{n \times \bar{m}}$ is the gadget matrix of \cite{MP12}. Given that $\mathbf{G} $ has a publicly known
trapdoor allowing to sample short vectors in $\Lambda_q^{\perp}(\mathbf{G})$, the user can use his private key $\mathbf{T}_{\mathsf{U}} \in \ZZ^{m \times \bar{m}}$ to decrypt
by running the $\mathsf{SampleRight}$ algorithm of Lemma \ref{lem:sampler}.

Having encrypted the witness $\mathbf{w} \in \{0,1\}^m$ by running the ABB encryption algorithm, the sender proceeds by encrypting a hash value of $\mathbf{B}_{\mathsf{U}} \in \Zq^{n \times \bar{m}}$ under the public key $\mathbf{B}_{\OA} = \bar{\mathbf{A}} \cdot \mathbf{T}_{\OA} \in \Zq^{n \times \bar{m}}$ of the opening authority.  The latter hash value
is obtained as a bit-wise decomposition of $\mathbf{F} \cdot \mathsf{mdec}_{n,m,q}(\mathbf{B}_{\mathsf{U}}^T) \in \Zq^{2n}$, where $\mathbf{F} \in \Zq^{2n \times n \bar{m} \lceil \log q \rceil}$
is a random public matrix and $\mathsf{mdec}_{n,m,q}(\mathbf{B}_{\mathsf{U}}^T) \in \{0,1\}^{n \bar{m} \lceil \log q \rceil}$ denotes an entry-wise binary decomposition of the matrix $\mathbf{B}_{\mathsf{U}} \in \Zq^{n \times \bar{m}}$.

By combining our new argument for quadratic relations and the extensions of Stern's protocol suggested in \cite{LNW15,LLM+16},
we are able to prove that some component of the ciphertext is of the form $\mathbf{c}=\mathbf{B}_{\mathsf{U}}^{T} \cdot \mathbf{s} + \mathbf{e} \in \Zq^{\bar{m}}$, for some $\mathbf{s} \in \Zq^n$
 and a small-norm $\mathbf{e} \in \ZZ^{\bar{m}}$ while also arguing possession of a signature on the binary decomposition
$\mathsf{mdec}_{n,m,q}(\mathbf{B}_{\mathsf{U}}^T) \in \{0,1\}^{n \bar{m} \lceil \log q \rceil}$ of $\mathbf{B}_{\mathsf{U}}^T$. For this purpose, we use a variant of a signature scheme
due to B\"ohl \textit{et al.}'s
signature \cite{BHJ+15}  which was described in \cref{ch:gs-lwe}
(and of which a description is given in \cref{se:gs-lwe-sigep}).
 At the same time, the prover $\mathcal{P}$ can also
argue that a hash value of $\mathsf{mdec}_{n,m,q}(\mathbf{B}_{\mathsf{U}}^T)  $ is properly
encrypted under the $\OA$'s public key using the ABB encryption scheme.




\subsection{Description of the Scheme}

Our $\mathsf{GE}$ scheme allows encrypting witnesses for the \ISIS relation (as in \cref{de:sis}) $ \mathrm{R}_{\ISIS}(n,m,q,1)$, which
 consists of pairs $((\mathbf{A}_R, \mathbf{u}_R), \mathbf{w}) \in (\Zq^{n \times m} \times \Zq^n) \times \{0,1\}^m $ satisfying $\mathbf{u}_R=\mathbf{A}_R \cdot \mathbf{w} \bmod q$.
This relation is in the same spirit as the one of Kiayias, Tsiounis and Yung \cite{KTY07}, who consider the verifiable encryption of discrete logarithms. While the construction of
\cite{KTY07} allow verifiably encrypting discrete-logarithm-type secret keys under the public key of some anonymous trusted third party, our construction makes it possible to encrypt GPV-type secret keys \cite{GPV08}.   \smallskip







\begin{description}
  \item[$\mathsf{SETUP_{init}}(1^\lambda)$:] This algorithm performs the following: \smallskip
  \begin{itemize}
    \item[1.] Choose integers $n = \mathcal{O}(\lambda)$, prime $q = \widetilde{\mathcal{O}}(n^4)$, and let $k = \lceil \log_2 q\rceil$, $\bar{m}=nk$ and $m =2\bar{m}= 2nk$. Choose a $B$-bounded distribution
		$\chi$ over $\ZZ$ for some $B= \sqrt{n}\omega(\log n)$.
    \item[2.] Choose a Gaussian parameter $\sigma = \Omega(\sqrt{n \log q}\log n)$. Let $\beta = \sigma\omega(\log n)$ be the upper bound of samples from $D_{\mathbb{Z}, \sigma}$.
    \item[3.] Select integers $\ell = \ell(\lambda)$ which determines the maximum expected group size $2^\ell$, and $\kappa = \omega(\log \lambda)$ (the number of protocol repetitions).
   % \item  Pick $3$ hash functions to be modelled as random oracles: $$\mathcal{H}_1: \{0,1\}^* \rightarrow \mathbb{Z}_q^{n \times L}, \mathcal{H}_2: \{0,1\}^* \rightarrow \mathbb{Z}_q^{n \times n}, \mathcal{H}_\textsf{FS}: \{0,1\}^* \rightarrow \{1,2,3\}^\kappa.$$
    \item[4.] Select a strongly unforgeable one-time signature $\mathcal{OTS} = (\mathsf{Gen}, \mathsf{Sig}, \mathsf{Ver})$. We assume that the verification keys live in $\mathbb{Z}_q^n$.
		\item[5.] Select public parameters $\compar$ for a statistically-hiding commitment scheme like \cite{KTX08}. This commitment will serve as a building block for the zero-knowledge argument
		system used in $\langle \mathcal{P}, \mathcal{V} \rangle $.
    \item[6.] Let $\mathsf{FRD}: \mathbb{Z}_q^{n} \rightarrow \mathbb{Z}_q^{n \times n}$ be the full-rank difference mapping from~\cite{ABB10}.
    \item[7.] Pick a random matrix $\mathbf{F} \leftarrow \mathbb{Z}_q^{2n \times n \bar{m}k}$, which will be used to hash users' public keys from $\Zq^{n \times \bar{m}}$ to $\mathbb{Z}_q^n$.
%    \item[7.] Pick matrices $\mathbf{A}_0,\mathbf{A}_1,\ldots,\mathbf{A}_{\ell}, \mathbf{D}_1 \xleftarrow{\$} \Zq^{n \times m}$, $\mathbf{D}, \mathbf{D}_0 \xleftarrow{\$} \mathbb{Z}_q^{n \times %nk}$ and vector $\mathbf{u} \xleftarrow{\$} \Zq^n$. These objects will be used for verifying the membership certificates issued by GM.
    \item[8.] Let $\mathbf{G} \in \Zq^{n \times \bar{m}}$ be the gadget matrix $\mathbf{G}= \mathbf{I}_n \otimes \begin{bmatrix} 1 & 2 & \ldots & 2^{k-1} \end{bmatrix}$ of \cite{MP12}.  Pick matrices $\bar{\mathbf{A}},  \mathbf{U} \sample \U(\mathbb{Z}_q^{n \times m})$ and $\mathbf{V} \sample \U(\mathbb{Z}_q^{n \times m})$. Looking ahead, $\mathbf{U}$ will be used to encrypt for the receiver while $\mathbf{V}$ will be used
			to encrypt the user's public key under the $\OA$'s public key. As for $\bar{\mathbf{A}}$, it will be used in two instances of the ABB encryption scheme \cite{ABB10}. \smallskip \smallskip
		  \end{itemize}
   Output
    \begin{eqnarray*} %\label{eq:param}
    \param &=& \big\{\lambda, n, q, k, m, B, \chi, \sigma, \beta, \ell, \kappa,  \mathcal{OTS}, \compar, \mathsf{FRD},   \bar{\mathbf{A}}, \mathbf{G}, \mathbf{F}, \mathbf{U}, \mathbf{V}  \big\}.
		%\\ & & ~ \mathbf{F}, \{\mathbf{A}_i\}_{i=0}^\ell,\mathbf{D}, \mathbf{D}_0, \mathbf{D}_1,\mathbf{u}, \mathbf{B}_0, \mathbf{B}_1, \mathbf{G}_{re}, \mathbf{G}_{op}  \big\}.
    \end{eqnarray*}
\item[$\langle
    \mathcal{G}_r, \mathsf{sample}_{R}
  \rangle$:]  Algorithm $\mathcal{G}_r(1^\lambda,1^n,1^m)$ proceeds by sampling   a   random matrix $\mathbf{A}_R \sample \U(\Zq^{n \times m})$ and outputting
	$(\pk_{R},\sk_{R})=(\mathbf{A}_R,\varepsilon)$. On input of a public key
	$\pk_{R}=\mathbf{A}_R \in \Zq^{n \times m}$ for the relation $\mathrm{R}_{\ISIS}$, algorithm
	$\mathsf{sample}_{R}$ picks  $\mathbf{w} \sample \U(\{0,1\}^m)$ and outputs a pair $((\mathbf{A}_R,\mathbf{u}_R),\mathbf{w})$, where $\mathbf{u}_R =\mathbf{A}_R \cdot \mathbf{w} \in \Zq^n$.


\item[$\mathsf{SETUP_{\GM}}(\param)$:] The $\GM$ generates   $(\sk_\GM,\pk_\GM) \leftarrow \mathsf{Keygen}(1^\lambda,q,n,m,\ell,\sigma)$ as a key pair  for the $\SIS$-based signature scheme of \cite{LLM+16} (as recalled in \cref{se:gs-lwe-sigep}).  This key pair
consists of $\sk_{\GM}:=
 \mathbf{T}_{\mathbf{A}}   $ and
\begin{eqnarray} \label{PK-GM}
\hspace*{-25pt} \pk_{\GM}:=\Bigl( \mathbf{A}, \mathbf{A}_0,\ldots, \mathbf{A}_{\ell} \in \Zq^{n \times m},
  ~  \mathbf{D}_0  , \mathbf{D}_1 \in \Zq^{n \times m}, \mathbf{D} \in \Zq^{n \times \bar{m}},  \mathbf{u} \in \Zq^n \Bigr).
	\end{eqnarray}


\item[$\mathsf{SETUP_{\OA}}(\param)$:] The $\OA$ samples a small-norm matrix $\mathbf{T}_{\OA} \leftarrow D_{\ZZ^m,\sigma}^{\bar{m}}$  in $\ZZ^{m \times \bar{m}}$ to obtain a statistically uniform $\mathbf{B}_{\OA} = \bar{\mathbf{A}}
 \cdot \mathbf{T}_{\OA}
 \in \Zq^{n \times \bar{m}}$. The $\OA$'s key pair consists of
$(\sk_{\OA},\pk_{\OA})=(\mathbf{T}_{\OA},\mathbf{B}_{\OA})$.
  \medskip

  \item[$\mathsf{JOIN}$:] The prospective user $\mathsf{U}$ and the $\GM$ interact in the following   protocol. \smallskip  \smallskip
	\begin{itemize}
	\item[1.]
	$\mathsf{U}$  first samples $\mathbf{T}_{\mathsf{U}} \leftarrow D_{\ZZ^m,\sigma}^{\bar{m}} $ in $\ZZ^{m \times \bar{m}}$ to compute a statistically uniform matrix $\mathbf{B}_{\mathsf{U}} = \bar{\mathbf{A}} \cdot \mathbf{T}_{\mathsf{U}} \in \Zq^{n \times \bar{m}}$.
	The prospective user defines his key pair as
	$(\pk_{\mathsf{U}},\sk_{\mathsf{U}})=(\mathbf{B}_{\mathsf{U}},\mathbf{T}_{\mathsf{U}})$ and sends $\pk_{\mathsf{U}}=\mathbf{B}_{\mathsf{U}}$ to the $\GM$.
	%In addition, $\mathsf{U}$ proves to $\GM$ that $\mathbf{B}_{\mathsf{U}} \in \Zq^{n \times m} $ is
	%	a valid public key by demonstrating possesion of a short basis of  $\Lambda_q^{\perp}(\mathbf{B}_{\mathsf{U}})$. This can be achieved via \cite{PV08}.
	\smallskip
	\item[2.] Upon receiving a public key $\mathsf{pk}_{\mathsf{U}} = \mathbf{B}_{\mathsf{U}} \in \mathbb{Z}_q^{n \times \bar{m}}$ from the user,  the $\GM$ certifies $\pk_U$ via the following steps:
	 \smallskip
     \begin{enumerate}
        \item[a.] Compute  $\mathbf{h}_{\mathsf{U}} = \mathbf{F}\cdot \mathsf{mdec}_{n,\bar{m},q}(\mathbf{B}_{\mathsf{U}}^T) \in \mathbb{Z}_q^{2n}$ as a  hash value
			 of the public key $\pk_{\mathsf{U}}=\mathbf{B}_{\mathsf{U}} \in \Zq^{n \times \bar{m}}$. \smallskip   %and let $\mathbf{e_M} = \mathsf{encode}(\mathsf{bin}(\mathbf{h_M})) \in \{0,1\}^{m}$.
        \item[b.] Use the trapdoor $\sk_{\GM} = \mathbf{T_A}$ to generate a signature
        \begin{eqnarray}\label{eq:cert-description}
        \crt_{\mathsf{U}} = \big( \tau, \mathbf{d}, \mathbf{r} \big) \in \{0,1\}^\ell \times [-\beta,\beta]^{2m} \times [-\beta,\beta]^m,
        \end{eqnarray}
            satisfying
            \begin{multline}\label{eq:cert-verification}
           \big[\mathbf{A}  ~|~ \sum_{j=1}^\ell \tau[j]\mathbf{A}_j\big] \cdot  \mathbf{d}  \\ = \mathbf{u} +
					\mathbf{D}\cdot \mathsf{vdec}_{n,q-1}( \mathbf{D}_0 \cdot \mathbf{r}  + \mathbf{D}_1 \cdot \mathsf{vdec}_{n,q-1}(\mathbf{h}_{\mathsf{U}}) ) \bmod q,
            \end{multline}
           where $\tau= \tau[1] \ldots \tau[\ell] \in \{0,1\}^{\ell}$,   as in the scheme of \cref{se:gs-lwe-sigep}. \smallskip
      \end{enumerate}
			$\mathsf{U}$ verifies that $\crt_{\mathsf{U}}$ is tuple of the form (\ref{eq:cert-description}) satisfying (\ref{eq:cert-verification}) and returns~$\perp$ if it is not the case.
      The $\GM$ stores $(\pk_{\mathsf{U}},\crt_\mathsf{U})$ in the user database $\mathsf{database}$ and returns the certificate $\crt_\mathsf{U}$ to the new user $\mathsf{U}$. \medskip
      %  \begin{eqnarray}\label{eq:cert-pk}
      %\mathsf{cert}_{\mathsf{pk}} = (\mathbf{h_M}, sig_{\mathbf{M}}).
      %\end{eqnarray}
		\end{itemize}
			
  \item[$\mathsf{ENC}(\pk_{\GM}, \pk_{\OA}, \pk_\mathsf{U}, \crt_\mathsf{U}, \mathbf{w}, L)$:] To encrypt a witness $\mathbf{w} \in \{0,1\}^m$
	for $((\mathbf{A}_R, \mathbf{u}_R), \mathbf{w})$ in relation $\mathrm{R}_{\mathsf{ISIS}}(n,m,q,1)$ (i.e., $\mathbf{A}_R \cdot \mathbf{w} = \mathbf{u}_R \bmod q$), parse $\pk_{\GM}$
	as in (\ref{PK-GM}), $\pk_{\OA}$ as $\mathbf{B}_{\OA} \in \Zq^{n \times \bar{m}}$, $\pk_{\mathsf{U}}$ as  $\mathbf{B}_\mathsf{U} \in \Zq^{n \times \bar{m}}$ and $\crt_{\mathsf{U}}$ as in~(\ref{eq:cert-description}).   \smallskip \smallskip
      \begin{enumerate}
        \item[1.] Generate a one-time  key-pair $(\sk, \vk) \leftarrow \mathsf{Gen}(1^\lambda)$, where $\vk \in \mathbb{Z}_q^n$.
        \item[2.] Compute a full-rank-difference hash $\mathbf{H}_{\vk}= \mathsf{FRD}(\vk) \in \mathbb{Z}_q^{n \times n}$ of the one-time verification key $\vk \in \Zq^n$.
				%Define $\mathbf{B}_{\vk} = \mathbf{B}_\mathsf{U} + \mathbf{H}_{\vk}\cdot \mathbf{G} \in \mathbb{Z}_q^{n \times m}$.
        \item[3.] Encrypt the witness $\mathbf{w} \in \{0,1\}^m$ under $\mathsf{U}$'s public key $\mathbf{B}_\mathsf{U} \in \Zq^{n \times \bar{m}}$ using the tag $\vk$ by taking the following steps: \smallskip
            \begin{enumerate}
                \item[a.] Sample $\mathbf{s}_{\rec} \leftarrow \U(\mathbb{Z}_q^n)$, $\mathbf{R}_{\rec} \leftarrow D_{\ZZ,\sigma}^{m \times \bar{m}} $ and
								$\mathbf{x}_{\rec}, \mathbf{y}_{\rec} \leftarrow \chi^m$. Compute $\mathbf{z}_{\rec} = \mathbf{R}_{\rec}^T\cdot \mathbf{y}_{\rec} \in \mathbb{Z}^{\bar{m}}$.
                \item[b.] Compute
                \begin{eqnarray}\label{eq:c-recipient}
                    \begin{cases}
                        \mathbf{c}_{\rec}^{(1)} = \bar{\mathbf{A}}^T\cdot \mathbf{s}_{\rec} +  \mathbf{y}_{\rec} \bmod q \\
                        \mathbf{c}_{\rec}^{(2)}  %= \mathbf{B}_{\vk}^T \cdot \mathbf{s}_{\rec}  + \mathbf{z}_{\rec} \bmod q
												= (\mathbf{B}_\mathsf{U} + \mathbf{H}_{\vk} \cdot \mathbf{G})^T \cdot \mathbf{s}_{\rec} + \mathbf{z}_{\rec} \bmod q ; \\
                        \mathbf{c}_{\rec}^{(3)} = \mathbf{U}^T \cdot \mathbf{s}_{\rec} + \mathbf{x}_{\rec} + \mathbf{w}\cdot \Big\lfloor\frac{q}{2}\Big\rfloor,
                    \end{cases}
                \end{eqnarray}
                and let  $\mathbf{c}_{\rec} = \big(\mathbf{c}_{\rec}^{(1)}, \mathbf{c}_{\rec}^{(2)}, \mathbf{c}_{\rec}^{(3)}\big)
								\in \mathbb{Z}_q^m \times \mathbb{Z}_q^{\bar{m}} \times \mathbb{Z}_q^m$, which forms an ABB ciphertext \cite{ABB10} for the tag $\vk \in \Zq^n$.

            \end{enumerate}
            \medskip

        \item[4.] Encrypt the decomposition $\mathsf{vdec}_{n,q-1}(\mathbf{h_\mathsf{U}}) \in \{0,1\}^{m}$ of the hashed $\pk_\mathsf{U}$ under
           the $\OA$'s public key $\mathbf{B}_{\OA} \in \Zq^{n \times \bar{m}}$ w.r.t. the tag $\vk \in \Zq^n$. Namely, conduct the following steps: \smallskip
            \begin{enumerate}
                \item[a.] Sample $\mathbf{s}_{\mathsf{oa}} \leftarrow \U( \mathbb{Z}_q^n)$, $\mathbf{R}_{\mathsf{oa}} \leftarrow D_{\ZZ,\sigma}^{m \times \bar{m}}$,
								$\mathbf{x}_{\mathsf{oa}} \leftarrow \chi^{m}, \mathbf{y}_{\mathsf{oa}} \leftarrow \chi^m$. Set $\mathbf{z}_{\mathsf{oa}} = \mathbf{R}_{\mathsf{oa}}^T\cdot \mathbf{y}_{\mathsf{oa}} \in \ZZ^{\bar{m}}$.
                \item[b.] Compute
                \begin{eqnarray}\label{eq:c-open}
                    \begin{cases}
                        \mathbf{c}_{\mathsf{oa}}^{(1)} = \bar{\mathbf{A}}^T \cdot \mathbf{s}_{\mathsf{oa}} +  \mathbf{y}_{\mathsf{oa}} \bmod q; \\
                        \mathbf{c}_{\mathsf{oa}}^{(2)} =  (\mathbf{B}_\OA + \mathbf{H}_{\vk} \cdot \mathbf{G})^T  \cdot \mathbf{s}_{\mathsf{oa}}  + \mathbf{z}_{\mathsf{oa}} \bmod q; \\
                        \mathbf{c}_{\mathsf{oa}}^{(3)} = \mathbf{V}^T \cdot \mathbf{s}_{\mathsf{oa}} + \mathbf{x}_{\mathsf{oa}} + \mathsf{vdec}_{n,q-1}(\mathbf{h_\mathsf{U}})\cdot \Big\lfloor\frac{q}{2}\Big\rfloor,
                    \end{cases}
                \end{eqnarray}
                and let  $\mathbf{c}_{\mathsf{oa}} = \big(\mathbf{c}_{\mathsf{oa}}^{(1)}, \mathbf{c}_{\mathsf{oa}}^{(2)}, \mathbf{c}_{\mathsf{oa}}^{(3)}\big) \in \mathbb{Z}_q^m \times \mathbb{Z}_q^{\bar{m}} \times \mathbb{Z}_q^{m}$.

            \end{enumerate}
        \medskip

        \item[5.] Compute a one-time signature $\Sigma = \mathsf{Sig}(\sk, (\mathbf{c}_{\rec}, \mathbf{c}_{\mathsf{oa}},L))$. \medskip

      \end{enumerate}

 Output the ciphertext
        \begin{eqnarray}\label{eq:final-ciphertext}
        \mathbf{\Psi} = (\vk,\mathbf{c}_{\rec}, \mathbf{c}_{\mathsf{oa}}, \Sigma).
        \end{eqnarray}
and the state information $coins_{\mathbf{\Psi}}=\big( \mathbf{s}_{\rec}, \mathbf{R}_{\rec} , \mathbf{x}_{\rec}, \mathbf{y}_{\rec}, \mathbf{s}_{\mathsf{oa}}, \mathbf{R}_{\mathsf{oa}} ,\mathbf{x}_{\mathsf{oa}}, \mathbf{y}_{\mathsf{oa}} \big) $.
\medskip

  \item[$\mathsf{DEC}(\mathsf{sk}_\mathsf{U}, \mathbf{\Psi},L)$]: The decryption algorithm proceeds as follows: \smallskip \smallskip
  \begin{enumerate}
    \item[1.]    If $\mathsf{Ver}\big(\vk, \Sigma, ( \mathbf{c}_{\rec}, \mathbf{c}_{\mathsf{oa}},L)\big) = 0$,  return $\bot$. Otherwise, parse the secret key $\mathsf{sk}_\mathsf{U}$ as $\mathbf{T}_\mathsf{U} \in \ZZ^{m \times \bar{m}}$ and the ciphertext $\mathbf{\Psi}$ as in~(\ref{eq:final-ciphertext}).
		Define the matrix $\mathbf{B}_{\vk} = \mathbf{B}_\mathsf{U} +
				\mathsf{FRD}(\vk) \cdot \mathbf{G} \in \mathbb{Z}_q^{n \times \bar{m}}$.
    \item[2.]
     Decrypt $\mathbf{c}_{\rec}$ using a decryption key for the tag $\vk \in \ZZ^n$. Namely, \smallskip
		\begin{itemize}
		\item[a.]  Define $\mathbf{B}_{\mathsf{U},\vk} = [ \bar{\mathbf{A}} | \mathbf{B}_\vk  ] = [ \bar{\mathbf{A}} | \bar{\mathbf{A}} \cdot \mathbf{T}_{\mathsf{U}} + \mathsf{FRD}(\vk) \cdot \mathbf{G} ] \in \Zq^{n \times (m+\bar{m})} $. Using $\mathbf{T}_\mathsf{U}$ and the publicly known trapdoor $\mathbf{T}_{\mathbf{G}}$ of $\mathbf{G}$,
		compute  a small-norm matrix $\mathbf{E}_{\vk} \in \ZZ^{(m+ \bar{m}) \times m} $ such that $\mathbf{B}_{\mathsf{U},\vk} \cdot \mathbf{E}_{\vk} = \mathbf{U} \bmod q$ by running the $\mathsf{SampleRight}$
		algorithm of Lemma \ref{lem:sampler}.
		\item[b.] Compute
		\begin{eqnarray*}
		\mathbf{w} = \left\lfloor \Bigl( \mathbf{c}_{\rec}^{(3)} - \mathbf{E}_{\vk}^T \cdot \begin{bmatrix} \mathbf{c}_{\rec}^{(1)} \\ \mathbf{c}_{\rec}^{(2)} \end{bmatrix}  \Bigr) /  \left\lfloor \frac{q}{2} \right\rfloor   \right\rceil  \in \ZZ^m
		\end{eqnarray*}
		and return the obtained $\mathbf{w}  \in \{0,1\}^m$.
			\end{itemize}
  \end{enumerate}

  \medskip
  \item[$\mathsf{OPEN}(\mathsf{sk}_{\OA}, \mathbf{\Psi},L)$]:
  The opening algorithm proceeds as follows: \smallskip \smallskip
  \begin{enumerate}
    \item[1.] If $\mathsf{Ver}\big(\vk, \Sigma, (\mathbf{c}_{\rec}, \mathbf{c}_{\mathsf{oa}}),L\big) = 0$, then return $\bot$. Otherwise, parse $\mathsf{sk}_{\OA}$ as $\mathbf{T}_\OA \in \ZZ^{m \times \bar{m}}$ and the ciphertext $\mathbf{\Psi}$ as in~(\ref{eq:final-ciphertext}).
    \item[2.]    Decrypt $\mathbf{c}_{\mathsf{oa}}$ using a decryption key for the tag $\vk \in \Zq^n$ in the same way as in the decryption algorithm.
		That is, do the following: \smallskip
		\begin{itemize}
		\item[a.] Define the matrix $\mathbf{B}_{\OA,\vk} = [  \bar{\mathbf{A}} | \mathbf{B}_\OA + \mathsf{FRD}(\vk) \cdot \mathbf{G} ] \in \Zq^{n \times (m+\bar{m})} $. Use $\mathbf{T}_\OA$ to
		compute
		a small-norm  $\mathbf{E}_{\OA,\vk} \in \ZZ^{(m+\bar{m}) \times m} $ satisfying   $\mathbf{B}_{\OA,\vk} \cdot \mathbf{E}_{\OA,\vk} = \mathbf{V} \bmod q$.
		\item[b.] Compute
		\begin{eqnarray*}
		\mathbf{h} = \left\lfloor \Bigl( \mathbf{c}_{\mathsf{oa}}^{(3)} - \mathbf{E}_{\OA,\vk}^T \cdot \begin{bmatrix} \mathbf{c}_{\mathsf{oa}}^{(1)} \\ \mathbf{c}_{\mathsf{oa}}^{(2)} \end{bmatrix}  \Bigr) /
		\left\lfloor \frac{q}{2} \right\rfloor   \right\rceil  \in \{0,1\}^{m}
		\end{eqnarray*}
		and $\mathbf{h}_\mathsf{U}'=\mathbf{H}_{2n,q-1} \cdot \mathbf{h} \in \Zq^{2n}$.
			\end{itemize}
		
	\item[3.] Look up $\mathsf{database}$ to find a public key $\pk_\mathsf{U}=\mathbf{B}_\mathsf{U} \in \Zq^{n \times \bar{m}}$ that hashes
	to $\mathbf{h}_\mathsf{U}' \in \Zq^{2n}$ (i.e., such that $\mathbf{h}_\mathsf{U}'= \mathbf{F} \cdot \mathsf{mdec}_{n,\bar{m},q}(\mathbf{B}_\mathsf{U}^T)$). If more than one such  key exists, return
	$\perp$.
	If only one key $\pk_\mathsf{U}=\mathbf{B}_{\mathsf{U}} \in \Zq^{n \times \bar{m}}$ satisfies $\mathbf{h}_\mathsf{U}'= \mathbf{F} \cdot \mathsf{mdec}_{n,\bar{m},q}(\mathbf{B}_\mathsf{U}^T)$,   return that key $\pk_\mathsf{U}$.
	 In any other situation, return $\bot$.
  \end{enumerate}




  \bigskip
  \item[$\langle \mathcal{P}, \mathcal{V}\rangle$:] The common input consists of $\param$ and $\mathsf{pk}_{\GM} $ as specified above, and $(\mathbf{A}_R, \mathbf{u}_R)$ in $\mathbb{Z}_q^{n \times m} \times \mathbb{Z}_q^n$,  $\mathsf{pk}_{\OA} = \mathbf{B}_{\OA} \in \Zq^{n \times \bar{m}}$,
	 and a ciphertext $\mathbf{\Psi}$ as in~(\ref{eq:final-ciphertext}).
	Both parties compute     $ \mathbf{B}_{\OA,\vk} =   [  \bar{\mathbf{A}} | \mathbf{B}_\OA + \mathsf{FRD}(\vk) \cdot \mathbf{G} ] $ as specified above.
      The prover's secret input consists of a witness $\mathbf{w} \in \{0,1\}^m$,
           $\mathsf{pk}_\mathsf{U}= \mathbf{B}_\mathsf{U}$, $\crt_\mathsf{U} = (\tau,\mathbf{d},\mathbf{r}) \in \{0,1\}^\ell \times \ZZ^{2m} \times \ZZ^m$, and the random coins
					$coins_{\mathbf{\Psi}}=\big( \mathbf{s}_{\rec}, \mathbf{R}_{\rec} , \mathbf{x}_{\rec}, \mathbf{y}_{\rec}, \mathbf{s}_{\mathsf{oa}}, \mathbf{R}_{\mathsf{oa}} ,\mathbf{x}_{\mathsf{oa}}, \mathbf{y}_{\mathsf{oa}} \big) $ used to generate $\mathbf{\Psi}$. \smallskip

      The prover's goal is to convince the verifier in zero-knowledge that his secret input satisfies the following: \smallskip
      \begin{enumerate}
        \item $\mathbf{A}_R \cdot \mathbf{w} = \mathbf{u}_R \bmod q$.
        \item $\mathbf{h_M} = \mathbf{F}\cdot \mathsf{mdec}_{n,m,q}(\mathbf{M}) \bmod q$.
        \item Conditions (\ref{eq:cert-description}) and (\ref{eq:cert-verification}) hold.
        \item Vectors $\mathbf{x}_{\rec}, \mathbf{y}_{\rec}, \mathbf{x}_{\mathsf{oa}}, \mathbf{y}_{\mathsf{oa}}$ have infinity norms bounded by $B$, and vectors $\mathbf{z}_{\rec}, \mathbf{z}_{\mathsf{oa}}$ have infinity norms bounded by $\beta mB$.
        \item Equations in (\ref{eq:c-recipient}) and (\ref{eq:c-open}) hold.
      \end{enumerate}
			
			\medskip 	\smallskip 	
	To this end $\mathcal{P}$ conducts the following steps. \medskip 	\smallskip 	 	\smallskip 	
	\begin{itemize}
	\item[1.]  Decompose the matrix $\mathbf{B}_\mathsf{U} \in \Zq^{n \times \bar{m}}$ into  $\mathbf{b}_{\mathsf{U}} = \mathsf{mdec}_{n,\bar{m},q}(\mathbf{B}_{\mathsf{U}}^T) \in \{0,1\}^{n\bar{m}k}$ and  the vectors $\mathbf{s}_{\rec} ,\mathbf{s}_{\mathsf{oa}} \in \Zq^n$ into $\mathbf{s}_{0,\rec} = \mathsf{vdec}_{n,q-1}(\mathbf{s}_{\rec}) \in \{0,1\}^{nk}$ and $\mathbf{s}_{0,\mathsf{oa}} = \mathsf{vdec}_{n,q-1}(\mathbf{s}_{\mathsf{oa}}) \in \{0,1\}^{nk}$. Combine the first two binary vectors into $\mathbf{z}_{\mathbf{\Psi}}   =  \expandtimes (\mathbf{b}_{\mathsf{U}},\mathbf{s}_{0,\rec}) \in \{0,1\}^{4n \bar{m} k^2}
$. Define
	$$\mathbf{Q} = \mathbf{H}_{\bar{m},q-1} \cdot [\overbrace{\mathbf{Q}_0 | \ldots | \mathbf{Q}_0}^{n \text{ times }}] \in \Zq^{\bar{m} \times 4n \bar{m} k^2} ,$$ where
	$\mathbf{Q}_0 = \mathbf{I}_{\bar{m} k} \otimes \mathbf{g}' \in \Zq^{\bar{m}k \times 4 \bar{m} k^2}$ is the matrix defined as in (\ref{Q0-def}).
		\smallskip
	\item[2.] Generate a zero-knowledge argument of knowledge of
\begin{eqnarray*}
\left\{
\begin{array}{l}
\tau \in \{0,1\}^\ell,   ~\mathbf{d}=[\mathbf{d}_1^T | \mathbf{d}_2^T ]^T \in [-\beta,\beta]^{2m},~\mathbf{r} \in [-\beta,\beta]^m \\
\mathbf{t}_{\mathsf{U}} \in \{0,1\}^{m},~\mathbf{w}_{\mathsf{U}} \in \{0,1\}^{\bar{m}}  \\
  \mathbf{b}_{\mathsf{U}} \in \{0,1\}^{n \bar{m} k},   ~\mathbf{s}_{0,\rec} \in \{0,1\}^{nk}, ~
	\mathbf{z}_{\mathbf{\Psi}}  = \expandtimes (\mathbf{b}_{\mathsf{U}},\mathbf{s}_{0,\rec})
   \\
 \mathbf{x}_{\rec}, ~\mathbf{y}_{\rec} \in [-B,B]^m,  ~ \mathbf{z}_{\rec} \in [-\beta mB, \beta mB]^{\bar{m}} ,  ~\mathbf{w} \in \{0,1\}^m, \\
\mathbf{s}_{0,\mathsf{oa}} \in \{0,1\}^{nk},~\mathbf{x}_{\mathsf{oa}}, ~\mathbf{y}_{\mathsf{oa}} \in  [-B,B]^m,   ~\mathbf{z}_{\mathsf{oa}} \in [-\beta mB,\beta mB]^{\bar{m}}
\end{array}
 \right.
\end{eqnarray*}

\end{itemize}
\end{description}

\begin{comment}
			\begin{eqnarray} \label{rel-un}
		 \left[ \begin{array}{c|c|c|c|c}  \mathbf{A}  ~& ~\mathbf{A}_0  ~&~ \mathbf{A}_1 ~ & ~\ldots & \mathbf{A}_\ell  \end{array} \right] \cdot \begin{bmatrix}  \mathbf{d}_1 \\ \hline
		 \mathbf{d}_2 \\ \hline \tau[1] \cdot \mathbf{d}_2 \\ \hline \vdots \\ \hline \tau[\ell] \cdot \mathbf{d}_2 \end{bmatrix}  &=& \mathbf{u} + \mathbf{D} \cdot \mathbf{w}_\mathsf{U} \bmod q  \qquad  \qquad \\ \nonumber
	\mathbf{H}_{n,q-1} \cdot \mathbf{w}_\mathsf{U} &=& \mathbf{D}_0 \cdot \mathbf{r} + \mathbf{D}_1 \cdot \mathbf{t}_{\mathsf{U}} 	 \\  \nonumber
  	\mathbf{H}_{2n,q-1} \cdot \mathbf{t}_{\mathsf{U}} & =&  \mathbf{F} \cdot \mathbf{b}_{\mathsf{U}} \bmod q
\end{eqnarray}
as well as
\begin{eqnarray} \nonumber
\mathbf{c}_{\rec}^{(1)} &=&  \left[ \begin{array}{c|c} \bar{\mathbf{A}}^T \cdot \mathbf{H}_{n,q-1} ~ &~ \mathbf{I}_m  \end{array} \right]  \cdot \begin{bmatrix} \mathbf{s}_{0,\rec}  \\ \hline \mathbf{y}_{\rec} \end{bmatrix}  , \qquad \quad  \\  \label{rel-deux}
\mathbf{z}_{\mathbf{\Psi}}   &= & \expandtimes (\mathbf{b}_{\mathsf{U}},\mathbf{s}_{0,\rec})  \qquad  \qquad  \\			\nonumber
\mathbf{c}_{\rec}^{(2)} &=&
\left[ \begin{array}{c|c|c} \mathbf{Q} ~&~  \mathbf{G}^T \cdot \mathbf{H}_\vk^T \cdot \mathbf{H}_{n,q-1} ~& ~     \mathbf{I}_{\bar{m}}
\end{array} \right]  \cdot \begin{bmatrix} \mathbf{z}_{\mathbf{\Psi}}  \\ \hline  \mathbf{s}_{0,\rec}  \\ \hline \mathbf{z}_{\rec} \end{bmatrix}  , \qquad \quad \\ \nonumber
\mathbf{c}_{\rec}^{(3)} &=&
\left[ \begin{array}{c|c|c}  ~ \mathbf{U}^T \cdot \mathbf{H}_{n,q-1} ~~& ~~     \mathbf{I}_m ~~& ~~\mathbf{I}_m \cdot \lfloor \frac{q}{p} \rfloor  ~
\end{array} \right]  \cdot \begin{bmatrix} \mathbf{s}_{0,\rec}  \\ \hline \mathbf{x}_{\rec} \\ \hline \mathbf{w}  \end{bmatrix}  , \\ \nonumber
  \mathbf{u}_R &=&  \mathbf{A}_R \cdot \mathbf{w}  \bmod q
\end{eqnarray}
and
\begin{eqnarray} \nonumber
\mathbf{c}_{\mathsf{oa}}^{(1)} &=&   \left[ \begin{array}{c|c}    \bar{\mathbf{A}}^T \cdot \mathbf{H}_{n,q-1} ~& ~     \mathbf{I}_m
\end{array} \right]  \cdot \begin{bmatrix} \mathbf{s}_{0,\mathsf{oa}}  \\ \hline \mathbf{y}_{\mathsf{oa}} \end{bmatrix}  , \qquad \quad  \\  \label{rel-trois}
\mathbf{c}_{\mathsf{oa}}^{(2)} &=&
\left[ \begin{array}{c|c}  (\mathbf{B}_\OA + \mathbf{H}_\vk \cdot \mathbf{G} )^T \cdot \mathbf{H}_{n,q-1} ~& ~     \mathbf{I}_{\bar{m}}
\end{array} \right]  \cdot \begin{bmatrix} \mathbf{s}_{0,\mathsf{oa}}  \\ \hline \mathbf{z}_{\mathsf{oa}} \end{bmatrix}  , \qquad \quad \\ \nonumber
\mathbf{c}_{\mathsf{oa}}^{(3)} &=&
\left[ \begin{array}{c|c|c}  ~ \mathbf{V}^T \cdot \mathbf{H}_{n,q-1} ~~& ~ ~    \mathbf{I}_m~~& ~~\mathbf{I}_m \cdot \lfloor \frac{q}{2} \rfloor  ~
\end{array} \right]  \cdot \begin{bmatrix} \mathbf{s}_{0,\mathsf{oa}}  \\ \hline \mathbf{x}_{\mathsf{oa}} \\ \hline \mathbf{t}_{\mathsf{U}}  \end{bmatrix}   .
\end{eqnarray}
	     The protocol is repeated $\kappa$ times to make the soundness error negligibly small.
\end{itemize}


\end{description}
\medskip
\end{comment}
such that the following system of $10$ equations holds:
\begin{eqnarray}\label{eq:big-system-main-scheme}
\begin{cases}
    \mathbf{u}= [\mathbf{A} | \mathbf{A}_0 | \mathbf{A}_1 | \ldots | \mathbf{A}_\ell]\cdot \left(
                                                                                             \begin{array}{c}
                                                                                               \mathbf{d}_1 \\
                                                                                               \mathbf{d}_2 \\
                                                                                               \tau[1]\cdot \mathbf{d}_2 \\
                                                                                               \vdots \\                                                                                               \tau[\ell]\cdot \mathbf{d}_2\\
                                                                                             \end{array}
                                                                                           \right)
     + (-\mathbf{D})\cdot\mathbf{w}_\textsf{U} \bmod q, \\[5pt]
    \mathbf{0} = \mathbf{H}_{n, q-1}\cdot \mathbf{w}_\textsf{U} + (-\mathbf{D}_0)\cdot \mathbf{r} +  (-\mathbf{D}_1)\cdot \mathbf{t}_\textsf{U} \bmod q, \\[5pt]
    \mathbf{0} = \mathbf{H}_{2n,q-1}\cdot \mathbf{t}_\textsf{U} + (-\mathbf{F})\cdot\mathbf{b}_\textsf{U}\bmod q, \\[5pt]
    \mathbf{c}_{\rec}^{(1)} = (\bar{\mathbf{A}}^T\cdot \mathbf{H}_{n,q-1}) \cdot \mathbf{s}_{0,\rec} + \mathbf{I}_m\cdot \mathbf{y}_{\rec} \bmod q, \\[5pt]
    \mathbf{c}_{\rec}^{(2)} = \mathbf{Q}\cdot \mathbf{z}_{\mathbf{\Psi}} + (\mathbf{G}^T\cdot \mathbf{H}_{\vk}^T \cdot \mathbf{H}_{n,q-1})\cdot \mathbf{s}_{0,\rec} + \mathbf{I}_{\bar{m}} \cdot \mathbf{z}_{\rec} \bmod q, \\[5pt]
    \mathbf{c}_{\rec}^{(3)} = (\mathbf{U}^T\cdot \mathbf{H}_{n,q-1})\cdot \mathbf{s}_{0,\rec} + \mathbf{I}_m\cdot \mathbf{x}_{\rec} + (\lfloor \frac{q}{2}\rfloor\cdot \mathbf{I}_m)\cdot \mathbf{w} \bmod q, \\[5pt]
    \mathbf{u}_R = \mathbf{A}_R\cdot \mathbf{w} \bmod q, \\[5pt]
    \mathbf{c}_{\mathsf{oa}}^{(1)} = (\bar{\mathbf{A}}^T \cdot \mathbf{H}_{n,q-1})\cdot \mathbf{s}_{0,\mathsf{oa}} + \mathbf{I}_m\cdot \mathbf{y}_{\mathsf{oa}} \bmod q, \\[5pt]
    \mathbf{c}_{\mathsf{oa}}^{(2)} = [(\mathbf{B}_{\OA} + \mathbf{H}_{\vk}\cdot \mathbf{G})^T\cdot \mathbf{H}_{n,q-1}]\cdot \mathbf{s}_{0,\mathsf{oa}} + \mathbf{I}_{\bar{m}}\cdot \mathbf{z}_{\mathsf{oa}} \bmod q, \\[5pt]
     \mathbf{c}_{\mathsf{oa}}^{(3)} = (\mathbf{V}^T\cdot \mathbf{H}_{n,q-1})\cdot \mathbf{s}_{0, \mathsf{oa}} + \mathbf{I}_m\cdot \mathbf{x}_{\mathsf{oa}} + (\lfloor \frac{q}{2}\rfloor\cdot \mathbf{I}_m)\cdot \mathbf{t}_{\mathsf{U}} \bmod q.
\end{cases}
\end{eqnarray}
Let $\mathbf{w}_1 = \mathbf{b}_{\mathsf{U}}$, $\mathbf{w}_2 = \mathbf{s}_{0,\rec}$, $\mathbf{w}_3 = \mathbf{z}_{\mathbf{\Psi}} = \expandtimes (\mathbf{b}_{\mathsf{U}},\mathbf{s}_{0,\rec})$, $\mathbf{w}_4 = \mathbf{w}_{\mathsf{U}}$, $\mathbf{w}_5 = \mathbf{t}_{\mathsf{U}}$,
$\mathbf{w}_6 = \mathbf{s}_{0,\mathsf{oa}}$, $\mathbf{w}_7 = \mathbf{w}$, $\mathbf{w}_8 = \mathbf{x}_{\rec}$, $\mathbf{w}_9 = \mathbf{y}_{\rec}$, $\mathbf{w}_{10} = \mathbf{z}_{\rec}$, $\mathbf{w}_{11} = \mathbf{r}$, $\mathbf{w}_{12} = \mathbf{x}_{\mathsf{oa}}$, $\mathbf{w}_{13} = \mathbf{y}_{\mathsf{oa}}$, $\mathbf{w}_{14}= \mathbf{z}_{\mathsf{oa}}$ and $$\mathbf{w}_{15}= \big(\hspace*{1.5pt}\mathbf{d}_1^T \hspace*{1.5pt}\|\hspace*{1.5pt} \mathbf{d}_2^T \hspace*{1.5pt}\|\hspace*{1.5pt} \tau[1]\hspace*{-1.5pt}\cdot\hspace*{1.5pt}\mathbf{d}_2^T \hspace*{1.5pt}\| \ldots \|\hspace*{1.5pt} \tau[\ell]\hspace*{-1.5pt}\cdot\hspace*{1.5pt}\mathbf{d}_2^T\hspace*{1.5pt}\big)^T.$$
Then system (\ref{eq:big-system-main-scheme}) can be rewritten as:
\begin{eqnarray}\label{eq:big-system-main-scheme-2}
    \begin{cases}
        \mathbf{v}_1 = \mathbf{M}_{1,1}\cdot \mathbf{w}_1 + \mathbf{M}_{1,2}\cdot \mathbf{w}_2 + \ldots + \mathbf{M}_{1,15} \cdot \mathbf{w}_{15}  \bmod q, \\[2.5pt]
        \mathbf{v}_2 = \mathbf{M}_{2,1}\cdot \mathbf{w}_1 + \mathbf{M}_{2,2}\cdot \mathbf{w}_2 + \ldots + \mathbf{M}_{2,15} \cdot \mathbf{w}_{15}  \bmod q, \\
        \hspace{.35\textwidth}\vdots\\
        %\hdotsfor{0} \\
         \mathbf{v}_{10} = \mathbf{M}_{10,1}\cdot \mathbf{w}_1 + \mathbf{M}_{10,2}\cdot \mathbf{w}_2 + \ldots + \mathbf{M}_{10,15} \cdot \mathbf{w}_{15}\bmod q,
    \end{cases}
\end{eqnarray}
where $\{\mathbf{M}_{i,j}\}_{(i,j) \in [10] \times [15]}$, $\{\mathbf{v}_i\}_{i \in [10]}$ are public matrices and vectors (which are possibly zero).

The argument system is obtained by invoking the protocol from \cref{sse:stern}. The protocol is repeated $\kappa$ times to make the soundness error negligibly small.

\subsection{Efficiency and Correctness} \label{correctness}

\paragraph{Efficiency.}
It can be seen that the given group encryption scheme can be implemented in polynomial time. We now will evaluate the bit-sizes of keys and ciphertext, as well as the communication cost of the protocol $\langle \mathcal{P}, \mathcal{V}\rangle$.
\begin{itemize}
\item The public key of \textsf{GM}, as in~\eqref{PK-GM}, has bit-size $\mathcal{O}(\ell n^2 \log^2 q) = \widetilde{\mathcal{O}}(\ell \lambda^2)$.
\item The public keys of \textsf{OA} and each user both have  bit-size $n\bar{m}\lceil\log_2 q\rceil =  \widetilde{\mathcal{O}}(\lambda^2)$.
\item The secret key of each party in the scheme is a trapdoor of bit-size $\widetilde{\mathcal{O}}(\lambda^2)$. The user's certificate $\mathsf{cert}_{\USR}$ has bit-size $\widetilde{\mathcal{O}}(\lambda)$.
\item The ciphertext $\mathbf{\Psi}$ consists of $\vk \in \ZZ_q^n$, two ABB ciphertexts of total size $2(2m + \bar{m})\lceil\log_2 q\rceil$ and a one-time signature $\Sigma$. Thus, its bit-size is $\widetilde{\mathcal{O}}(\lambda) + \big|\Sigma\big|$.
\item The communication cost of the protocol $\langle \mathcal{P}, \mathcal{V}\rangle$ is largely dominated by the bit-size of the witness $\mathbf{z}_{\mathbf{\Psi}}   =  \expandtimes (\mathbf{b}_{\USR},\mathbf{s}_{0,\rec}) \in \{0,1\}^{4n \bar{m} k^2}$. The total cost is $\kappa\cdot \mathcal{O}(n^2 \log^4 q) = \widetilde{\mathcal{O}}(\lambda^2)$ bits.
\end{itemize}



\paragraph{Correctness.}
The given group encryption scheme is correct with overwhelming probability.
We first remark that the scheme parameters are set up so that the two instances of the ABB identity-based encryption~\cite{ABB10} are correct. Indeed, during the decryption procedure of $\mathsf{DEC}(\mathsf{sk}_\USR, \mathbf{\Psi},L)$, we have:
\[
 \mathbf{c}_{\rec}^{(3)} - \mathbf{E}_{\vk}^T \cdot \begin{bmatrix} \mathbf{c}_{\rec}^{(1)} \\ \mathbf{c}_{\rec}^{(2)} \end{bmatrix} = \mathbf{x}_{\rec} - \mathbf{E}_{\vk}^T \cdot \begin{bmatrix} \mathbf{y}_{\rec} \\ \mathbf{z}_{\rec} \end{bmatrix} + \mathbf{w}\cdot \left\lfloor \frac{q}{2} \right\rfloor.
\]
Note that $\|\mathbf{x}_{\rec}\|_\infty$ and $\|\mathbf{y}_{\rec}\|_\infty$ are bounded by $B$, and $\|\mathbf{z}_{\rec}\|_\infty = \|\mathbf{R}_{\rec}^T\cdot \mathbf{y}_{\rec}\|_\infty \leq \beta m B = \widetilde{\mathcal{O}}(n^2)$. Furthermore, the entries of the discrete Gaussian matrix $\mathbf{E}_{\vk}^T$ are bounded by $\widetilde{\mathcal{O}}(\sqrt{n})$. Hence, the error term $\mathbf{x}_{\rec} - \mathbf{E}_{\vk}^T \cdot \begin{bmatrix} \mathbf{y}_{\rec} \\ \mathbf{z}_{\rec} \end{bmatrix}$ is bounded by $\widetilde{\mathcal{O}}(n^{3.5})$ which is much smaller than $q/4 = \widetilde{\mathcal{O}}(n^4)$. As a result, the decryption algorithm returns $\mathbf{w}$ with overwhelming probability. The correctness of algorithm $\mathsf{OPEN}(\mathsf{sk}_{\OA}, \mathbf{\Psi},L)$ also follows from a similar argument.

Finally, we note that if a certified group user honestly follows all the prescribed algorithms, then he should be able to compute valid witness-vectors to be used in the protocol $\langle \mathcal{P}, \mathcal{V}\rangle$, and he should be accepted by the verifier, thanks to the perfect completeness of the argument system in \cref{sse:stern}.

\subsection{Security}  \label{security}

Our  scheme is proven secure under the $\SIS$ and $\LWE$ assumptions using   classical reduction techniques. 
The security results are explicited in the following theorems.
%, for which some proofs have been deferred to Appendix~\ref{proofs-GE}.

\subsubsection{Anonymity}

\begin{theorem} \label{anon-thm}
  The scheme provides anonymity if the $\LWE$ assumption holds and if $\mathcal{OTS}$ is a strongly unforgeable one-time signature.
%   \textnormal{(The proof is given in Appendix~\ref{proof-anon-thm}.)}
\end{theorem}

%%%%%%%%%
% Proof %
%%%%%%%%%
\begin{proof}
  We consider a sequence of games where the first game is the real experiment of definition \ref{anonymity-def} while, in the final game, the adversary $\adv$ is essentially  an adversary against the anonymity of the Agrawal-Boneh-Boyen IBE scheme \cite{ABB10}.
  In Game $i$, we call $W_i$ the event that the challenger outputs $1$.
  \smallskip  \smallskip

  \noindent \textbf{Game $1$:} The challenger $\bdv$ generates public parameters $\param$, which include matrices $ \bar{\mathbf{A}},  \mathbf{U}, \mathbf{V} \in \ZZ_q^{n \times m} $ and
  $\mathbf{F} \in \ZZ_q^{2n \times n \bar{m} k} $. The opening authority's public
  key $\pk_{\OA} = \mathbf{B}_{\OA} \in \ZZ_q^{n \times \bar{m}}$ is given to $\adv$ who generates a group manager's public key
  $\pk_{\GM}$ of its own. By invoking the $\mathsf{USER}$ oracle, $\adv$
  registers two distinct receivers' public keys
  $\pk_{\USR,0}=\mathbf{B}_{\USR,0} \in \ZZ_q^{n \times \bar{m}}$, $\pk_{\USR,1}=\mathbf{B}_{\USR,1} \in \ZZ_q^{n \times \bar{m}}$
  chosen by the challenger. It also  makes a number of opening queries and decryption
  queries, which the challenger handles using $\sk_{\OA}=\mathbf{T}_{\OA}$ and $\sk_{\USR,0}=\mathbf{T}_{\USR,0} $, $\sk_{\USR,1}=\mathbf{T}_{\USR,1}$,
  respectively. After a while, the adversary outputs $((\mathbf{A}_R,\mathbf{u}_R),\mathbf{w},L)$ such
  that $\mathbf{u}_R= \mathbf{A}_R \cdot \mathbf{w} \bmod q$, with $\mathbf{A}_R \in \ZZ_p^{n \times m}$, $\mathbf{u}_R \in \ZZ_q^n$ and $\mathbf{w} \in \{0,1\}^m$. In return, $\adv$  obtains, as a challenge, a
  group encryption
  $\Psi^\star= (\vk^\star,\mathbf{c}_{\rec}^\star, \mathbf{c}_{\oa}^\star, \Sigma^\star).$
  of the witness $\mathbf{w}$ under $\pk_{\USR,b} =\mathbf{B}_{\USR,b}$, for some random bit $b \leftarrow \U( \{0,1\})$ of the challenger's
  choice. Then, the adversary  obtains proofs $\pi_{\Psi^\star}^\star$ for
  $\Psi^\star$ and makes further opening and decryption queries under the
  natural restrictions of Definition \ref{anonymity-def}. When the adversary $\adv$ halts, it
  outputs a bit $b' \in \{0,1\}$ and the challenger outputs $1$ if and  only if  $b'=b$. % We call $W_1$ the latter event.
  \smallskip \smallskip

  \noindent \textbf{Game $2$:} This game is like Game $1$ except  the challenger aborts in
  the event that the adversary $\adv$ queries the opening of a ciphertext
  $\Psi= (\vk,\mathbf{c}_{\rec}, \mathbf{c}_{\oa}, \Sigma)$ such that $\vk=\vk^\star$ and $\sigma$ is valid (we
  assume w.l.o.g. that $\vk^\star$ is generated ahead of time). If
  this event occurs, the adversary $\adv$ is necessarily able to break the strong unforgeability
  of $\mathcal{OTS}$ (note that,  if the query occurs before the challenge phase, it means that
  $\adv$ has forged a signature without seeing a signature at all). There thus exist a one-time signature forger $\bdv$ such that
  $|\Pr[W_2]-\Pr[W_1]|  \leq \mathbf{Adv}_\bdv^{\mathrm{ots}}(\lambda)$, which means that Game $2$ is identical to Game $1$
  so long as $\mathcal{OTS}$ is a strongly unforgeable one-time signature. \smallskip  \smallskip

  \noindent \textbf{Game $3$:}  In this game, we modify the generation of proofs $\pi_{\Psi^\star}^\star$:
  instead of generating proofs using the real witnesses, we appeal to the zero-knowledge simulator of the argument system of \cref{sse:stern-abstraction} at each invocation
  of $\mathcal{P}$ after the challenge phase. Note that, since we assume public parameters generated by a trusted party, the statistical ZK simulator is allowed to use
  a trapdoor embedded in $\param$ to generate simulated proofs (using, e.g., Damg\aa rd's technique \cite{Dam00}).
  The statistical zero-knowledge property of the argument system ensures that $\adv$'s view remains statistically close to that of Game $2$: we have
  $|\Pr[W_3]-\Pr[W_2] | \leq \mathsf{negl}(\lambda)$. \smallskip  \smallskip

  \noindent \textbf{Game $4$:} We now modify the generation of the challenge ciphertext $\Psi^\star$.
  In this game, the challenger computes the ciphertext
  $\mathbf{c}_{\oa}^\star$ as an ABB encryption under the identity $\vk^\star$ of  a random $m$-bit string  instead of a decomposition
  $\mathsf{vdec}_{n,q-1}(\mathbf{h}_{\USR,b}) \in \{0,1\}^m$ of $\mathbf{h}_{\USR,b} = \mathbf{F} \cdot \mathsf{mdec}_{n,\bar{m},q}(\mathbf{B}_{\USR,b}^T) \in \ZZ_q^{2n}$. Since
  the random encryption coins $\mathbf{s}_{\oa}^\star, \mathbf{R}_{\oa}^\star ,\mathbf{x}_{\oa}^\star, \mathbf{y}_{\oa}^\star $   are no longer used to generate proofs
  $\pi_{\Psi^\star}$, we can show that  any noticeable change  in $\adv$'s output distribution implies
  a selective adversary against the ABB IBE, as established by Lemma \ref{ABB-un}, which would contradict the $\LWE$ assumption.
  The result of Agrawal \textit{et al.}~\cite[Theorem~23]{ABB10} (recalled in Theorem~\ref{ABB-pseudorand-prop}) indeed implies
  $|\Pr[W_4]-\Pr[W_3]| \leq \mathbf{Adv}^{\mathsf{LWE}}(\lambda) $.  \smallskip  \smallskip

  In Game $4$, we can show that, if the adversary $\adv$ has noticeable advantage in the anonymity game, we can break the anonymity of the ABB IBE system, as shown
  in the proof of Lemma \ref{ABB-deux}.
  From the result of \cite[Theorem 23]{ABB10}, we deduce that $|\Pr[W_4]-1/2| \leq \mathbf{Adv}^{\mathsf{LWE}}(\lambda)$,
  which implies the announced result.
\end{proof}

\begin{lemma}\label{ABB-un}
  Any PPT adversary such that $\Pr[W_4]$ is noticeably different from $\Pr[W_3]$ implies a selective adversary against the ABB IBE scheme.
\end{lemma}

\begin{proof}
  Let $\adv$ be a PPT adversary for which $\left| \Pr[W_4] - \Pr[W_3] \right| = \varepsilon$ is non-negligible. We use $\adv$ to build a selective adversary against the  ABB IBE system.

  At the outset of the game, the reduction $\bdv$ generates a one-time signature key pair $(\vk^\star, \sk^\star)$ and declares
	$\vk^\star$ as the target identity to its challenger for the selective security game, and obtains in return the IBE public parameters
  $$\mathsf{PP} = \big(\bar{\mathbf A}, \mathbf B, \mathbf V \big) \in \Zq^{n \times m} \times \Zq^{n \times \bar{m}} \times \Zq^{n \times m}.$$

	Next, the reduction runs the appropriate steps of the actual $\textsf{SETUP}_\textsf{init}$ algorithm to obtain
	$\mathsf{COM}_{\mathsf{par}}$,  $\mathbf{F} \in \ZZ_q^{2n \times n\bar{m}k}$ and $\mathbf{U} \in \ZZ_q^{n \times m}$.
  Namely, $\bdv$ samples $\mathbf F \sample \U(\Zq^{2m\times n \bar{m} k})$ and $\mathbf U \sample \U(\Zq^{n \times m})$ like in the $\mathsf{SETUP}_\mathsf{init}$ algorithm and sends
  $$ \param = \big\{\lambda, n, q, k, m, B, \chi, \sigma, \beta, \ell, \kappa,  \mathcal{OTS}, \compar, \mathsf{FRD},   \bar{\mathbf{A}}, \mathbf{G}, \mathbf{F}, \mathbf{U}, \mathbf{V}  \big\} $$
  along with $\pk_\OA = \mathbf B \in \ZZ_q^{n \times \bar{m}}$ to the adversary $\adv$.

  In return, the adversary $\adv$ chooses $\pk_\GM$, which allows it to enroll two users for whom $\bdv$ faithfully
	generates $(\pk_{\USR, i}, \sk_{\USR, i})_{i \in \bit}$. Knowing both private keys $\{\sk_{\USR,i} = \mathbf{T}_{\USR,i} \}_{i \in \bit}$, $\bdv$ is   able to perfectly simulate  the $\mathsf{DEC}(\cdot)$
	oracle.%	Further decryption queries will use $\pk_{\USR, i}$ to decrypt $\mathbf c_\rec$.
  \medskip

  \noindent \paragraph{Open Queries.} To answer   opening queries for ciphertexts $\mathbf{\Psi} = (\vk, \mathbf c_\rec, \mathbf c_\OA, \Sigma)$ and labels $L$, $\bdv$
	first checks that $\mathsf{Ver}(\vk, \Sigma, (\mathbf c_\rec, \mathbf c_\OA, L)) = 1$. If this test fails, $\bdv$ returns $\bot$. Otherwise, $\bdv$ queries its IBE challenger
	to obtain a IBE private key $\mathbf T_{\OA, \vk} \in \ZZ^{(m+ \bar{m}) \times m}$ for identity $\vk \neq \vk^\star$. The IBE challenger's response allows $\bdv$  to  decrypt $\mathbf c_\OA$ and figure out the identity of the receiver  by looking up $\mathsf{database}$.
  The result of the opening operation is then returned  to~$\adv$.
  \medskip

  After a number of queries, $\adv$ decides to move to the challenge phase and  sends a challenge query $\big( (\mathbf{A}_R,\mathbf{u}_R), \mathbf w^\star, L^\star \big)$ such that
	$\mathbf{u}_R=\mathbf{A}_R \cdot \mathbf{w}^\star \bmod q$. The reduction
	 handles this query  by requesting a challenge ciphertext for the IBE security game with the messages $\mathbf{m}_0=\mathsf{vdec}_{n,q-1}(\mathbf h_{\mathsf U,b})$, for some random bit $b \sample \U(\bit)$
	 and $\mathbf{m}_1 \sample \U(\{0,1\}^m)$. In return, $\bdv$ obtains
	 a challenge ciphertext $\mathbf c^\star_\OA$ under identity $\vk^\star$, which is embedded in  $\adv$'s challenge ciphertext. Namely,
	$\mathbf{\Psi}^\star = (\vk^\star, \mathbf c_\rec^\star, \mathbf c_\OA^\star, \Sigma^\star)$ is obtained by computing   $\mathbf c_\rec^\star$   as an ABB encryption
	of the witness $\mathbf w^\star$ using the matrix $\mathbf{B}_{\mathsf U,b} \in \ZZ_q^{n \times \bar{m}}$ as in~\eqref{eq:c-recipient}
	and $\Sigma^\star = \Sign(\sk^\star, (\mathbf c_\rec^\star, \mathbf c_\OA^\star, L^\star))$.
All queries to the proving oracle  $\mathcal{P}$ are replied by returning a simulated ZK argument as in Game $3$.

  When $\adv$ halts, it outputs a bit $b' \in \{0,1\}$. If $b = b'$, $\bdv$ returns the bit $0$ as a guess that  the selective security challenger encrypted $\mathbf{m}_0=\mathsf{vdec}_{n,q-1}(\mathbf h_{\mathsf U,b})$.
 Otherwise, $\bdv$ outputs $1$ meaning that the IBE challenger chose to encrypt $\mathbf{m}_1$, which was chosen independently of the value of $b \in \{0,1\}$.
 If we call \textsf{Random} (resp. \textsf{Real}) the event that the IBE challenger chooses to encrypt $\mathbf{m}_1$ (resp. $\mathbf{m}_0$),
we can assess % corresponds to \SFGame 3.
  the advantage of the reduction $\bdv$ as
  \begin{align*}
  \mathbf{Adv}_{\bdv}^{\textrm{sID-CPA}}(\lambda) &= \left|\Pr[b=b'\mid \textsf{Random}] - \Pr[b=b' \mid\textsf{Real}]\right|\\
  &= \left|\Pr[W_4] - \Pr[W_3]\right|\\
  &= \varepsilon,
  \end{align*}
	which proves the result.
\end{proof}

\begin{lemma} \label{ABB-deux}
  In Game $4$, the adversary's advantage is negligible assuming that the ABB IBE has pseudo-random ciphertexts.
\end{lemma}

\begin{proof}
  Let us assume the existence of a PPT adversary $\adv$ with non negligible advantage $\varepsilon$ in Game $4$. From $\adv$,
  we construct a selective adversary $\bdv$ that can distinguish  ABB ciphertexts from random elements of the ciphertext space with non-negligible advantage in the game described in Definition~\ref{de:pseudorand-cipher}.

  First, $\bdv$ generates $(\sk^\star, \vk^\star)$ via the key generation algorithm of the one-time-signature \textsf{OTS} and hands $\vk^\star$ to its pseudo-randomness challenger. In return,
  $\bdv$ receives
  \[ \mathsf{PP} = \big(\bar{\mathbf A}, \mathbf B, \mathbf U \big) \in \Zq^{n \times m} \times \Zq^{n \times \bar{m}} \times \Zq^{n \times m} \]
  from its real-or-random (ROR) challenger.

  Our reduction  uses $\mathsf{PP}$  to compute public parameters for our $\mathsf{GE}$ scheme. To this end, it samples $\mathbf F \sample \U(\Zq^{2n \times n \bar{m}k})$,
  $\mathbf V \sample \U(\Zq^{n\times m})$ as in the real $\mathsf{SETUP}_\mathsf{init}$ algorithm.
  The reduction $\bdv$ also computes $\mathbf B_\OA =\bar{\mathbf{A}} \cdot  \mathbf T_\OA \bmod q $,
  where the small-norm matrix $\mathbf{T}_\OA$ is sampled from $D_{\ZZ,\sigma}^{m \times \bar{m}}$, and sends $\adv$ the parameters
  \[ \mathsf{param}= \big\{\lambda, n, q, k, m, \sigma, \beta, \ell, \kappa,  \mathcal{OTS}, \compar, \mathsf{FRD},   \bar{\mathbf{A}}, \mathbf{G}, \mathbf{F}, \mathbf{U}, \mathbf{V}  \big\}, \]
  where $\bar{\mathbf{A}}$ is taken from $\mathsf{PP}$,
  along with $\pk_\OA = \mathbf B_\OA$. The rest of the keys are generated as in Game $4$.

  The reduction $\bdv$ then tosses a coin $b \sample \U(\bit)$. When the adversary $\adv$ triggers an execution of the join protocol,
  $\bdv$ generates the public keys $(\pk_i)_{i\in \bit}$ by	defining $\pk_{\USR,b} = \mathbf B$ using  the matrix $\mathbf B \in \ZZ_q^{n \times \bar{m}}$ supplied by
  the ROR challenger as part of $\mathsf{PP}$ and generates $(\pk_{\USR,1-b},\sk_{1-b}) =(\mathbf{B}_{\USR,1-b} = \bar{\mathbf{A}} \cdot  \mathbf T_{1-b}, \mathbf{T}_{1-b})$ for
	a secret key $\mathbf{T}_{1-b} \sample D_{\ZZ^m,\sigma}^{\bar{m}}$ of  its own.
  The two public keys  $(\pk_{\USR,i})_{i\in \bit}$ are then certified by the adversarially-controlled $\GM$.
  Notice that in the adversary's view, both public  keys $\pk_{\USR,b}$ and $\pk_{\USR,1-b}$ are identically distributed.

  To answer  decryption queries $(\mathbf{\Psi} = (\vk, \mathbf c_\rec, \mathbf c_\OA, \Sigma),L)$, for any query pertaining to   $\pk_{\USR,b}$, the reduction invokes its ROR challenger to obtain an IBE
  private key for the identity $\vk \neq \vk^\star$ and uses the result to decrypt $\mathbf c_\rec$. For any decryption query involving
  $\pk_{\USR,1-b}$,  the reduction can faithfully run the actual decryption algorithm  using its trapdoor $\mathbf T_{1-b}$.
  Open queries are answered using $\mathbf T_\OA$ as in the real \textsf{Open} algorithm.

  When the adversary $\adv$ decides to do so, it queries a challenge for a triple $((\mathbf A_R, \mathbf{u}_R), \mathbf w, L)$ of its choice subject to the constraint
  $\mathbf{u}_R = \mathbf A_R \cdot \mathbf{w}$. At this point, $\bdv$ queries a challenge to
  its own challenger for the message $\mathbf w$ and obtains a ciphertext  $\mathbf c$, which is  embedded in
  $\mathbf{\Psi}^\star=(\vk^\star, \mathbf c, \mathbf c_\OA^\star, \Sigma^\star)$  while $\mathbf c_\OA^\star$ and $\Sigma^\star$ are generated as
  in Game $3$ (in particular, $\mathbf c_\OA^\star$ encrypts a random string instead of a hash value of $\pk_{\USR,b}$). After the challenge phase, all queries to the proving oracle  $\mathcal{P}$ are replied by returning a simulated ZK argument as in Game $3$.

  When  $\adv$ ends, it  outputs a bit $b' \in \{0,1\}$. If $b' = b$,  the reduction outputs  \textsf{Real}. Otherwise, it outputs \textsf{Random}.
  Indeed, if the ROR challenger is playing the real game, we are exactly in Game $4$: we have $\Pr[b'=b | \mathsf{Real}] = \Pr[W_4]$.
  Otherwise, the  challenge ciphertext $\mathbf{\Psi}^\star$ is completely independent of $b \in \{0,1\}$
  so that we can only have $b'=b$ with probability $\Pr[b'=b| \mathsf{Random}]=1/2$. It follows that $\advantage{\mathrm{ROR}}{\bdv}(\lambda) \geq | \Pr[W_4] -1/2 |$.
\end{proof}

\subsubsection{Message Secrecy}

\begin{theorem} \label{secrecy-thm}
  The scheme provides message secrecy assuming that  the $\LWE$ assumption holds and that $\mathcal{OTS}$ is a strongly unforgeable one-time signature.
%   \textnormal{(The proof is presented in Appendix~\ref{proof-secrecy-thm}.)}
\end{theorem}

\begin{proof}
  We proceed via a sequence of games. The first one corresponds to the experiment of Definition \ref{security-def} when the challenger's bit $b$ is $1$ and the
  adversary obtains an actual encryption of the witness $\mathbf{w} \in \{0,1\}^m$ and real proofs at each invocation of the $\mathsf{PROVE}(.)$ oracle. In the last
  game, the adversary $\adv$ is given an encryption of some random plaintext whereas $\mathsf{PROVE}(.)$ returns simulated zero-knowledge arguments which are
  generated a simulator $\mathcal{P}'$ that does not use any witness. In Game $i$, $W_i$ stands for the event that the adversary $\adv$ outputs the bit $b'=1$.
  \smallskip  \smallskip

  \noindent \textbf{Game} $1$: This is the real game, where the challenger feeds $\adv$ with
  public parameters $\param$ containing $ \bar{\mathbf{A}},  \mathbf{U}, \mathbf{V} \in \ZZ_q^{n \times m} $ and
  $\mathbf{F} \in \ZZ_q^{2n \times n \bar{m}k} $. The adversary produces  public
  keys $\pk_{\OA} = \mathbf{B}_{\OA} \in \ZZ_q^{n \times \bar{m}}$
  and $\pk_{\GM}= ( \mathbf{A}, \{\mathbf{A}_i\}_{i=0}^{\ell},
  \mathbf{D}_0  , \mathbf{D}_1, \mathbf{D},  \mathbf{u}  )$ on behalf of the opening authority and the group manager which are both under its control.
  The challenger and $\adv$ run an execution of the $\mathsf{JOIN}$ protocol
  which allows $\adv$ to register and certify  the public  key $\pk_{\USR}=\mathbf{B}_{\USR} \in \ZZ_q^{n \times \bar{m}}$ of some
  honest receiver chosen by the challenger. Then, the adversary $\adv$  makes a polynomial number of
  decryption queries which the challenger faithfully handles using the private key  $\sk_{\USR} = \mathbf{T}_{\USR} \in \ZZ^{m \times \bar{m}} $
  for which $\mathbf{B}_{\USR} \cdot \mathbf{T}_{\USR} = \mathbf{0}^{n \times \bar{m}} $. At some point, the adversary $\adv$ outputs a triple  $((\mathbf{A}_R,\mathbf{u}_R),\mathbf{w},L)$ such
  that $\mathbf{u}_R= \mathbf{A}_R \cdot \mathbf{w} \bmod q$, with $\mathbf{A}_R \in \ZZ_p^{n \times m}$, $\mathbf{u}_R \in \ZZ_q^n$ and $\mathbf{w} \in \{0,1\}^m$.
  At this point, the challenger generates  a challenge ciphertext
  $\Psi^\star= (\vk^\star,\mathbf{c}_{\rec}^\star, \mathbf{c}_{\oa}^\star, \Sigma^\star)$ consisting of a group encryption of
  the real witness $\mathbf{w}$ under $\pk_{\USR} =\mathbf{B}_{\USR}$. Then, the adversary  obtains a polynomial number of proofs $\pi_{\Psi^\star}^\star$ related to the
  challenge ciphertext
  $\Psi^\star$ and is granted further access to the   decryption oracle under the obvious
  restrictions. When $\adv$ halts, it
  outputs a bit $b' \in \{0,1\}$. % and we call $W_1$ the event that $b'=1$.
  \smallskip  \smallskip

  \noindent \textbf{Game} $2$: In this game, we modify the $\mathsf{DEC}(.)$ oracle and have the challenger reject any ciphertext of the form
  $\Psi= (\vk,\mathbf{c}_{\rec}, \mathbf{c}_{\oa}, \Sigma)$
  such that $\vk=\vk^\star$
  (note that $\vk^\star$ can be generated at the outset of the game w.l.o.g.).
  Clearly Game $2$ is identical to Game $1$ until the event that the challenger rejects a ciphertext that would not have been rejected in Game $1$.
  This can only occur if $\adv$ is able to break the strong unforgeability of the one-time signature $\mathcal{OTS}$.
  As in the proof of Theorem~\ref{anon-thm}, we have  $|\Pr[W_2]-\Pr[W_1]| \leq \mathbf{Adv}^{\mathrm{ots}}(\lambda)$, which is negligible
  if $\mathcal{OTS}$ is strongly unforgeable.
  \smallskip  \smallskip

  \noindent \textbf{Game} $3$:  We now modify the generation of proofs
  $\pi_{\Psi^\star}^\star$. Instead of generating them using the  witnesses used in the generation of $\Psi^\star$, we rely on
  the zero-knowledge simulator of the argument system of \cref{sse:stern-abstraction} at each invocation
  of  $\mathsf{PROVE}^b_{\mathcal{P},\mathcal{P}'}$ after the challenge phase (note that, since we assume trusted public parameters, the simulator can use   techniques \cite{Dam00} to achieve statistically
  perfect simulation without increasing the number of rounds). The statistical ZK property of the argument system ensures that  this change will remain unnoticed, even in the
  view of an all powerful adversary: we have
  $|\Pr[W_3]-\Pr[W_2] | \in \mathsf{negl}(\lambda)$. From now onwards, the random coins  $coins_{\mathbf{\Psi}}^\star=\big( \mathbf{s}_{\rec}^\star, \mathbf{R}_{\rec}^\star , \mathbf{x}_{\rec}^\star, \mathbf{y}_{\rec}^\star, \mathbf{s}_{\oa}^\star, \mathbf{R}_{\oa}^\star ,\mathbf{x}_{\oa}^\star, \mathbf{y}_{\oa}^\star \big) $ are no longer used by the $\mathsf{PROVE}$ oracle.
  \smallskip  \smallskip

  \noindent \textbf{Game} $4$: In the generation of $\Psi^\star$, we
  set  $\mathbf{c}_{\rec}^\star$ as an encryption of a random element of $\ZZ_p^m$. Since
  the random encryption coins $  \mathbf{s}_{\rec}^\star, \mathbf{R}_{\rec}^\star , \mathbf{x}_{\rec}^\star, \mathbf{y}_{\rec}^\star $ are not used in Game $3$, Lemma \ref{ABB-simple} gives a
  simple
  reduction showing that
  any
  significant change in  $\adv$'s behavior would imply  a selective adversary against the ABB identity-based encryption scheme.
  The result of
  \cite{ABB10} tells us  that, under the $\LWE$ assumption, Game $4$ is computationally indistinguishable from Game $3$ in the adversary's view: we have
  $|\Pr[W_4]-\Pr[W_3]| \leq
  \mathbf{Adv}^{\mathsf{LWE}}(\lambda) $.
  \smallskip  \smallskip

  \noindent \textbf{Game} $5$: We bring a last modification to  the $\mathsf{DEC}(.)$
  oracle and now refrain from applying  the rejection rule of Game $2$. If
  $\mathcal{OTS}$ is strongly unforgeable, the distance $|\Pr[W_5]-\Pr[W_4]|
  \leq  \mathbf{Adv}^{\mathrm{ots}}(\lambda) $ must be negligible.
  \medskip

  In the last game, the  oracle  $\mathsf{PROVE}(.)$
  does not need to know any witness. It thus mirrors  the experiment of Definition \ref{security-def} where the
  challenger's bit is $b=0$.  Putting everything altogether, we get
  $|\Pr[W_5]-\Pr[W_1]| \in \mathsf{negl}(\lambda) $, which yields the claimed result.
\end{proof}

\begin{lemma} \label{ABB-simple}
Any PPT adversary that can distinguish Game $4$ from Game $3$ implies a selective adversary against the ABB IBE scheme.
\end{lemma}

\begin{proof}
  Let us assume  a $\ppt$ adversary $\adv$ such that $\varepsilon = \bigl| \Pr[W_4] - \Pr[W_3] \bigr|$ is noticeable. We use $\adv$ to construct a PPT
	adversary $\bdv$ that breaks the IND-sID-CPA security 
%(as defined in Definition~\ref{de:IND-sID-CPA}) 
of the ABB scheme, which would contradict the $\LWE$ assumption, as established
		in~\cite[Th. 23]{ABB10}.

  At the very beginning of the IND-sID-CPA game, the reduction $\bdv$ generates   a one-time signature key pair $(\sk^\star, \vk^\star)$ and hands $\vk^\star$ to its selective security challenger as the target identity under which the challenge ciphertext will later be computed. In response, $\bdv$ receives the public parameters
	$$\mathsf{PP} = (\bar{\mathbf A}, \mathbf B, \mathbf U) \in \Zq^{n \times m} \times \Zq ^{n \times \bar m} \times \Zq^{n \times m}$$ from its IBE challenger.

  The reduction then runs the missing steps of the actual $\Setup_{\mathsf{init}}$ algorithm: namely, $\bdv$ samples $\mathbf F \sample \U(\Zq^{2m \times n\bar{m}k}), \mathbf V \sample \U(\Zq^{n \times m})$ and generates $\compar$ before sending the common public parameters
  $$\param = \bigl\{\lambda, n, q, k, m, B, \chi, \sigma, \beta, \ell, \kappa,  \mathcal{OTS}, \compar, \mathsf{FRD},   \bar{\mathbf{A}}, \mathbf{G}, \mathbf{F}, \mathbf{U}, \mathbf{V}  \bigr\}$$
  to the adversary $\adv$.

  At this point, the adversary $\adv$ chooses the public keys $\pk_\OA = \mathbf B_\OA \in \Zq^{n \times \bar{m}}$ and $\pk_\GM = (\mathbf A, \{ \mathbf A_i \}_{i=0}^\ell, \mathbf D_0, \mathbf D_1, \mathbf D, \mathbf{u})$ on
	behalf of the opening authority and the group manager. It also  starts an execution of the joining protocol in which the reduction $\bdv$
	defines $\pk_\USR = \mathbf B \in \Zq^{n \times \bar m}$ as the honest receiver's public key, where $\mathbf B \in \ZZ_q^{n \times \bar{m}}$ is taken from the public parameters
	$\mathsf{PP}$ supplied by its IBE challenger. Note that $\pk=\mathbf{B} \in \ZZ_q^{n \times \bar{m}}$ is  distributed as a real key in $\adv$'s view.
	This public key is certified by $\mathbf A$ which controls the $\GM$.

  In the next stage, $\adv$ makes a number of decryption queries for ciphertexts of the form $\mathbf{\Psi} = (\vk, \mathbf c_\rec, \mathbf c_\OA, \Sigma)$. To answer these,
	the reduction invokes its IBE challenger so as  to obtain an IBE private key $\mathbf E_\vk \in \ZZ^{(m+\bar m) \times m}$ for the identity
	$\vk \neq \vk^\star$. The resulting $\mathbf E_\vk$ is used it to IBE-decrypt $\mathbf c_\rec$ and return the corresponding witness $\mathbf w$ to $\mathbf A$ .

  At some point, the adversary $\adv$ queries a challenge ciphertext by outputting a triple $((\mathbf A_R, \mathbf u_R), \mathbf w, L)$ such that $\mathbf{w} \in \{0,1\}^m$  satisfies
	  $\mathbf u_R = \mathbf A_R \cdot \mathbf{w} \bmod q$. Then, the reduction $\bdv$ requests a challenge ciphertext $\mathbf c^\star_\rec$ to its IBE
	challenger by sending it  the messages $\mathbf m_1 = \mathbf{w} \in \{0,1\}^m$ and $\mathbf m_0 \sample \U(\{0,1\}^m)$. The resulting ciphertext $\mathbf c^\star_\rec$
	is embedded  in $\mathbf{\Psi}^\star = (\vk^\star, \mathbf c_\rec^\star, \mathbf c_\OA^\star, \Sigma^\star)$ by faithfully computing
	$\mathbf c_\OA^\star$ and $\Sigma^\star$ as in the actual $\textsf{Enc}$ algorithm.

  After the challenge phase, $\adv$ keeps sending  decryption queries for ciphertexts $\Psi^\star$ containing one-time verification keys
	$\vk \neq \vk^\star$ and these decryption queries are answered as before. In addition, $\adv$ is granted access to the stateful oracle $\mathsf{PROVE}^b_{\mathcal{P},\mathcal{P}'}$.
	Recall that, from Game $3$ onwards, all these queries are answered by returning simulated zero-knowledge arguments.
	%The following proof queries are answered as in Game~3, meaning that they are simulated by the zero-knowledge simulator of the underlying argument system.
  Eventually $\adv$ outputs a bit $b' \in \{0,1\}$ which is also returned by $\bdv$  to its own challenger.

  If the IBE challenger provides  a challenge  $\mathbf c^\star_\rec$ that encrypts a random message (i.e., by encrypting $\mathbf{m}_0$), then we are exactly in the setting of Game $4$.
	 In the even that $\mathbf c^\star_\rec$ rather encrypts  $\mathbf{m}_1=\mathbf{w} \in \{0,1\}^m$, $\adv$'s view is exactly the same as in Game $3$. If we denote by
	\textsf{Random} (resp. \textsf{Real}) the event that the IBE challenger chooses to encrypt $\mathbf m_0$ (resp. $\mathbf m_1$), the advantage of the reduction $\bdv$ as an IND-sID-CPA
	adversary is
  \begin{align*}
  \advantage{\textrm{sID-CPA}}{\bdv}(\lambda) &= \left| \Pr[b'=1 | \textsf{Real}] - \Pr[b'=1 | \mathsf{Random}] \right| = \left|\Pr[W_3] - \Pr[W_4] \right|\\
  &= \varepsilon,
\end{align*}
  which concludes our proof.
\end{proof}




 %\subsection{ Soundness (Proof of Theorem~\ref{soundness-thm})} \label{proof-soundness-thm}
 


\subsubsection{Soundness}


\begin{theorem} \label{soundness-thm}
  The scheme provides soundness under the $\SIS$ assumption.
 %\textnormal{(The proof is detailed in Appendix~\ref{proof-soundness-thm}.)}
\end{theorem}


\begin{proof}
To prove the result, we observe that, in order to break the soundness property, the adversary must come up with a relation
$\pk_{\mathcal{R}}=(\mathbf{A}_{{R}},\mathbf{u}_{R}) \in \ZZ_q^{n \times m} \times \ZZ_q^n$,  a ciphertext
$\Psi^\star= (\vk^\star,\mathbf{c}_{\rec}^\star, \mathbf{c}_{\oa}^\star, \Sigma^\star)$, a label $L$ and produce a convincing proof $\pi_{\Psi^\star}$ such that either
\begin{enumerate}
  \item $\mathbf{c}_{\oa}^\star$ does not decrypt to a string $\mathbf{h} \in \{0,1\}^m$ such that $\mathbf{h}_{\USR} = \mathbf{H}_{2n,q-1} \cdot \mathbf{h} \in \ZZ_q^{2n}$ coincides
    with $\mathbf{h}_{\USR} = \mathbf{F} \cdot \mathsf{mdec}_{n,m,q}(\mathbf{B}_{\USR}^T)$ for some $\pk_{\USR}=\mathbf{B}_{\USR} \in \ZZ_q^{n \times \bar{m}}$ appearing in $\mathsf{database}$.
  \item  $\mathbf{c}_{\oa}^\star$  opens to a certified public key $\pk_{\USR}=\mathbf{B}_{\USR} \in \ZZ_q^{n \times \bar{m}}$, which belongs to  $\mathsf{database}$ (and for which a certificate
    was issued), but $\mathbf{B}_{\USR} $ is outside the language $\mathcal{PK}$ of valid public keys. This case is immediately ruled out
    by the density of the public key space.
    Namely, all matrices
    $\mathbf{B}_{\USR} \in \ZZ_q^{n \times \bar{m}} $ are potentially valid public keys as there always exist a small-norm matrix $\mathbf{T}_{\USR} \in \ZZ^{m \times \bar{m}}$
    such that $\mathbf{B}_{\USR} = \bar{\mathbf{A}} \cdot \mathbf{T}_{\USR} \bmod q$.
  \item $\mathbf{c}_{\oa}^\star$  opens to a certified   key $\pk_{\USR}=\mathbf{B}_{\USR}$ for which
    $\Psi^\star= (\vk^\star,\mathbf{c}_{\rec}^\star, \mathbf{c}_{\oa}^\star, \Sigma^\star)$ is not a valid encryption of $\mathbf{w} \in \{0,1\}^m$ such that
    $\mathbf{u}_R= \mathbf{A}_R \cdot \mathbf{w} \bmod q$.
  \item The opening algorithm fails to uniquely identify the receiver. This occurs if  $\mathbf{c}_{\oa}^\star$ decrypts to a string $\mathbf{h} \in \{0,1\}^m$ such that $\mathbf{h}_{\USR}' = \mathbf{H}_{2n,q-1} \cdot \mathbf{h} \in \ZZ_q^{2n}$ corresponds to
    at least two distinct public keys $\mathbf{B}_{\USR,0} ,\mathbf{B}_{\USR,1} \in \ZZ_q^{n \times \bar{m}}$ which satisfy
    $$\mathbf{h}_{\USR}' = \mathbf{F} \cdot \mathsf{mdec}_{n ,\bar{m},q}(\mathbf{B}_{\USR,0}^T )  \bmod q=\mathbf{F} \cdot \mathsf{mdec}_{n ,\bar{m},q}(\mathbf{B}_{\USR,1}^T )  \bmod q.  $$
    Since $\mathsf{mdec}_{n,\bar{m},q}(.) : \ZZ_q^{\bar{m} \times n} \rightarrow \{0,1\}^{n \bar{m} k}$ is an injective function, the above equality necessarily implies a
    collision for the $\mathsf{SIS}$-based hash function built upon $\mathbf{F} \in \ZZ_q^{2n \times n \bar{m} k}$: namely,
    $$ \mathsf{mdec}_{n ,\bar{m},q}(\mathbf{B}_{\USR,0}^T )  -  \mathsf{mdec}_{n ,\bar{m},q}(\mathbf{B}_{\USR,1}^T ) ~~\in \{-1,0,1\}^{n\bar{m} k}  $$
    is a short non-zero vector of $\Lambda_q^\perp (\mathbf{F})$.
\end{enumerate}
Having shown that cases \textit{b} and \textit{d} cannot occur if the $\mathsf{SIS}$ assumption holds, we only need to consider cases \textit{a}  and \textit{c}. The computational soundness of the argument system ensures that, by replaying
the soundness adversary a sufficient number of times,
the knowledge extractor will be  able to extract either:
(i) A breach in the computational soundness of the argument system and thus the binding property of the commitment scheme $\mathsf{COM}$ (which relies on  the $\mathsf{SIS}$ assumption with the commitment scheme of \cite{KTX08}). Note that this situation covers case (\textit{c}.) above.
(ii) A set of witnesses
\begin{eqnarray*}
  \left\{
    \begin{array}{l}
      \tau \in \{0,1\}^\ell,   ~\mathbf{d}=[\mathbf{d}_1^T | \mathbf{d}_2^T ]^T \in [-\beta,\beta]^{2m},~\mathbf{r} \in [-\beta,\beta]^m \\
      \mathbf{t}_{\USR} \in \{0,1\}^{m},~\mathbf{w}_{\USR} \in \{0,1\}^{\bar{m}}  \\
      \mathbf{b}_{\USR} \in \{0,1\}^{n \bar{m} k},   ~\mathbf{s}_{0,\rec} \in \{0,1\}^{nk}, ~
      \mathbf{z}_{\mathbf{\Psi}}   \in \{0,1\}^{4n \bar{m} k^2}
      \\
      \mathbf{x}_{\rec}, ~\mathbf{y}_{\rec} \in [-B,B]^m,  ~ \mathbf{z}_{\rec} \in [-\beta mB,\beta mB]^{\bar{m}} ,  ~\mathbf{w} \in \{0,1\}^m, \\
      \mathbf{s}_{\oa} \in \{0,1\}^{nk},~\mathbf{x}_{\oa}, ~\mathbf{y}_{\oa} \in  [-B,B]^m,   ~\mathbf{z}_{\oa} \in [-\beta mB,\beta mB]^{\bar{m}}
    \end{array}
  \right.
\end{eqnarray*}
satisfying relations \eqref{eq:big-system-main-scheme}.
%(\ref{rel-un})-(\ref{rel-trois}).
Given that witnesses $\tau \in \{0,1\}^\ell$, $\mathbf{d} \in [-\beta,\beta]^{2m}$, $\mathbf{r} \in [- \beta,\beta]^m$ and $\mathbf{t}_{\USR} \in \{0,1\}^m$ satisfy \eqref{eq:big-system-main-scheme}, %(\ref{rel-un}),
it comes that $(\tau,\mathbf{d},\mathbf{r})$ form a valid signature for the message $\mathbf{t}_{\USR} \in \{0,1\}^m$. At this point,  case \textit{a} implies that no matrix
$\mathbf{B}_{\USR} \in \ZZ_q^{n \times \bar{m}} $ of
$\mathsf{database}$ decomposes to a string $\mathbf{h}_\USR \in \{0,1\}^{n\bar{m}k}$ such that $\mathbf{t}_{\USR} = \mathsf{vdec}_{n,q-1}(\mathbf{F} \cdot \mathbf{h}_{\USR} \bmod q )$ was
signed by the reduction during  an execution of $\mathsf{JOIN}$. This implies that
the pair
$\big(\mathbf{t}_{\USR} ,  (\tau,\mathbf{d},\mathbf{r})  \big)$ forms a forgery for the $\mathsf{SIS}$-based signature scheme of Section~\ref{se:gs-lwe-sigep}. The reduction is straightforward
and omitted.
\end{proof}

