Résumé en français
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\chapter*{Remerciements}


\addcontentsline{toc}{chapter}{Remerciements}


\chapter*{Remerciements/Acknowledgements}


\addcontentsline{toc}{chapter}{Acknowledgements}


\addcontentsline{tof}{chapter}{Remerciements}




I would like to thank


\begin{otherlanguage}{french}


Je tiens tout d'abord à remercier mon directeur de thèse Benoît Libert, qui m'a accompagné durant toute la durée de ma thèse et m'a gratifié de sa présence et de sa disponibilité tout au long de cette aventure. Je le remercie aussi de son support lors des phases de soumissions, des possibles \textit{rebuttals} qui s'en suivaient et lors des présentations en conférences.


Ses enseignements et ses conseils m'ont aidé à avoir une vision plus claire de la recherche et me permettront désormais de prendre mon envol.


\end{otherlanguage}




I would also like to thank Dario Catalano and David Pointcheval, who accepted to review this thesis during the summer and granted me with their useful remarks. I also thank the rest of the committee: Shweta Agrawal, who also invited me for a visit in the nice city of Chennai and taught me to be cautious with monkeys; PierreAlain Fouque, who also accepted to be part of my halfway thesis committee; Philippe Gaborit and our discussions at \textit{LaLondelèsMaures}; and Carla Ràfols, who I met in several conferences and for the organization of the insightful \textsc{CostIacr} School on Randomness in Cryptography.




I also want to thank my coauthors for their ideas and our useful discussions: Benoît Libert, San Ling, Khoa Nguyen, Thomas Peters, Huaxiong Wang and Moti Yung. Without you, my research would have be less productive!




For inviting me and their warm reception, I would like to express my gratitude to Shweta Agrawal from IIT Madras; Adeline Langlois and PierreAlain Fouque from Rennes; Satya Lokam from Microsoft Bangalore; Abderrahmane Nitaj and Brigitte Vallée from Caen; and Ali El Kaafarani from Oxford.


It was the opportunity for me to discover other research environments and our interesting discussions.




Without having to move so far, I also want to thank the AriC team in Lyon and especially the crypto part with its dynamic that is very supportive to research:


Damien, Benoît, Fabien,


Alexandre, Alice, Alice, Alonso, Benjamin, Chen, Chitchanok, Elena, Gottfried, Ida, Jiangtao, Jie, Junqing, Laurent, Marie, Miruna, Radu, Sanjay, Shi, Somindu and Weiqiang.


I especially want to thank to Damien Stehlé, who introduced me to cryptography with his course in master, and helped me going further in this topic by introducing me to Fré Vercauteren from KU Leuven who was my first contact with research in cryptography.


Where the boundaries are blurry, I would like to thank Guillaume Hanrot, who was there to guide me in solving the administrative conundrum and also for showing his interest for my research.


From the rest of the permanent members of AriC team, I want to thank


Nicolas Brisebarre, ClaudePierre Jeannerod, Vincent Lefèvre, Nicolas Louvet, JeanMichel Muller, Nathalie Revol, Bruno Salvy and Gilles Villard.




\begin{otherlanguage}{french}


Je tenais aussi à remercier mes collègues, à commencer par ceux qui ont eu la chance de partager mon bureau et mes lubies: Valentina pour sa joie de vivre


\end{otherlanguage}




\begin{otherlanguage}{french}


\end{otherlanguage}





@ 2,7 +2,7 @@ In the last fifty years, the use of cryptography has shifted from military and c


For instance, the Enigma machine had a design for military purposes, and another one for companies (Enigma A26).


As of today, about $60\%$ of the first million most visited websites propose encrypted and authenticated communications (via \texttt{https}), and so are most of the communications channels used by electronic devices (like \textit{Wifi Protected Access}).




At the same time, the growth of exchanged data and the sensitivity of transferred information make the urge of procecting these data efficiently even more critical.


At the same time, the growth of exchanged data and the sensitivity of transferred information make the urge of protecting these data efficiently even more critical.


While we are reaching the Moore's law barrier, other threats exist against nowadays' cryptosystems.


For instance, the existence of a quantum computer with sufficient memory~\cite{Sho99} would break most of realworld cryptographic designs, which mostly rely on modular arithmetic assumptions.


In this context, it is crucial to design cryptographic schemes that are believed to be quantumresistant.





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Les cinquante dernière années, l'utilisation de la cryptographie s'est éloignée de ses origines militaires et de son usage pour le secret commercial pour se démocratiser à un public plus large.


Par exemple, la machine Enigma initialement conçue pour un usage militaire a été déclinée pour un usage commerciale (la machine Enigma A26).


Aujourd'hui, environ $60\%$ du premier million des sites les plus visités propose une connexion chiffrée et authentifiée (à l'aide du protocole \texttt{https}), tout comme les canaux de communication des appareils électronique portatifs (comme la norme \texttt{WPA}, en anglais \textit{Wifi Protected Access}).




Dans le même temps, la croissance des données échangée en ligne et la sensibilité de ces informations rend de plus en plus urgent la protection de ces canaux.


Pendant que la loi de Moore\footnote{La loi qui prédit la puissance de calcul des processeurs modernes.} atteint ses limites, d'autres menaces existent sur nos cryptosystèmes actuels.


Par exemple, l'existence d'un ordinateur quantique possédant suffisamment de mémoire~\cite{Sho99} casserait la majorité des constructions cryptographiques utilisées en pratique, puisqu'elles reposent sur des hypothèses d'arithmétique modulaire dont la structure peutêtre exploitée par un adversaire quantique.


Dans cette situation, il est crucial de construire des schémas cryptographiques qui résisteraient à une menace quantique.




Pour répondre à ce problème, la \textit{cryptographie postquantique} est née au début des années 2000.


Les différents candidats reposent sur différents objets mathématiques, comme les réseaux euclidiens, les codes correcteurs d'erreurs, les systèmes polynomiaux multivariés, etc.


Récemment, le NIST (\textit{National Institude of Standards and Technology}) a organisé une compétition pour évaluer les différentes solutions postquantiques en terme de chiffrement et de signature~\cite{NIS17}.


Dans cette compétition, 82 protocoles ont été proposés, parmi lesquels 28 reposent sur les réseaux euclidiens, 24 sur les codes correcteurs, 13 sur des systèmes multivariés, 4 sur des fonctions de hachages et 13 sur d'autres objets.




Si la cryptographie pratique vise principalement à fournir des schémas de signatures et de chiffrement, comme l'atteste la compétition du NIST,


la recherche théorique propose des solutions à des problèmes plus spécifiques, comme la construction de systèmes de monnaie électronique\footnote{À ne pas confondre avec les cryptomonnaies\ldots}~\cite{CFN88}, qui sont l'équivalent numérique de notre monnaies échangée. Les pièces sont délivrées par une autorité centrale (la banque), et les dépenses restent intraçables. En cas de comportement malhonnête (comme une doubledépense), l'identité de l'utilisateur malicieux est révélée.




Les constructions cryptographiques doivent en plus vérifier des propriétés de sécurités.


Par exemple, un schéma de chiffrement doit pouvoir cacher un message en présence d'un attaquant passif voire actif (c'estàdire qui peut modifier certains messages).


Pour garantir ces exigences, les cryptographes fournissent des preuves de sécurité dans le cadre de modèles de sécurité précis.


Une preuve de sécurité précise principalement qu'une construction cryptographique est au moins aussi difficile qu'un problème supposé dur par la littérature.




Finalement, l'importance de la préservation de la vie privé et la protection des données a été un sujet qui a fait couler beaucoup d'encres, comme en atteste le développement de la règlementation générale sur la protection des données (RGPD) en 2016, mise en application ce 25 mai.


Il est donc intéressant pour les cryptographes de fournir des solutions qui resisteraient, dans le meilleurs des mondes, à un adversaire quantique.


Néanmoins, la construction de ces protocoles reposent de manière décisive sur les «\,preuves à divulgation nulles de connaissances\,». Ce sont des protocoles interactifs entre un prouveur et un vérificateur où le prouveur cherche à convaincre le vérificateur de la véracité d'une affirmation sans rien divulguer de plus que la valeur de vérité de cette affirmation.


Dans le contexte de la cryptographie postquantique, de tels systèmes de preuves sont limités en expressivité ou en terme de coût de calcul (en temps ou en mémoire).




\begin{comment}


\section{PrivacyPreserving Cryptography}


\label{se:privacypreservingcrypto}




In this context, `privacypreserving' refers to the ability of a primitive to provide some functionalities while holding sensitive information private.


An example of such primitives are \textit{anonymous credentials}~\cite{Cha85,CL01}.


Informally, this primitive allows users to prove themselves to some verifiers without telling their identity, nor the pattern of their authentications.


To realize this, this system involves one (or more) credential issuer(s) and a set of users who have their own secret keys and pseudonyms that are bound to their secret.


Users can dynamically obtain credentials from an issuer that only knows users' pseudonyms and obliviously sign users' secret key as well as a set of attributes.


Later on, users can make themselves know to verifiers under a different pseudonym and demonstrate possession of a certification from the issuer, without revealing neither the signature nor the secret key.


This primitive thus allows a user to authenticate to a system (e.g., in anonymous access control) while retaining its anonymity.


In addition, the system is guaranteed that users indeed possess a valid credential.




Interests in privacybased cryptography date back to the beginning of publickey cryptography~\cite{Rab81,Cha82,GM82,Cha85}.


A reason for that could be the similarities between the motivations of cryptography and the requirements of privacy protection.


Additionally, cryptographers' work in this field may have direct consequences in term of services that could be developed in the realworld.


Indeed, having a practical anonymous credential scheme will enable its use for access control in a way that limits security flaws.


Whereas, nowadays' implementations are based on more elementary building blocks, like signatures, whose manipulations may lead to different security holes~\cite{VP17}.




Similarly, \textit{advanced primitives} often involve simpler building blocks in their design.


The difference lies in that provable security conveys security guarantees for the construction.


As explained before, these proofs make the security of a set of schemes rely on hardness assumptions.


Thus, the security relies on the validity of those assumptions, which are independently studied by cryptanalysts.


Hence, security is guaranteed by the study of those assumptions.


For example, the security analysis of multilinear maps in~\cite{CHL+15} made obsolete a large amount of candidates at this time.


This example reflects the importance of relying on wellstudied and simple assumptions as we will explain in~\cref{ch:proofs}.




In the context of this thesis, the developed cryptographic schemes rely on lattices and bilinear maps over cyclic groups.


Latticebased cryptography is used to step towards postquantum cryptography, while the latter proves useful in the design of practical schemes.


The details of these two structures are given in~\cref{ch:structures}.




\subsection{ZeroKnowledge Proofs}




As explained before, zeroknowledge proofs are a basic building block for privacypreserving cryptography.


They require completeness, soundness and zeroknowledge properties.


Completeness captures the correctness of the protocol if everyone is honest. In the case of a dishonest prover, soundness asks the probability that the verifier is convinced to be negligible.


On the contrary, if the verifier is cheating, the zeroknowledge property guarantees that the prover's secret remains hidden.




In the case of identification schemes, the nature of the secret remains simple and solutions exist under multiple assumptions~\cite{Sch96,Ste96,KTX08,Lyu08}.


For more complex statements, such as proving correct computation, a gap appears between postquantum schemes and modular arithmeticbased schemes.


In the case of pairingbased cryptography, there exist noninteractive zeroknowledge proofs which can prove a large variety of statements~\cite{GOS06,GS08} without idealized assumptions.


Such proofs are still missing in the context of postquantum cryptography so far.




In the lattice world, there are two main families of proof systems: Schnorrlike proofs~\cite{Sch96,Lyu09} and Sternlike proofs~\cite{Ste96}, named after their respective authors.


The first family works on some structured lattices. Exploiting this structure allows for more compact proofs, while the expressiveness of statements is quite restricted.


The second kind of proofs is combinatorial and works on the representation of lattice elements (as matrix and vectors).


By nature, these proofs are quite expensive in term of communication complexity.


However, they can be used to prove a wide variety of statements as we will explain in more details along this thesis and especially in~\cref{sse:stern}.


More generally, zeroknowledge proofs are detailed in~\cref{ch:zka}.




\subsection{Signatures with Efficient Protocols}




To enable privacypreserving functionalities, a possible avenue is to couple zeroknowledge proofs with signature schemes.


One of such signatures are \textit{signatures with efficient protocols}.


This primitive extends the functionalities of ordinary digital signature schemes in two ways: (i)~It provides a protocol to allow a signer to obliviously sign a hidden message and (ii)~Users are able to prove knowledge of a hidden messagesignature pair in a zeroknowledge fashion.




These two properties turn out to be extremely useful when it comes designing efficient anonymityrelated protocols such as anonymous credentials or ecash.


The design of effective signatures with efficient protocols is thus important for privacypreserving cryptography.




In this thesis, we provide two of these signature schemes.


One of them, described in~\cref{ch:sigmasig}, based on pairings, shifts the~\cite{LPY15} signature scheme to an idealized but practically acceptable model, aiming at efficiency.


The other, described in~\cref{ch:gslwe}, adapts a variant of Boyen's signature~\cite{Boy10,BHJ+15} along with the KawachiTanakaXagawa commitment scheme~\cite{KTX08} to provide a latticebased signature schemes that is compatible with Sternlike proofs.


This scheme has also been relaxed in the context of adaptive oblivious transfer where, in some places, it is only required to have randommessage security instead of security against chosenmessage security as described in~\cref{ch:otlwe}.




\section{Pairings and Lattices}




In this thesis, the proposed constructions rely on the assumed hardness of assumptions over pairingfriendly groups and lattices.


These two objects have widely been used in cryptography since the early 2000s~\cite{SOK00,Reg05}.


Even since, they attracted much attention from cryptographers, leading to multiple constructions in advanced cryptography (as in~\cite{Jou00,BBS04,BN06,GS08,LYJP14,LPQ17} for pairings, and~\cite{GPV08,ABB10,BV11,GSW13,dPLNS17} for lattices).




\subsection{PairingBased Cryptography}




A pairing is a bilinear map from two cyclic source groups to a target group.


This bilinear property takes advantage of a rich structure to groups that are compatible with such a map.


It is then not surprising to see the variety of schemes that stems from pairingbased cryptography.


In the context of privacybased cryptography, an important breakthrough was the introduction of GrothSahai proofs~\cite{GOS06,GS08} that allow proving in a noninteractive zeroknowledge fashion a large class of statements in the standard model.


For instance, GrothSahai proofs have been used in group signatures and anonymouscredential schemes~\cite{Gro07,BCKL08,BCC+09}, or ecash systems in the standard model~\cite{BCKL09}.




In this thesis, however, our pairingbased constructions focus on practicality.


Thus, they are instantiated in the random oracle model, where Schnorr's proof are made noninteractive through the FiatShamir transform when the statement to prove is simple enough.




A recent line of work in cryptanalysis of bilinear maps~\cite{KB16,MSS17,BD18} led to a change in the panorama of practical pairingbased cryptography.


This affects us in the sense that security parameter has to be increased in order to achieve the same security level.




Nevertheless, pairingbased cryptography offers a nice tradeoff between its capabilities and efficiency.


As an example, we can cite the work of Döttling and Garg~\cite{DG17}, who closed the problem of providing an identitybased encryption scheme which only relies on the DiffieHellman assumption (it is construction on cyclic groups that does not need pairings, as defined in~\cref{de:DDH}).


While their construction relies on a simpler mathematical object, it does not reach the efficiency of pairingbased ones~\cite{BB04}.




\subsection{LatticeBased Cryptography}




From an algebraic point of view, a lattice is a discrete subgroup of $\RR^n$,


which leads to a simple additive structure.


The core difference with numbertheoretic cryptography, such as discretelogarithmbased cryptography, is the existence of the geometrical structure of the lattice.


From this geometry rises some problems that are believed to withstand quantum computers.


Despite this apparently simple structure, some advanced primitives are only known, as of today, to be possible under lattice assumptions, such as fullyhomomorphic encryption~\cite{Gen09,GSW13}.




The versatility of latticebased cryptography is enabled by the existence of lattice trapdoors~\cite{GPV08,CHKP10,MP12}, as we explain in~\cref{sse:latticetrapdoors}.


Informally, the knowledge of a short basis for a lattice allows sampling short vectors, which is believed to be hard without such a short basis.


Furthermore, knowing a short basis for the lattice $\{\mathbf{v} \in \ZZ^m \mid \mathbf{A} \mathbf{z} = 0 \bmod q\}$ described by matrix $\mathbf{A} \in \ZZ_q^{n \times m}$ makes it possible to generate a short basis for a related lattice described by $[ \mathbf{A} \mid \mathbf{B}] \in \ZZ_q^{n \times m'}$.


An application for this property is Boyen's signature scheme~\cite{Boy10}.


In this scheme, a signature for message $m$ is a short vector in the orthogonal lattice of the matrix $\mathbf{A}_m = [\mathbf{A} \mid \mathbf{B}_m]$, where $\mathbf{B}_m$ is publicly computable.


Hence, knowing a trapdoor for $\mathbf{A}$ makes the computation of this short vector possible, and the message is bound to the description of the lattice $\mathbf{A}_m$.


Indeed, some extra care has to be taken to avoid multiplicative attacks.




Still, the use of lattice trapdoors comes at a price, as it significantly decreases the efficiency of cryptographic designs that use them~\cite{Lyu12,LLNW16}.


Given that we provide the first latticebased construction for the scheme we present, we focused on designing provablysecure scheme under wellstudied assumptions.




\section{Our Results}




In this thesis, we present several cryptographic constructions that preserve privacy.


These constructions are the result of both improvements we made in the use of zeroknowledge proofs and the ability to prove the security of our constructions under standard assumptions.


We believe that these advances on zeroknowledge proofs are of independent interest and that the given schemes are a step towards quantumsecure privacypreserving cryptography.


In the following, we detail four contributions that are developed in this thesis.


These results are taken from four published articles: \cite{LMPY16,LLM+16,LLM+16a,LLM+17}.




\subsection{Dynamic Group Signatures and Anonymous Credentials}




In~\cref{pa:gsac}, we present two primitives: dynamic group signatures and anonymous credentials.


We already described the behavior of anonymous credential in~\cref{se:privacypreservingcrypto}.


As for dynamic group signatures, they are a primitive that allows a group of users to authenticate messages on behalf of the group while remaining anonymous inside this group.


The users still remain accountable for their actions, as another authority knows a secret information that gives it the ability to lift anonymity of misbehaving users.




By itself, this primitive can be used to provide anonymous authentications while providing accountability (which is not the case with anonymous credentials).


For instance, in the Internet of things, such as smart cars, it is important to provide authenticated communication channels as well as anonymity. For car communications, if exchanged data may not be sensitive by themselves, the identity of the driver could be.


We can imagine a scenario where some burglars eavesdrop a specific car to know whenever a house is empty.




In this thesis, we present in~\cref{ch:sigmasig} pairingbased group signatures that aims at efficiency while relying on simple assumptions.


The resulting scheme shows competitive signature size with other schemes that rely on more adhoc assumptions, and its practicality is supported by an implementation.


This scheme is presented in~\cite{LMPY16}, which is joint work with Benoît Libert, Thomas Peters an Moti Yung presented at AsiaCCS'16.




\cref{ch:gslwe} presents the first \textit{dynamic} group signature scheme relying on lattice assumptions.


This has been made possible by adapting Sternlike proofs to properly interact with a signature scheme: a variant of Boyen's signature~\cite{Boy10,BHJ+15}.


It results in a \textit{signature with efficient protocols} that is of independent interest.


Later, it has been adapted in the design dynamic group encryption~\cite{LLM+16a} and adaptive oblivious transfer~\cite{LLM+17}.


This work is described in~\cite{LLM+16}, made with Benoît Libert, San Ling, Khoa Nguyen and Huaxiong Wang and presented at Asiacrypt'16.




\subsection{Group Encryption}




Group encryption schemes~\cite{KTY07} are the encryption analogue of group signatures.


In this setting, a user is willing to send a message to a group member, while keeping the recipient of the message hidden inside the group.


In order to keep user accountable for their actions, an opening authority is further empowered with some secret information allowing it to unanonymize ciphertexts.




More formally, a group signature scheme is a primitive allowing the sender to generate publicly verifiable proofs that: (1) The ciphertext is wellformed and intended to some registered group member who will be able to decrypt; (2) The opening authority will be able to 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, Tsiounis and Yung~\cite{KTY07}, the message secrecy and anonymity properties are required to withstand active adversaries, which are granted access to decryption oracles in all security definitions.




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


Furthermore, group encryption schemes are motivated by privacy applications such as anonymous trusted third parties, key recovery mechanisms or oblivious retriever storage system.


In cloud storage services, group encryption enables privacypreserving 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 to some anonymous certified client who has a valid account to the storage provider.


In case of suspicions on the archive's content, a judge should be able do identify the recipient of the archive.




To tackle the problem of designing latticebased group encryption, we needed to handle ``quadratic relations''.


Indeed, latticebased zeroknowledge proof systems were able to handle only relations where witnesses are multiplied by a public value.


Let us recall that, in LearningWithErrors schemes, an encryption have the form $\mathbf{A} \cdot \mathbf{s} + \mathbf{e} + \mathbf{m} \lceil \frac{q}{2} \rceil \bmod q$, where $\mathbf{A}$ is the recipient publickey.


As group encryption requires this publickey $\mathbf{A}$ to be private, a way to achieve this is to have a zeroknowledge proof system which handles relations where the witness is multiplied with a private matrix.




We address this issue introducing new technique to handle this kind of relations.


These techniques, based on a \textit{divideandconquer} strategy, are described in~\cref{ch:gelwe}, as well as the construction of the group signature scheme proven fullysecure in the standard model.


This work have been presented at Asiacrypt'16~\cite{LLM+16a} and have been done with Benoît Libert, San Ling, Khoa Nguyen and Huaxiong Wang.




\subsection{Adaptive Oblivious Transfer}




Oblivious transfer is a primitive coined by Rabin~\cite{Rab81} and later extended by Even, Goldreich and Lempel~\cite{EGL85}.


It involves a server with a database of messages indexed from $1$ to $N$ and a receiver with a secret index $\rho \in \{1,\ldots,N\}$.


The protocol allows the receiver to retrieve the $\rho$th message from the receiver without letting him infer anything on his choice.


Furthermore, the receiver only obtains the $\rho$th message and learns nothing about the other messages.




In its adaptive flavor~\cite{NP99}, oblivious transfer allows the receiver to interact $k$ times with the server to obtain $k$ messages in such a way that, each request may depend on the previously retrieved messages.




From a theoretical point of view, oblivious transfer is known to be a \textit{complete building block} for cryptography in the sense that, if it can be realized, then any secure multiparty computation can be.


In its adaptive variant, oblivious transfer has applications in privacypreserving access to sensitive databases (such as medical records or financial data) stored in an encrypted form on a remote server.




In its basic form, (adaptive) oblivious transfer does not restrict in any way the population of users who can obtain specific records.


In many sensitive databases (e.g., DNA samples or patients' medical history), however, not all users should be able to access the whole database.


It is thus crucial to protect the access to certain entries conditioned on the receiver holding suitable credentials delivered by authorities.


At the same time, privacy protection requires that authorized users should be able to query database records while leaking as little as possible about their interests or activities.




This requirements is handled by endowing oblivious transfer with access control, as stated by Camenish, Dubovitskaya and Neven~\cite{CDN09}.


In this variant, each database record is protected by a different access control policy.


Based on their attributes, users can obtain credentials from predetermined authorities, which entitle them to anonymously retrieve database records of which the access policy accepts their certified attributes.


During the transfer phase, the user demonstrates, in a zeroknowledge manner, possession of an attribute string compatible with the policy of a record in the database, as well as a credential for this attribute.


The only information that the database holder eventually learns is that some user retrieved some record which he was authorized to obtain.




To achieve this, an important property is the expressiveness of such access policies.


In other words, the system should be able to handle complex attribute policies while keeping time and memory consumption reasonable\footnote{Here, ``\textit{reasonable}'' means (probabilistic) polynomial time.}.


In this thesis, we propose in~\cref{ch:otlwe} a zeroknowledge protocol to efficiently handle any access policy that can be described with a logarithmicdepth boolean circuit, also known as $\mathsf{NC}1$, based on lattices.


In the context of adaptive oblivious transfer with access control, most of the schemes (based on pairing assumptions) manage to handle the case of conjunctions under reasonable assumptions~\cite{CDN09,CDNZ11,ACDN13}. Under strong assumptions, however, the case of $\mathsf{NC}1$ can be taken care of~\cite{ZAW+10}.




This joint work with Benoît Libert, San Ling, Khoa Nguyen and Huaxiong Wang was presented at Asiacrypt'17~\cite{LLM+17}.


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Delerabl\'eePointcheval & $4$ & $5$ & $2304$ bits & $q$\textsf{SDH} + \textsf{XDH} & Dynamic & CCA \\ \hline


Bichsel {\em et al.} & $3$ & $2$ &$1280$ bits & \textsf{LRSW} + \textsf{SDL} & Dynamic & CCA \\ \hline


PointchevalSanders & $2$ & $2$ & $1024$ bits & \textsf{LRSW} & Dynamic & CCA \\ \hline


PointchevalSanders & $2$ & $2$ & $1024$ bits & \textsf{LRSW}like & Dynamic & CCA \\ \hline




\end{tabular}


\caption{Comparison between different group signature schemes}





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\textbf{École Doctorale ED512\\% rectifier le numéro d'accréditation


\textbf{École Doctorale N$^{\mathrm{o}}$512\\% rectifier le numéro d'accréditation


École Doctorale en Informatique et Mathématiques de Lyon}% nom complet de l'école doctorale




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Soutenue publiquement le jj/mm/aaaa, par :\\


Soutenue publiquement le 18/10/2018, par :\\


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\textbf{Fabrice Mouhartem}






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\chapter{Introduction}


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