Start structure

abstract.tex

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+\chapter*{Résumé}
+\addcontentsline{toc}{chapter}{Résumé}

acknowledgements.tex

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+\thispagestyle{empty}
 \chapter*{Remerciements}
+\addcontentsline{toc}{chapter}{Remerciements}
 
-
+\begin{otherlanguage}{french}
+\end{otherlanguage}

chap-introduction.tex

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+\chapter{Introduction}
+\pagenumbering{arabic}
+\pagestyle{fancy}

chap-lattices.tex

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+\chapter{Lattices}
+
+A (full-rank) lattice~$L$ is defined as the set of all integer linear
+combinations of some linearly independent basis
+vectors~$(\mathbf{b}_i)_{i\leq n}$ belonging to some~$\RR^n$. We work with $q$-ary lattices, for some prime $q$.
+\begin{definition} \label{de:qary-lattices}
+  Let~$m \geq n \geq 1$, a prime~$q \geq 2$, $\mathbf{A} \in \ZZ_q^{n \times m}$ and   $\mathbf{u} \in \ZZ_q^n$, define
+$\Lambda_q(\mathbf{A}) := \{ \mathbf{e} \in \ZZ^m \mid \exists \mathbf{s} \in \ZZ_q^n ~\text{ s.t. }~\mathbf{A}^T \cdot \mathbf{s} = \mathbf{e} \bmod q \}$ as well as
+\begin{align*}
+  \Lambda_q^{\perp} (\mathbf{A}) &:= \{\mathbf{e} \in \ZZ^m \mid \mathbf{A} \cdot \mathbf{e} = \mathbf{0}^n \bmod q \},&
+  \Lambda_q^{\mathbf{u}} (\mathbf{A}) &:=  \{\mathbf{e} \in \ZZ^m \mid \mathbf{A} \cdot \mathbf{e} = \mathbf{u} \bmod q \}
+\end{align*}
+For any $\mathbf{t} \in \Lambda_q^{\mathbf{u}} (\mathbf{A})$,  $\Lambda_q^{\mathbf{u}}(\mathbf{A})=\Lambda_q^{\perp}(\mathbf{A}) + \mathbf{t}$ so that $\Lambda_q^{\mathbf{u}} (\mathbf{A}) $
+is a shift of   $\Lambda_q^{\perp} (\mathbf{A})$.
+\end{definition}
+
+
+\noindent For a lattice~$L$, a vector $\mathbf{c} \in \RR^n$ and a real~$\sigma>0$, define the function
+$\rho_{\sigma,\mathbf{c}}(\mathbf{x}) = \exp(-\pi\|\mathbf{x}- \mathbf{c} \|^2/\sigma^2)$.
+The discrete Gaussian distribution of support~$L$, parameter~$\sigma$ and center $\mathbf{c}$ is defined as
+ $D_{L,\sigma,\mathbf{c}}(\mathbf{y}) = \rho_{\sigma,\mathbf{c}}(\mathbf{y})/\rho_{\sigma,\mathbf{c}}(L)$ for any $\mathbf{y} \in L$.
+We denote by  $D_{L,\sigma }(\mathbf{y}) $ the distribution centered in $\mathbf{c}=\mathbf{0}$.
+We will extensively use the fact that samples
+from~$D_{L,\sigma}$ are short with overwhelming probability.
+
+\begin{lemma}[{\cite[Le.~1.5]{Bana93}}]
+\label{le:small}
+For any lattice~$L \subseteq
+  \RR^n$ and positive real number~$\sigma>0$,
+  we have  $\Pr_{\mathbf{b} \sample D_{L,\sigma}} [\|\mathbf{b}\|
+    \leq \sqrt{n} \sigma] \geq 1-2^{-\Omega(n)}.$
+\end{lemma}
+
+\noindent As shown by Gentry {\em et al.}~\cite{GePeVa08}, Gaussian
+distributions with lattice support can be sampled efficiently
+given a sufficiently short basis of the lattice.
+
+\begin{lemma}[{\cite[Le.~2.3]{BLPRS13}}]
+\label{le:GPV}
+There exists a $\PPT$ (probabilistic polynomial-time) algorithm $\textsf{GPVSample}$ that takes as inputs a
+basis~$\mathbf{B}$ of a lattice~$L \subseteq \ZZ^n$ and a
+rational~$\sigma \geq  \|\widetilde{\mathbf{B}}\| \cdot \Omega(\sqrt{\log n})$,
+and outputs vectors~$\mathbf{b} \in L$ with distribution~$D_{L,\sigma}$.
+\end{lemma}
+
+%We
+%use an algorithm that jointly samples a uniform~$\mathbf{A}$ and a short
+%basis of~$\Lambda_q^{\perp}(\mathbf{A})$.
+
+\begin{lemma}[{\cite[Th.~3.2]{AlPe09}}]
+\label{le:TrapGen}
+There exists a $\PPT$ algorithm $\TrapGen$ that takes as inputs $1^n$,
+$1^m$ and an integer~$q \geq 2$ with~$m \geq \Omega(n \log q)$, and
+outputs a matrix~$\mathbf{A} \in \ZZ_q^{n \times m}$ and a
+basis~$\mathbf{T}_{\mathbf{A}}$ of~$\Lambda_q^{\perp}(\mathbf{A})$ such
+that~$\mathbf{A}$ is within statistical distance~$2^{-\Omega(n)}$
+to~$U(\ZZ_q^{n \times m})$, and~$\|\widetilde{\mathbf{T}_{\mathbf{A}}}\| \leq
+\bigO(\sqrt{n \log q})$.
+\end{lemma}
+
+\noindent Lemma~\ref{le:TrapGen} is often combined with the sampler from Lemma~\ref{le:GPV}. Micciancio and Peikert~\cite{MiPe12} recently proposed a more efficient
+approach for this combined task, which should be preferred in practice but, for the sake of simplicity, we present our schemes using~$\TrapGen$.
+
+We also make use of an algorithm that extends a trapdoor for~$\mathbf{A} \in \ZZ_q^{n \times m}$ to a trapdoor of any~$\mathbf{B} \in \ZZ_q^{n \times m'}$ whose left~$n \times m$
+submatrix is~$\mathbf{A}$.
+
+\begin{lemma}[{\cite[Le.~3.2]{CaHoKiPe10}}]\label{lem:extbasis}
+  There exists a $\PPT$ algorithm $\ExtBasis$ that takes as inputs a
+  matrix~$\mathbf{B} \in \ZZ_q^{n \times m' }$ whose first~$m$ columns
+  span~$\ZZ_q^n$, and a basis~$\mathbf{T}_{\mathbf{A}}$
+  of~$\Lambda_q^{\perp}(\mathbf{A})$ where~$\mathbf{A}$ is the left~$n \times
+  m$ submatrix of~$\mathbf{B}$, and outputs a basis~$\mathbf{T}_{\mathbf{B}}$
+  of~$\Lambda_q^{\perp}(\mathbf{B})$ with~$\|\widetilde{\mathbf{T}_{\mathbf{B}}}\|
+  \leq \|\widetilde{\mathbf{T}_{\mathbf{A}}}\|$.
+\end{lemma}
+
+\noindent In our security proofs, analogously to \cite{Boy10,BHJKS15} we also use a technique due to Agrawal, Boneh and Boyen~\cite{ABB1} that implements
+an all-but-one trapdoor mechanism (akin to the one of Boneh and Boyen \cite{BB04}) in the lattice setting.
+%In other words we need the following algorithm:
+
+\begin{lemma}[{\cite[Th.~19]{ABB1}}]\label{lem:sampler}
+  There exists a $\PPT$ algorithm $\SampleR$ that takes as inputs matrices $\mathbf A, \mathbf C \in \ZZ_q^{n \times m}$, a low-norm matrix $\mathbf R \in \ZZ^{m \times m}$,
+  a short basis $\mathbf{T_C} \in \ZZ^{m \times m}$ of $\Lambda_q^{\perp}(\mathbf{C})$, a vector $\mathbf u \in \ZZ_q^{n}$ and a rational $\sigma$ such that $\sigma \geq \|
+  \widetilde{\mathbf{T_C}}\| \cdot \Omega(\sqrt{\log n})$, and outputs a short vector $\mathbf{b} \in \ZZ^{2m}$ such that $\left[ \begin{array}{c|c} \mathbf A ~ &~ \mathbf A
+  \cdot \mathbf R + \mathbf C \end{array} \right]\cdot \mathbf b = \mathbf u \bmod q$ and with distribution statistically close to $D_{L,\sigma}$ where $L$ denotes the shifted
+  lattice $\Lambda^\mathbf{u}_q \left( \left[ \begin{array}{c|c} \mathbf A ~&~ \mathbf A \cdot \mathbf R + \mathbf C \end{array} \right] \right)$.
+  %$\{ \mathbf x \in \ZZ^{2 m} : \left[ \begin{array}{c|c} \mathbf A ~&~ \mathbf A \cdot \mathbf R + \mathbf C \end{array} \right] \cdot \mathbf x = \mathbf u \bmod q \}$.
+\end{lemma}
+
+

chap-pairings.tex

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+\chapter{Pairing-based cryptography}

garde.tex

@@ -85,7 +85,7 @@ Nom Prénom, grade/qualité, établissement/entreprise \hfill Examinateur/trice
 \bigskip
 
 %Nom Prénom, grade/qualité, établissement/entreprise \hfill Directeur/trice de thèse
-Benoît Libert, Directeur de Recherche, CNRS et École Normale Supérieure de Lyon\hfill Directeur de thèse
+Benoît Libert, Directeur de Recherche, CNRS et ENS de Lyon\hfill Directeur de thèse
 
 %Nom Prénom, grade/qualité, établissement/entreprise \hfill Co-directeur/trice de thèse % le cas échéant
 

macros.tex

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+% Abbreviations
+%% Usual
+\newcommand{\PPT}{\textsf{PPT}\xspace}
+%% Algorithms
+\newcommand{\TrapGen}{\textsf{TrapGen}\xspace}
+\newcommand{\ExtBasis}{\textsf{ExtBasis}\xspace}
+\newcommand{\SampleR}{\textsf{SampleR}\xspace}
+
+% Operators
+\newcommand{\sample}{\xspace\ensuremath{\hookleftarrow}\xspace}
+\newcommand{\bigO}{\ensuremath{\mathcal{O}}}
+
+% Sets
+%% Usual sets
+\newcommand{\RR}{\xspace\ensuremath{\mathbb{R}}\xspace}
+\newcommand{\ZZ}{\xspace\ensuremath{\mathbb{Z}}\xspace}
+\newcommand{\CC}{\xspace\ensuremath{\mathbb{C}}\xspace}
+\newcommand{\QQ}{\xspace\ensuremath{\mathbb{Q}}\xspace}

main.tex

@@ -1,26 +1,66 @@
-\documentclass[a4paper]{book}
+\documentclass[a4paper, 11pt]{memoir}
 
-\usepackage[utf8]{inputenc}
-\usepackage[french]{babel}
+\usepackage[utf8x]{inputenc}
+\usepackage[french,english]{babel}
 %\usepackage[UKenglish]{babel}
 \usepackage[T1]{fontenc}
 \usepackage{libertine}
 
+\usepackage{fancyhdr}
+\pagestyle{fancy}
+
+\usepackage[pagebackref]{hyperref}
+\renewcommand*{\backref}[1]{}
+\renewcommand*{\backrefalt}[4]{\small Citations: \S{} #4}
+\hypersetup{colorlinks=true, linkcolor=black!50!blue, citecolor=black!50!green, breaklinks=true}
+
 \usepackage{amsmath, amssymb, mathrsfs}
 \usepackage{amsthm}
 
-\usepackage{pdfpages}
+\newtheorem{theorem}{Theorem}
+\newtheorem{lemma}{Lemma}
+\theoremstyle{definition}
+\newtheorem{definition}{Definition}
 
-\title{}
-\author{}
-\date{}
+\usepackage{pdfpages}
+\usepackage{xspace}
+
+\input macros
+
+\title{Cryptographie protégeant la vie privée avec des fonctionnalité avancées}
+\author{Fabrice Mouhartem}
+\date{\today}
 
 \begin{document}
 \pagenumbering{roman}
 \includepdf{garde.pdf}
-
 \pagestyle{empty}
 
+%%%%%%%%%%%%%
+% Décidaces %
+%%%%%%%%%%%%%
+\cleardoublepage
+\vspace*{\stretch{1}}
+\begin{flushright}
+  À \ldots
+\end{flushright}
+\vspace*{\stretch{2}}
+
 \input acknowledgements
 
+\input abstract
+
+\tableofcontents
+\addcontentsline{toc}{chapter}{Contents}
+
+\input chap-introduction 
+
+\part{Background and definitions}
+\input chap-lattices
+
+\input chap-pairings
+
+\bibliographystyle{alpha}
+\bibliography{these.bib}
 \end{document}
+% vim: spl=en