Update chapters

chap-conclusion.tex

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+\chapter*{Conclusion}
+\addcontentsline{toc}{part}{Conclusion}

chap-proofs.tex

@@ -1,7 +1,22 @@
 \chapter{Security Proofs in Cryptography}
 
+Provable security is a subfield of cryptography where constructions are proven secure with regards to a security model.
+To illustrate this notion, let us take the example of public-key encryption schemes.
+This primitive consists in three algorithms:~key generation, encryption and decryption.
+These algorithms acts according to their names.
+Then, the question of ``how to define the security of this set of algorithms'' rises.
+To answer this question, we have to define the power of the adversary, and its goal.
+To model those two notions, cryptographers uses security games.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%
+% Security Reductions %
+%%%%%%%%%%%%%%%%%%%%%%%
 \section{Security Reductions}
 
+Provable security focuses on providing constructions for which the security is guaranteed by a security proof, or security reduction.
+These proofs consist in polynomial reductions from difficult problems: the hardness assumptions.
+The quality of a proof depends on the security of the hardness assumption, and the tightness of the proof.
 
 
 

chap-structures.tex

@@ -1,7 +1,12 @@
 \chapter{Underlying Structures}
+\label{chap:structures}
 
 \section{Pairing-Based Cryptography}
+\label{se:pairing}
+
+\input sec-pairings
 
 \section{Lattice-Based Cryptography}
+\label{se:lattices}
 
-\input sec-lattices.tex
+\input sec-lattices