Intro chap-structures

chap-structures.tex

@@ -1,6 +1,13 @@
 \chapter{Underlying Structures}
 \label{chap:structures}
 
+In the previous chapter, we saw that theoretical cryptography has to rely on \emph{computational hardness assumptions}.
+Beside \emph{information theory-base cryptography}, most hardness assumptions are built on top of algebraic structures.
+For instance the discrete logarithm assumption (Definition~\ref{de:DLP}) is based on a cyclic group structure.
+That is, in some groups it is assumed that computing the discrete logarithm is an intractable problem for any probabilistic polynomial time algorithms.
+
+In this chapter, we focus on describing the different structures on which the cryptography we build in this thesis are based on, namely bilinear groups and lattices.
+
 \section{Pairing-Based Cryptography}
 \label{se:pairing}