\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}

\input sec-pairings

\section{Lattice-Based Cryptography}
\label{se:lattices}

\input sec-lattices
