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.
The existence of these structures proves useful when it comes to design protocols.
For that, constructions takes advantage of the mathematical properties of the structure to allow the functionality.
An example is the multiplicative homomorphism of the ElGamal cryptosystem which is possible using the structure of the underlying cyclic group $\GG$ on which the scheme is built upon.
%Namely, an El Gamal encryption of a message $M$ under the public key $h = g^\alpha_{} \in \GG$ is a couple $(c_1^{}, c_2^{}) = (g^r_{}, M \cdot h^r_{}) \in \GG^2_{}$, which can be decrypted with the knowledge of the secret key $\alpha \in \Zp$: $M = c_2^{} \cdot c_1^{-\alpha}$.
%Then, the cyclic group structure of $\GG$ leads to the ability to compute a valid ciphertext for $M \cdot M'$ given ciphertexts $(c_1^{}, c_2^{})$ and $(c'_1, c'_2)$ of $M$ and $M'_{}$ respectively.
In this chapter, we describe the different structures on which the cryptography primitives we design in this thesis are based on, namely bilinear groups and lattices.
