In this thesis, we presented new cryptographic schemes that relies on lattice or pairing assumptions.
These contributions focus on the design and analysis of new cryptographic schemes that target privacy-preserving applications.
In pairing-related cryptography, we propose a practical dynamic group signature scheme, for which security is well understood.
It relies on broadly used assumptions with simple statements that exists for more than ten years.
This work is also supported by a practical implementation in C.
Our work in the lattice work give rise of three fundamental schemes that were missing in the landscape of lattice-based privacy-preserving primitives.
Even if these schemes suffer from some efficiency issues due to their novelty, we do believe that it's one step toward a quantum-secure privacy-friendly world.
In the way of doing it, improvements have been made in the state of zero-knowledge proofs in the lattice setting as well as providing building blocks that, we believe, are of independent interest.
As of our signature with efficient protocols, which have been used to provide a lattice-based e-cash system~\cite{LLNW17}.
All these works are proven under strong security model within simple assumptions.
This made a breeding ground for new theoretical constructions, as well as going toward practicality.
\section*{Open Problems}
The path of providing new cryptographic primitives and proving them is sometimes uneven.
The most obvious questions that stem from this work are about how to tackle the compromises we made in the design of those primitives.
\begin{question}
Is it possible to build an adaptive oblivious transfer with access control with polynomially-large approximation factor?
\end{question}
In other words, is it possible to avoid smudging to keep message-privacy in the oblivious transfer scheme of~\cref{ch:ot-lwe}.
