\chapter*[List of Symbols]{List of Symbols}
\addcontentsline{toc}{chapter}{List of Symbols}
\addcontentsline{tof}{chapter}{Liste des symboles et abréviations}

\begin{longtable}{ll}
  \multicolumn{2}{l}{\scbf{General Notations}}                                               \\
  TM                                  & Turing Machine                                       \\
  $\ppt$                              & Probabilistic Polynomial Time                        \\
  $\epsilon$                          & empty word                                           \\
  $\mathbf{A}$                         & bold uppercase letters represent matrices            \\
  $\mathbf{b}$                         & bold lowercase letters represent column vectors      \\
  $\widetilde{\mathbf{A}}$             & Gram-Schmidt orthogonalization of matrix $\mathbf{A}$ \\
  $\mathbf{A}^T_{}, \mathbf{u}^T_{}$  & the transpose of a matrix or a vector respectively   \\
  $\mathbf{I}_n$ & the $n$ dimension identity matrix in $\RR^{n \times n}$ \\
  $\U(S)$                             & If $S$ is a finite set, $\U(S)$ denotes the uniform distribution over $S$\\
  $\Pr[E]$ & Probability that an event $E$ occurs \\
  [1ex] \multicolumn{2}{l}{\scbf{Usual sets}}                                                              \\
  $\QQ$                     & the set of rational numbers                                                  \\
  $\RR$                     & the set of real numbers                                                      \\
  $\ZZ$                     & the set of relative integers                                                 \\
  $\ZZ_q$                   & the field $\ZZ_{/q\ZZ}$, with $q$ prime                                      \\
  $\FF_2$                   & the field $\ZZ_{/2\ZZ}$                                                      \\
  $\mathbb{S}^d$            & the set of vectors of dimension $d$ in the set $\mathbb{S}$                  \\
  $\mathbb{S}^{n \times m}$ & the set of matrices with $n$ rows and $m$ columns in the set $\mathbb{S}$    \\
  [1ex] \multicolumn{2}{l}{\scbf{Protocols}}                                                               \\
  $\PKE$   & Public Key Encryption                                                                         \\
  $\ZK$    & Zero-Knowledge                                                                                \\
  $\ZKAoK$ & Zero-Knowledge Argument of Knowledge                                                          \\
  $\NIZK$  & Non-Interactive Zero-Knowledge                                                                \\
  $\QANIZK$  & Quasi-Adaptive Non-Interactive Zero-Knowledge                                                                \\
  $\OT$    & Oblivious Transfer                                                                            \\
  [1ex] \multicolumn{2}{l}{\scbf{Security Notions}}                                                         \\
  EU-CMA & Existentially Unforgeable under chosen-message attacks \\
  EU-RMA & Existentially Unforgeable under random-message attacks \\
  IND-CPA & Indistinguishable under chosen-plaintext attacks (passive adversary) \\
  IND-CCA1 & Indistinguishable under non-adaptive active adversary\\
  IND-CCA2 & Indistinguishable under adaptive active adversary\\
  [1ex] \multicolumn{2}{l}{\scbf{Security Models}}                                                         \\
  $\ROM$ & Random-Oracle Model                                                                             \\
  $\UC$  & Universal Composability                                                                         \\
  [1ex] \multicolumn{2}{l}{\scbf{Security Assumptions}}                                                    \\
  [.5ex] \multicolumn{2}{l}{\quad\textbf{Lattices}}                                                   \\
  $\SIS$  & Short Integer Solution                                                                         \\
  $\ISIS$ & Inhomogeneous Short Integer Solution                                                           \\
  $\LWE$  & Learning with Errors                                                                           \\
  $\SIVP$ & Shortest Independent Vectors Problem                                                           \\
  [.5ex] \multicolumn{2}{l}{\quad\textbf{Cyclic groups}}                                                   \\
  $\DLP$ & Discrete Logarithm Problem                                                                      \\
  $\DDH$ & Decisional Diffie-Hellman                                                                       \\
  [.5ex] \multicolumn{2}{l}{\quad\textbf{Bilinear groups}}                                                 \\
  $\SXDH$ & Symmetric eXternal Diffie-Hellman                                                              \\
  $\SDL$  & Symmetric Discrete Logarithm \\
  [1ex] \multicolumn{2}{l}{\scbf{Stern-like protocol}}                                                     \\
  $\mathsf{B}^2_{\mathfrak m}$ & The set of $\bit$ vector of hamming weight $\mathfrak m$                  \\
  $\mathsf{B}^3_{\mathfrak m}$ & The set of $\nbit$ vectors with $\mathfrak m$ elements in $-1$, $0$ and $1$ \\
\end{longtable}
