\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 $S$ \\
  $\Supp(D)$ & If $D$ is a probability distribution, $\Supp(D)$ denotes the support of $D$ \\
  $\Pr[E]$ & Probability that an event $E$ occurs \\
  $D \approx_s D'$ & The distribution $D$ is statistically close to the distribution $D'$ \\
  [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 $\mathbb{S}$ \\
  $\permutations_k$ & The set of all permutations over $\{1,\ldots, k\}$ \\
  [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 \\
  $\textsf{WI}$ & Witness Indistinguishable \\
  $\textsf{GS}$ & Group Signature \\
  $\GE$ & Group Encryption \\
  $\OT$ & Oblivious Transfer \\
  [1ex] \multicolumn{2}{l}{\scbf{Security Notions}} \\
  $\advantage{\mathrm{E}}{\adv}$ & Advantage of adversary $\adv$ for experiment $\mathrm{E}$ \\
  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 (\cref{de:sis}) \\
  $\ISIS$ & Inhomogeneous Short Integer Solution (\cref{de:sis}) \\
  $\LWE$ & Learning-with-Errors (\cref{de:lwe}) \\
  $\SIVP$ & Shortest Independent Vectors Problem (\cref{de:sivp}) \\
    [.5ex] \multicolumn{2}{l}{\quad\textbf{Cyclic groups}} \\
    $\DLP$ & Discrete Logarithm Problem (\cref{de:DLP}) \\
    $\DDH$ & Decisional Diffie-Hellman (\cref{de:DDH}) \\
    [.5ex] \multicolumn{2}{l}{\quad\textbf{Bilinear groups}} \\
    $\SXDH$ & Symmetric eXternal Diffie-Hellman (\cref{de:SXDH}) \\
    $\SDL$ & Symmetric Discrete Logarithm (\cref{de:SDL}) \\
    [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}