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 \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}