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