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 2018-04-12 16:42:39 +00:00 \chapter*[List of Symbols]{List of Symbols} 2018-01-27 20:27:06 +00:00 \addcontentsline{toc}{chapter}{List of Symbols} 2018-04-04 16:46:37 +00:00 \addcontentsline{tof}{chapter}{Liste des symboles et abréviations} 2018-01-27 20:27:06 +00:00   2018-04-06 13:46:55 +00:00 \begin{longtable}{ll} 2018-06-19 12:41:13 +00:00  \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 \\ 2018-04-30 12:54:55 +00:00  $\mathbf{I}_n$ & the $n$ dimension identity matrix in $\RR^{n \times n}$ \\ 2018-06-19 12:41:13 +00:00  $\U(S)$ & If $S$ is a finite set, $\U(S)$ denotes the uniform distribution over $S$ \\ 2018-04-30 14:12:03 +00:00  $\Supp(D)$ & If $D$ is a probability distribution, $\Supp(D)$ denotes the support of $D$ \\ 2018-04-12 16:42:39 +00:00  $\Pr[E]$ & Probability that an event $E$ occurs \\ 2018-06-19 12:41:13 +00:00  $D \approx_s D'$ & 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 \\ 2018-06-19 15:26:01 +00:00  $\textsf{WI}$ & Witness Indistinguishable \\ 2018-06-19 12:41:13 +00:00  $\textsf{GS}$ & Group Signature \\  $\GE$ & Group Encryption \\  $\OT$ & Oblivious Transfer \\  [1ex] \multicolumn{2}{l}{\scbf{Security Notions}} \\ 2018-06-12 12:16:16 +00:00  $\advantage{\mathrm{E}}{\adv}$ & Advantage of adversary $\adv$ for experiment $\mathrm{E}$ \\ 2018-04-12 16:42:39 +00:00  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\\ 2018-06-19 12:41:13 +00:00  [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 $1$ \\ 2018-04-06 13:46:55 +00:00 \end{longtable}