\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}