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% \section{Lattice-Based Cryptography} %
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\subsection{Lattices and Hard Lattice Problems}
\label{sse:lattice-problems}

\begin{figure}
  \centering
  \begin{tikzpicture}
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                         % Comment or uncomment crop %
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    \clip[use as bounding box]
    %\draw[thick, black]
    (2,3) rectangle (10,8);
    \foreach \i in {0,1,...,20} {
      \foreach \j in {0,1,..., 10} {
        \draw[fill=black] (.2*\j + .5*\i, \j + .1*\i) circle (1pt) coordinate (v-\i-\j);
      }
    }
    \draw[very thick, green!80!black, ->] (v-9-4) -- (v-8-4);
    \draw[very thick, green!80!black, ->] (v-9-4) -- (v-9-5);
    \draw[very thick, red!80!black, ->] (v-9-4) -- (v-15-5);
    \draw[very thick, red!80!black, ->] (v-9-4) -- (v-18-3);
    \foreach \i in {0,1,...,10} {
      \draw[dotted, color=black!70] (v-0-\i) -- (v-20-\i);
    }
    \foreach \i in {0,1,...,20} {
      \draw[dotted, color=black!70] (v-\i-0) -- (v-\i-10);
    }
  \end{tikzpicture}
  \caption{A lattice $\Lambda$ with two different basis.}
  \label{fig:lattice-basis}
\end{figure}

A (full-rank) lattice~$\Lambda$ is defined as the set of all integer linear combinations of some linearly independent basis vectors~$(\mathbf{b}_i)_{i\leq n}$ belonging to some~$\RR^n$.
We can notice that this basis is not unique, as illustrated in Figure~\ref{fig:lattice-basis}.
In the following, we work with $q$-ary lattices, for some prime $q$.

\begin{definition} \label{de:qary-lattices}
  Let~$m \geq n \geq 1$, a prime~$q \geq 2$, $\mathbf{A} \in \ZZ_q^{n \times m}$ and   $\mathbf{u} \in \ZZ_q^n$, define
\begin{align*}
  \Lambda_q(\mathbf{A}) &\triangleq \{ \mathbf{e} \in \ZZ^m \mid \exists \mathbf{s} \in \ZZ_q^n ~\text{ s.t. }~\mathbf{A}^T \cdot \mathbf{s} = \mathbf{e} \bmod q \} \text{ as well as}\\
  \Lambda_q^{\perp} (\mathbf{A}) &\triangleq \{\mathbf{e} \in \ZZ^m \mid \mathbf{A} \cdot \mathbf{e} = \mathbf{0}^n \bmod q \} \text{, and}\\
  \Lambda_q^{\mathbf{u}} (\mathbf{A}) &\triangleq  \{\mathbf{e} \in \ZZ^m \mid \mathbf{A} \cdot \mathbf{e} = \mathbf{u} \bmod q \}.
\end{align*}

For any lattice point $\mathbf{t} \in \Lambda_q^{\mathbf{u}} (\mathbf{A})$, it holds that $\Lambda_q^{\mathbf{u}}(\mathbf{A})=\Lambda_q^{\perp}(\mathbf{A}) + \mathbf{t}$. Meaning that $\Lambda_q^{\mathbf{u}} (\mathbf{A}) $
is a shift of   $\Lambda_q^{\perp} (\mathbf{A})$.
\end{definition}

\noindent For a lattice~$\Lambda$, a vector $\mathbf{c} \in \RR^n$ and a real~$\sigma>0$, define the distribution function
$\rho_{\sigma,\mathbf{c}}(\mathbf{x}) \triangleq \exp(-\pi\|\mathbf{x}- \mathbf{c} \|^2/\sigma^2)$.
The discrete Gaussian distribution of support~$L$, parameter~$\sigma$ and center $\mathbf{c}$ is defined as
 $D_{L,\sigma,\mathbf{c}}(\mathbf{y}) = \rho_{\sigma,\mathbf{c}}(\mathbf{y})/\rho_{\sigma,\mathbf{c}}(L)$ for any $\mathbf{y} \in L$.
We denote by  $D_{L,\sigma }(\mathbf{y}) $ the distribution centered in $\mathbf{c}=\mathbf{0}$.

In order to work with lattices in cryptography, it is useful to define hard lattice problems. In the following we define the shortest Independent Vectors Problem ($\SIVP$). This problem reduces to the Learning With Errors ($\LWE$) problems and the Short Integer Solution ($\SIS$) problem as explained later. These links are important because those are ``wost-case to average-case'' reductions. In other words, the $\SIVP$ assumption by itself is not very handy to manipulate in order to build new cryptographic designs, while the $\LWE$ and $\SIS$ assumptions are ``average-case'' assumptions, are are more suitable to design cryptographic schemes.

In order to define the $\SIVP$ problem and assumption, let us first define the successive minima of a lattice, a generalization of the minimum of a lattice (the length of a shortest non-zero vector in a lattice).

\begin{definition}[Successive minima] \label{de:lattice-lambda}
  For a lattice $\Lambda$ of dimension $n$, let us define for $i \in \{1,\ldots,n\}$ the $i$-th successive minimum as
  \[ \lambda_i(\Lambda) = \inf \bigl\{ r \mid \dim \left( \Span\left(\lambda \cap \mathcal B\left(\mathbf 0, r \right) \right)  \right) \geq i \bigr\}, \]
  where $\mathcal B(\mathbf c, r)$ denotes the ball of radius $r$ centered in $\mathbf c$.
\end{definition}

Which lead us to the $\SIVP$ problem, which is finding a set of sufficiently short linearly independent vectors given a lattice basis.

\begin{definition}[$\SIVP$] \label{de:sivp}
  For a dimension $n$ lattice described by a basis $\mathbf B \in \RR^{n \times m}$, and a parameter $\gamma > 0$, the shortest independent vectors problem is to find $n$ linearly independent vectors $v_1, \ldots, v_n$ such that $\| v_1 \| \leq \| v_2 \| \leq \ldots \leq \| v_n \|$ and $\|v_n\| \leq \gamma \cdot \lambda_n(\mathbf B)$.
\end{definition}

As explained before, we will rely on the assumption that both algorithmic problems below are hard. Meaning that no (probabilistic) polynomial time algorithms can solve them with non-negligible probability and non-negligible advantage, respectively.

\begin{definition}[The SIS problem]
  Let~$m,q,\beta$ be functions of~$n \in \mathbb{N}$. The Short Integer
  Solution problem $\SIS_{n,m,q,\beta}$ is, given~$\mathbf{A} \sample
  U(\Zq^{n \times m})$, find~$\mathbf{x} \in \Lambda_q^{\perp}(\mathbf{A})$
  with~$0 < \|\mathbf{x}\| \leq \beta$.
\end{definition}

If~$q \geq \sqrt{n} \beta$ and~$m,\beta \leq \mathsf{poly}(n)$, then $\SIS_{n,m,q,\beta}$ is at least as hard as
standard worst-case lattice problem $\mathsf{SIVP}_\gamma$ with~$\gamma = \softO(\beta\sqrt{n})$
(see, e.g., \cite[Se.~9]{GPV08}).
\begin{definition}[The LWE problem]
Let $n,m \geq 1$, $q \geq 2$, and let $\chi$ be a probability distribution on~$\mathbb{Z}$. For $\mathbf{s} \in \mathbb{Z}_q^n$, let $A_{\mathbf{s}, \chi}$ be the distribution obtained by sampling $\mathbf{a} \hookleftarrow  U(\mathbb{Z}_q^n)$ and $e \hookleftarrow \chi$, and outputting $(\mathbf{a}, \mathbf{a}^T\cdot\mathbf{s} + e) \in \mathbb{Z}_q^n \times \mathbb{Z}_q$. The Learning With Errors problem $\mathsf{LWE}_{n,q,\chi}$ asks to distinguish~$m$ samples chosen according to $\mathcal{A}_{\mathbf{s},\chi}$ (for $\mathbf{s} \hookleftarrow U(\mathbb{Z}_q^n)$) and $m$ samples chosen according to $U(\mathbb{Z}_q^n \times \mathbb{Z}_q)$.
\end{definition}

If $q$ is a prime power, $B \geq \sqrt{n}\omega(\log n)$, $\gamma= \widetilde{\mathcal{O}}(nq/B)$, then there exists an efficient sampleable $B$-bounded distribution~$\chi$ ({i.e.}, $\chi$ outputs samples with norm at most $B$ with overwhelming probability) such that $\mathsf{LWE}_{n,q,\chi}$ is as least as hard as $\mathsf{SIVP}_{\gamma}$ (see, e.g., \cite{Reg05,Pei09,BLP+13}).
 % (see~\cite{Pei09,BLPRS13} for classical analogues).


\subsection{Lattice Trapdoors}

\noindent As shown by Gentry {\em et al.}~\cite{GPV08}, Gaussian
distributions with lattice support can be sampled efficiently
given a sufficiently short basis of the lattice.

\begin{lemma}[{\cite[Le.~2.3]{BLP+13}}]
\label{le:GPV}
There exists a $\PPT$ (probabilistic polynomial-time) algorithm $\GPVSample$ that takes as inputs a
basis~$\mathbf{B}$ of a lattice~$L \subseteq \ZZ^n$ and a
rational~$\sigma \geq  \|\widetilde{\mathbf{B}}\| \cdot \Omega(\sqrt{\log n})$,
and outputs vectors~$\mathbf{b} \in L$ with distribution~$D_{L,\sigma}$.
\end{lemma}

%We
%use an algorithm that jointly samples a uniform~$\mathbf{A}$ and a short
%basis of~$\Lambda_q^{\perp}(\mathbf{A})$.

\begin{lemma}[{\cite[Th.~3.2]{AP09}}]
\label{le:TrapGen}
There exists a $\PPT$ algorithm $\TrapGen$ that takes as inputs $1^n$,
$1^m$ and an integer~$q \geq 2$ with~$m \geq \Omega(n \log q)$, and
outputs a matrix~$\mathbf{A} \in \ZZ_q^{n \times m}$ and a
basis~$\mathbf{T}_{\mathbf{A}}$ of~$\Lambda_q^{\perp}(\mathbf{A})$ such
that~$\mathbf{A}$ is within statistical distance~$2^{-\Omega(n)}$
to~$U(\ZZ_q^{n \times m})$, and~$\|\widetilde{\mathbf{T}_{\mathbf{A}}}\| \leq
\bigO(\sqrt{n \log q})$.
\end{lemma}

\noindent Lemma~\ref{le:TrapGen} is often combined with the sampler from Lemma~\ref{le:GPV}. Micciancio and Peikert~\cite{MP12} proposed a more efficient
approach for this combined task, which should be preferred in practice but, for the sake of simplicity,
schemes are presented using~$\TrapGen$ in this thesis.

We also make use of an algorithm that extends a trapdoor for~$\mathbf{A} \in \ZZ_q^{n \times m}$ to a trapdoor of any~$\mathbf{B} \in \ZZ_q^{n \times m'}$ whose left~$n \times m$
submatrix is~$\mathbf{A}$.

\begin{lemma}[{\cite[Le.~3.2]{CHKP10}}]\label{lem:extbasis}
  There exists a $\PPT$ algorithm $\ExtBasis$ that takes as inputs a
  matrix~$\mathbf{B} \in \ZZ_q^{n \times m' }$ whose first~$m$ columns
  span~$\ZZ_q^n$, and a basis~$\mathbf{T}_{\mathbf{A}}$
  of~$\Lambda_q^{\perp}(\mathbf{A})$ where~$\mathbf{A}$ is the left~$n \times
  m$ submatrix of~$\mathbf{B}$, and outputs a basis~$\mathbf{T}_{\mathbf{B}}$
  of~$\Lambda_q^{\perp}(\mathbf{B})$ with~$\|\widetilde{\mathbf{T}_{\mathbf{B}}}\|
  \leq \|\widetilde{\mathbf{T}_{\mathbf{A}}}\|$.
\end{lemma}

\noindent In our security proofs, analogously to \cite{Boy10,BHJ+15} we also use a technique due to Agrawal, Boneh and Boyen~\cite{ABB10} that implements
an all-but-one trapdoor mechanism (akin to the one of Boneh and Boyen \cite{BB04}) in the lattice setting.

\begin{lemma}[{\cite[Th.~19]{ABB10}}]\label{lem:sampler}
  There exists a $\PPT$ algorithm $\SampleR$ that takes as inputs matrices $\mathbf A, \mathbf C \in \ZZ_q^{n \times m}$, a low-norm matrix $\mathbf R \in \ZZ^{m \times m}$,
  a short basis $\mathbf{T_C} \in \ZZ^{m \times m}$ of $\Lambda_q^{\perp}(\mathbf{C})$, a vector $\mathbf u \in \ZZ_q^{n}$ and a rational $\sigma$ such that $\sigma \geq \|
  \widetilde{\mathbf{T_C}}\| \cdot \Omega(\sqrt{\log n})$, and outputs a short vector $\mathbf{b} \in \ZZ^{2m}$ such that $\left[ \begin{array}{c|c} \mathbf A ~ &~ \mathbf A
  \cdot \mathbf R + \mathbf C \end{array} \right]\cdot \mathbf b = \mathbf u \bmod q$ and with distribution statistically close to $D_{L,\sigma}$ where $L$ denotes the shifted
  lattice $\Lambda^\mathbf{u}_q \left( \left[ \begin{array}{c|c} \mathbf A ~&~ \mathbf A \cdot \mathbf R + \mathbf C \end{array} \right] \right)$.
  %$\{ \mathbf x \in \ZZ^{2 m} : \left[ \begin{array}{c|c} \mathbf A ~&~ \mathbf A \cdot \mathbf R + \mathbf C \end{array} \right] \cdot \mathbf x = \mathbf u \bmod q \}$.
\end{lemma}