Added contents on Jacobi/Legendre Symbol

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@@ -125,7 +125,7 @@ $$
 $$
 Let us characterize $\phi(n)$.
 \begin{lem}
- Let $m,n\in\mathbb{N}$, $p,p_1,p_2$ be prime, $k\in\mathbb{N}$.
+ Let $k,m,n\in\mathbb{N}$, $p,p_1,p_2$ be prime.
  \begin{itemize}
   \item $\phi(p) = p-1$;
   \item $\phi(p^k) = p^{k}-p^{k-1} = p^k \left(1-\frac{1}{p}\right)$;
@@ -133,14 +133,51 @@ Let us characterize $\phi(n)$.
   \item $\phi(n) = n\cdot \prod_{p|n}\limits\left(1-\frac{1}{p}\right)$ ;
   \item $\sum_{d|n}\limits \phi(d) = n$.
  \end{itemize}
- \subsubsection*{The Legendre/Jacobi Symbol}
- TODO
+Moreover if $a, n\in\mathbb{N}$ are relatively prime, then
+\begin{itemize}
+ \item $a^{\phi(n)} \equiv 1 \pmod n$, in particular $a^{p-1}\equiv 1\pmod p$ for all primes $p$.
+\end{itemize}
+The last point is known as the Euler-Fermat theorem.
+\end{lem}
+
+\subsubsection*{The Legendre/Jacobi Symbol}
+In cryptographic applications, the Legendre/Jacobi symbol appears. Let $p$ be a prime. An element $a\in\mathbb{Z}_p$ is
+called a \emph{quadratic residue modulo $p$} if and only if $x^2-a$ has a zero in $\mathbb{Z}_p$. Otherwise, $a$ is called
+a quadratic nonresidue modulo $p$.
+\begin{defn}[Legendre/Jacobi Symbol]~\\
+Let $p$ be a prime. The \emph{Legendre-Symbol} $\left(\frac{a}{p}\right)$ is defined as
+$$
+\left(\frac{a}{p}\right) =
+\begin{cases}
+ 0,\qquad & \text{if $p|a$}\\
+ 1,\qquad & \text{if $a$ is quadratic residue modulo $p$}\\
+ -1,\qquad & \text{if $a$ is quadratic nonresidue modulo $p$}
+\end{cases}
+$$
+Let $a$ be an integer and $n$ be a positive odd number such that $n = p_1^{\alpha_1}\ldots p_s^{\alpha_s}$. 
+The \emph{Jacobi-Symbol} $\left(\frac{a}{n}\right)$ is defined as
+$$
+\left(\frac{a}{n}\right) = \left(\frac{a}{p_1}\right)^{\alpha_1}\ldots \left(\frac{a}{p_s}\right)^{\alpha_s}.
+$$
+\end{defn}
+\begin{lem}
+ Let $p,q$ be a primes, $a,b\in\mathbb{Z}_p$. Then,
+\begin{itemize}
+ \item $\left(\frac{a}{p}\right) \equiv a^{(p-1)/2} \pmod p$;
+ \item $\left(\frac{ab}{p}\right) = \left(\frac{a}{p}\right)\left(\frac{b}{p}\right)$;
+ \item $\sum_{a=0}^{p-1}\limits\left(\frac{a}{p}\right) = 0$;
+ \item $\left(\frac{p}{q}\right) = (-1)^{(p-1)(q-1)/4} \left(\frac{q}{p}\right)$;
+ \item $\left(\frac{2}{p}\right) = (-1)^{(p^2-1)/8}$.
+\end{itemize}
+
+\end{lem}
+
+ 
  \subsubsection*{The discrete Logarithm problem}
  TODO 
  \subsection{The RSA cryptosystem}
  TODO
 
-\end{lem}