deletions | additions       diff --git a/section_Some_cryptographic_applications_This__.tex b/section_Some_cryptographic_applications_This__.tex  index 3fb0724..a0955ec 100644  --- a/section_Some_cryptographic_applications_This__.tex  +++ b/section_Some_cryptographic_applications_This__.tex 
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We now describe another cryptographic algorithm that provides public-key encryption:  \begin{itemize}  \item Bob chooses a prime $p$ and a primitive element $g$ of $Z_p$. $\mathbb{Z}_p$.  \item Bob chooses secretly a positive integer $x$ smaller than $p-1$.  \item Bob computes the exponentiation: $h=g^{x}$ in $Z_p$. $\mathbb{Z}_p$.  \item Bob sends $p$, $g$ and $h$ to Alice (the enemies may intercept them);  \item Alice has received $p$,$g$ and $h$; she has a message $m$ to send to Bob; she chooses secretely a positive integer $y$ smaller than $p-1$;  \item Alice computes two preliinary exponentiatos preliminary exponentiations  in $Z_p$\,: $\mathbb{Z}_p$\,:  $c_{1}c\,=\,g^{y}$ and $s\,=\,h^{y}$ (obviously $h=g^{xy}$). \item Alice finally encrypts her message $m$ by computing $c_{2} \,=\, m s$ and sends it to Bob together with $c_2$ (the enemies may intercept them);  \item Bob computes $s$ by an exponentiaitoin exponentiation  in $z_p$, since $s \,=\, {c_1}^{x}$; \item Finally, Bob computes the message $m$ by another exponentiaitoin exponentiation  in $Z_p$, $\mathbb{Z}_p$,  i.e. $m \, =\, c_{2}\cdot s^{{-1}}$ \end{itemize}