deletions | additions       diff --git a/section_Some_cryptographic_applications_This__.tex b/section_Some_cryptographic_applications_This__.tex  index e709178..a814a7a 100644  --- a/section_Some_cryptographic_applications_This__.tex  +++ b/section_Some_cryptographic_applications_This__.tex 
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The system we have described is still used today a lot, especially on the Internet.  We complete this subsection with some observations on the DH protocol as it used nowadays:  \begin{itemize}  \item the prime number $p$ is huge, at least $p > 2^{1024}$, 2^{2048}$,  \item the prime number $p$ has to be chosen with some specific algebraic properties, for example the factorization of $p-1$ in prime numbers  should contain one huge prime,  \item there are some primes that have been standardized and are used by many applications, as for instance those proposed by the National Institute of Standards in the U.S.  
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\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 preliminary exponentiations in $\mathbb{Z}_p$\,: $c_{1}\,=\,g^{y}$ and $s\,=\,h^{y}$ (obviously $h=g^{xy}$).  \item Alicefinally  encrypts her message $m$ by computing $c_{2} \,=\, m s$ and s$; finally, she  sends it $c_1$ and $ c_2$  to Bobtogether with $c_1$  (the enemies may intercept them); \item Bob computes $s$ by an exponentiation in $\mathbb{Z}_p$, since $s \,=\, {c_1}^{x}$;  \item Finally, Bob computes the message $m$ by another exponentiation in $\mathbb{Z}_p$, i.e. $m \, =\, c_{2}\cdot s^{{-1}}$  \end{itemize}  Thanks to the research in algebra developed so far, we observe that, even if the enemies simultaneously collect $p$, $g$, $h=g^x$ and $c_1=g^y$, $s\,=\,h^{y}$ and $c_{2} \,=\, m s$, then they have a negligible probability to reconstruct $m$ or $x$  and so the message remains hidden from them. The system we have described is still used.  We complete this subsection with some observations on the El-Gamal protocol as it used nowadays:  \begin{itemize}  \item the prime number $p$ is huge, at least $p > 2^{1024}$, 2^{2048}$,  \item the prime number $p$ has to be chosen with some specific algebraic properties, for example the factorization of $p-1$ in prime numbers  should contain one huge prime,  \item there are some primes that have been standardized and are used by many applications, as for instance those proposed by the National Institute of Standards in the U.S. IETF.  \end{itemize}