deletions | additions       diff --git a/section_Some_cryptographic_applications_This__.tex b/section_Some_cryptographic_applications_This__.tex  index 8980c41..f73553e 100644  --- a/section_Some_cryptographic_applications_This__.tex  +++ b/section_Some_cryptographic_applications_This__.tex 
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\item Bob has received $c$ and so he can compute another exponentiation $c^d$ in $\ZZ_p$, the results being $m$ thanks to Theorem ??.  \end{itemize}  Thanks to the research in number theory developed so far, even if the enemies simultaneously collect $N$, $e$ and $c$, then they have a negligible probability to reconstruct $m$ and so the message remains hidden from them.  The system we have described is still used today, especially for electronic payments.  We complete this subsection with some observations on the RSA protocol as it used nowadays:  \begin{itemize}  \item the prime numbers $p$ and $q$ are huge, at least $p,q > 2^{1024}$,  \item the prime numbers $p,q$ has to be chosen with care, for example their difference $|p-q|$ must exceed both their square roots,  \item the private exponent $d$ must be large, at least $^4\sqrt(N)$,  \item the public exponent $e$ should be at least $e \geq 65537$.  \end{itemize}  \subsection {El Gamal}