From: Christoph Fuerst <ch.fuerst@gmx.at>
Date: Tue, 28 Mar 2017 19:13:22 +0000 (+0200)
Subject: Content on RSA
X-Git-Url: http://git.risc.jku.at/gitweb/?a=commitdiff_plain;h=ae56bd554b66f1126cac38d9cce1d7412d05ea3b;p=cfuerst%2Fformal-numbers.git

Content on RSA
---

diff --git a/report/formal.pdf b/report/formal.pdf
index 603f73c..4ed7828 100644
Binary files a/report/formal.pdf and b/report/formal.pdf differ
diff --git a/report/formal.tex b/report/formal.tex
index f0d654a..1af3363 100644
--- a/report/formal.tex
+++ b/report/formal.tex
@@ -127,7 +127,7 @@ $$
 \phi(n) := |\{a\in\mathbb{N}: 1\leq a\leq n  \gcd(a,n) = 1\}|.
 $$
 Let us characterize $\phi(n)$.
-\begin{lem}
+\begin{lem}\label{lem:eulerfermat}
  Let $k,m,n\in\mathbb{N}$, $p,p_1,p_2$ be prime.
  \begin{itemize}
   \item $\phi(p) = p-1$;
@@ -191,7 +191,27 @@ TODO: Baby Step Giant Step Algorithm
 
 
 \subsection{The RSA cryptosystem}
-TODO
+At the RSA cryptosystem, named after its authors Rivest, Shamir and Adleman, the two
+protagonists \textbf{A}lice and \textbf{B}ob want to exchange secret messages. To that
+end, Alice generates two primes $p$ and $q$ which are approximately of the same size.
+Then, Alice calculates the product $n=pq$ and $\phi(n) = (p-1)(q-1)$, and proceeds by
+choosing $1<e<\phi(n)$ such that $\gcd(e,\phi(n)) = 1$. With the help of the Euclidean
+algorithm, Alice calculates $1<d<\phi(n)$ such that
+$$
+ed \equiv 1 \pmod {\phi(n)}.
+$$
+We call the pair $(n,e)$ the \emph{public key} of Alice and $d$ the \emph{private key}.
+If Bob now wants to transmit a secret message to Alice, Bob uses the public key $(n,e)$ 
+of Alice. He represents his (secret) message $m$ by digits $\{0,1,\ldots,n-1\}$, and
+computes the encrypted message $c\equiv m^e\pmod n$ by repeated squaring. When he is
+finished, Bob transmits the encrypted message $c$ to Alice.\\
+
+Alice, on her side, uses her private key $d$ to compute $c^d \pmod n$ which obviously gives
+$$
+c^d \equiv (m^e)^d \equiv m^{(ed)} = m^1 \mod n,
+$$
+by Euler-/Fermat's Theorem Lemma \ref{lem:eulerfermat},
+and translates the message $m$ back into their agreed alphabet.
 
 \subsection{The Diffie-Hellmann cryptosystem}
 TODO