crypto: rsa

crypto.bib

@@ -65,4 +65,18 @@
   year = "2026",
   url = "https://en.wikipedia.org/w/index.php?title=Galois/Counter_Mode&oldid=1352962810",
   note = "[Online; accessed 10-May-2026]"
+}
+@book{rijndael,
+  title={The design of Rijndael},
+  author={Daemen, Joan and Rijmen, Vincent},
+  volume={2},
+  year={2002},
+  publisher={Springer}
+}
+@misc{ enwiki:aes,
+  author = "{Wikipedia contributors}",
+  title = "Advanced Encryption Standard --- {Wikipedia}{,} The Free Encyclopedia",
+  year = "2026",
+  url = "https://en.wikipedia.org/w/index.php?title=Advanced_Encryption_Standard&oldid=1358692329",
+  note = "[Online; accessed 28-June-2026]"
 }

crypto.tex

@@ -19,6 +19,8 @@
 \usepackage{glossaries}
 
 \newcommand{\ub}[2]{\underbrace{\text{#1}}_{\text{#2}}}
+\newcommand{\secret}[1]{\textcolor{red}{#1}}
+\newcommand{\public}[1]{\textcolor{blue}{#1}}
 \makeglossaries
 \newacronym{DES}{DES}{Data Encryption Standard}
 \newacronym{AES}{AES}{Advanced Encryption Standard}
@@ -44,7 +46,7 @@ Much later, with the widespread adoption of the internet, the need for several c
 Due to its intended use as a trusted research network (ARPANET),
 almost none of the original protocols were 'secure' in any sense of the word.
 
-Most notably still today is SMTP, the \textit{Simple Mail Transfer Protocol}, used to send email to servers.
+One of the most notable examples still today is SMTP, the \textit{Simple Mail Transfer Protocol}, used to send email to servers.
 In its original implementation, it allowed attackers to intercept emails in transit to read and modify them
 and even spoof the sender address to impersonate others.
 SMTP today is secured using a combination of mitigations for these attacks, such as STARTTLS, SPF, DKIM and DMARC,
@@ -165,7 +167,8 @@ diffusion through permutation boxes (P-box).
 The \acrfull{DES} is a symmetric (or private-key) cipher developed in the 1970s at IBM as an archetypal block cipher.
 It takes in a block of 64 bits and transforms it to a ciphertext using a key of equal length.
 Despite suspicions of backdoors engineered into the algorithm due to the involvement of the NSA in the development of \acrshort{DES},
-it was approved as a federal standard in the USA in 1976 and only retired due to its short key length,
+only minor weaknesses were found.
+It was approved as a federal standard in the USA in 1976 and only retired due to its short key length,
 for which the NSA however was directly responsible as well.
 Nevertheless, it sparked public and scientific interest in the research of encryption algorithms, producing a large body of publications.
 Extended versions of \acrshort{DES} such as Triple-DES (or 3DES) are still in use in embedded applications (chipcards).
@@ -187,7 +190,21 @@ they naturally lend themselves to hardware implementations.
 
 \subsection{AES} %TODO
 The \acrfull{AES} superseded \acrshort{DES} in 2001 after an official selection process.
-Unlike its predecessor, it does not use a Feistel network.
+Standardized under \verb|ISO 18033-3|, \acrshort{AES} supports key lengths of 128, 192 and 256 bits,
+the underlying algorithm supports any key length that is a multiple of 32.
+Developed as Rijndael, contrary to its predecessor it does not employ a Feistel network,
+though it does use the familiar pattern of substitution and permutation boxes.
+
+The key size used also specifies the number of rounds, as a round key is derived for each transformation round.
+\begin{enumerate}
+    \item 
+\end{enumerate}
+
+
+Though minor theoretical attacks exist, such as reducing the key search space by a factor of 4,
+the algorithm can be considered secure even to quantum-computing supported attacks halving the key space,
+as simply increasing the key length is a viable mitigation.
+\cite{rijndael,enwiki:aes}
 
 
 
@@ -198,35 +215,53 @@ Symmetric encryption however historically suffered from a key exchange problem;
 because the same key is used for encryption and decryption, a secure channel is required to agree on a common key.
 This chicken-and-egg problem can be solved in two major ways, both typically relying on the mathematical theory of
 \textbf{galois fields}, employing either \textit{modular arithmetic} or \textit{elliptic curves}.
+%NOTE section on galois fields/ modulo/elliptic curve?
 
 \paragraph{The Difie-Hellman Key Exchange} is an algorithm allowing the communication parties to establish a shared secret using
 properties of the discrete logarithm.
+The algorithm is implemented by the following steps, with \public{public values} in blue and \secret{secret values} in red.
+\begin{enumerate}
+    \item First, Alice and Bob publicly agree on prime numbers \public{$p$} and \public{$g$}.
+    \item Alice (and Bob, each on their own) choose their secret keys \secret{$a$} (\secret{$b$})
+    \item They then exchange
+        $\public{A} = \public{g}^{\secret{a}} \mod \public{p}$ and $\public{B} = \public{g}^{\secret{b}} \mod \public{p}$, respectively.
+    \item Both arrive at the same key $\secret{s}$ by computing $\secret{s} = \public{B}^{\secret{a}} \mod \public{p} =
+        \public{A}^{\secret{b}} \mod \public{p}  $.
+\end{enumerate}
 
 Both methods however are still vulnerable to a \acrfull{mitm}, thus also requiring a trusted \acrfull{CA} for authentication. \cite{enwiki:mitm}
 
-\subsection{RSA}
+\subsection{RSA}\label{sec:rsa}
 \acrfull{RSA} is the first asymmetric (or public-key) cryptographic algorithm and can thus be used for encryption and digital signing.
 It was named after its eponymous inventors in \citeyear{rsa} after trying to disprove the existence of \textit{trapdoor functions},
-a concept introduced by \citeauthor{diffiehellman} in their appropriately named pivotal paper \citetitle{diffiehellman}.
+a concept introduced by \citeauthor{diffiehellman} in their appropriately named pivotal paper \citetitle{diffiehellman} \cite{diffiehellman}.
 
 
 The algorithm they came up with relies on modular arithmetic, which remains the most popular class of asymmetric cryptography.
 
 \begin{enumerate}
-    \item Choose randomly and stochastically independet primes $p,q$ of similar size so that
+    \item Choose randomly and stochastically independet primes $\secret{p,q}$ of similar size so that
         \newline$0.1 < | \log_2 p - \log_2 q | < 30 $.
-    \item Calculate $ N= p \cdot q $
+    \item Calculate $ \public{N}= \secret{p} \cdot \secret{q} $
-    \item Compute Euler's totient function of $ \varphi (N) = (p-1) \cdot (q-1)$ which is kept secret.
+    \item Compute Euler's totient function of $ \secret{\varphi (N)} = (\secret{p}-1) \cdot (\secret{q}-1)$ which is kept secret.
-    \item Choose an integer $e$ so that $ 1 < e < \varphi (N) $ and $\gcd(e, \varphi(N)) =1$, i.e. $e$ and $\varphi(N)$
+    \item Choose an integer $\public{e}$ so that $ 1 < \public{e} < \secret{\varphi (N)} $ and $\gcd(\public{e}, \secret{\varphi(N)}) =1$,
-        are coprime. The most common choice here is $ e= 2^(16) +1 = 65537 $, as $e$ is released as part of the public key.
+        i.e. $\public{e}$ and $\secret{\varphi(N)}$ are coprime.
-    \item For the private key, % TODO
+        The most common choice here is $ e= 2^(16) +1 = 65537 $, as $\public{e}$ is released as part of the public key.
+    \item Calculate $ \secret{d}  \equiv \public{e}^{-1}$, i.e. $\secret{d}\public{e} = 1\quad | \mod \secret{\varphi(N)}$
+        (modular multiplicative inverse).
 \end{enumerate}
+The \public{public key} consists of the modulus $N$ and the public exponent $e$.
+The \secret{private key} consists of the private exponent $d$, as a result of which $p,q,\varphi(N)$ have to be kept secret as well.
+
+In the encryption process, the ciphertext $c$ is obtained from the message $m$ by calculating $c \equiv m^e \mod N$.
+After transmission, the message can then be recovered using the the private key exponent $d$ as follows:
+\[ c^d \equiv (m^e)^d = m^{e*d} \equiv m^1 | \mod N \]
 \clearpage
 
 \section{Conclusion}
 Trust on the web with untrusted channels fundamentally remains an unsolved issue,
 though depending on the threat model, everyday communications can be considered relatively secure from non-APT actors.
-A typical cipher suite employed by TLS could look like the following:
+A typical cipher suite employed by TLS to achieve this security could look like the following:
 $$\ub{ECDHE}{Key exchange}-\ub{ECDSA}{authentication}-\ub{AES128}{encryption}-\ub{GCM}{Cipher operation mode}-\ub{SHA256}{hashing} $$
 \begin{itemize}
     \item \textbf{ECDHE} Elliptic Curve Diffie Hellman Exchange