diff --git a/crypto.bib b/crypto.bib index cca115e..e2521ce 100644 --- a/crypto.bib +++ b/crypto.bib @@ -24,4 +24,24 @@ year = "2025", url = "https://en.wikipedia.org/w/index.php?title=Confusion_and_diffusion&oldid=1307746165", note = "[Online; accessed 3-February-2026]" -} \ No newline at end of file +} +@ARTICLE{diffiehellman, + author={Diffie, W. and Hellman, M.}, + journal={IEEE Transactions on Information Theory}, + title={New directions in cryptography}, + year={1976}, + volume={22}, + number={6}, + pages={644-654}, + keywords={Cryptography;Receivers;Authentication;Eavesdropping;Costs;Business;Public key cryptography}, + doi={10.1109/TIT.1976.1055638}} +@article{rsa, + title={A method for obtaining digital signatures and public-key cryptosystems}, + author={Rivest, Ronald L and Shamir, Adi and Adleman, Leonard}, + journal={Communications of the ACM}, + volume={21}, + number={2}, + pages={120--126}, + year={1978}, + publisher={ACM New York, NY, USA} +} diff --git a/crypto.tex b/crypto.tex index 7dcca89..6ed7a2f 100644 --- a/crypto.tex +++ b/crypto.tex @@ -110,9 +110,9 @@ Early encryptions intuitively demonstrate two concepts that can be employed to e \paragraph{Substitution} is used by the simple Caesar cipher, often achieved by rotating two disks against each other, each with the alphabet written out on them. -\autoref{tab-caesar} shows a simple caesar cipher where the cipher alphabet is simply shifted by 3 positions from the plaintext alphabet. +\autoref{tab:caesar} shows a simple caesar cipher where the cipher alphabet is simply shifted by $+3$ positions from the plaintext alphabet. In the process of encoding, A is therefore replaced (substituted) with D, B with E, and so on. -Upon reception of the message, the same process is done in reverse. +Upon reception of the message, the same process is done in reverse, i.e. shifted by $-3$. \begin{table}[h] \resizebox{\textwidth}{!}{% @@ -124,15 +124,24 @@ Upon reception of the message, the same process is done in reverse. \end{tabular}% } \caption{A simple substitution cipher demonstrated by a 3-letter shift.} -\label{tab-caesar} +\label{tab:caesar} \end{table} +This simple encryption is easy to break however for several reasons. +Caesar ciphers in general only offer 26 different keys as further shifts only wrap around to $29 \mod 26 = 3$, with a shift of 26 +being equal to the cleartext. \newline +Furter, by shifting every letter by the same amount, +the properties of the source language such as word spacing and letter frequencies are retained in the ciphertext, +leaving it vulnerable to simple attacks. -\paragraph{Transposition} + +\paragraph{Transposition} is the process of reordering the plaintext to obtain a ciphertext. +Here, the key can be understood as instructions on how to re-order the ciphertext to obtain the original message. +The \textit{scytale} is one of the earliest implementations of a transposition cipher. \paragraph{Confusion and Diffusion} \cite{enwiki:confusion-diffusion} -\section{DES} +\section{DES}\label{sec:des} 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}, @@ -145,12 +154,15 @@ The \acrfull{AES} superseded \acrshort{DES} in 2001 after an official selection Unlike its predecessor, it does not use a Feistel network. \section{RSA} -\acrfull{RSA} is an asymmetric (or public-key) cryptographic algorithm used for encryption and digital signing. -It was named after its eponymous inventors in 1977 after trying to disprove the Diffie-Hellman key exchange. +\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}. + + The algorithm they came up with relies on modular arithmetic, which remains the most popular class of asymmetric cryptography. \begin{enumerate} - \item Choose and randomly and stochastically independet primes $p,q$ of similar size so that + \item Choose randomly and stochastically independet primes $p,q$ of similar size so that $0.1 < | \log_2 p - \log_2 q | < 30 $. \item Calculate $ N= p \cdot q $ \item Compute Euler's totient function of $ \varphi (N) = (p-1) \cdot (q-1)$ which is kept secret.