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 result*
 .direnv
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-main.pdf

bibliography.bib

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main.typ

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 #set page(
   paper: "a4",
-  //numbering: "1",
+  numbering: "-1-",
   margin: (top: 2.5cm, left: 2.5cm, right: 2.5cm, bottom: 2cm)
 )
 
-#if (context here().page()) != 1 [
-  #set page(
-    numbering: "1"
-  )
-]
-
-#set page(
-  footer: context {
-    if here().page() > 1 {
-      align(center)[#counter(page).display()]
-    }
-  }
-)
-
 #set text(
+  font: "Times New Roman",
   size: 12pt,
 )
 
 Marius Drechsler\
-Problem --- Solution Essay\
+Process Essay\
-July 5th, 2025
+May 17th, 2025
 
 #align(center, text(size: 17pt, weight: "bold")[
-  *Essay Title*
+  *Around the world in 133 ms*
 ])
 
 #set align(left)
@@ -42,57 +29,39 @@ July 5th, 2025
 
 #show: word-count
 
-In an increasing digital world, securing information through encryption methods has become a necessity.
+Have you ever wondered what really happens with your voice when you talking to someone on the phone?
-The rising trend of improvements in quantum computation poses a serious security vulnerability to information that is currently encrypted through classical encryption methods.
+From the instant the soundwaves leave your throat until they reach the ear of the person you are talking to,
-This essay will explain the risk of quantum computers regarding cryptography and present possible solutions for it.
+a series of analog and digital processes collaborate to carry your message.
-To properly understand the security vulnerability opened up by quantum computing, encryption methods in general will be investigated. 
+In fact, this whole process can be broken down into three major steps -- sampling, quantisation and modulation.
+In the course of this essay, we will investigate each of these steps in more depth to understand how modern
+communication works on a technical level.
+//To understand how we communicate across the globe on a technical level, we begin with the most primitive
+//instrument of all: the human voice.
 
-Current state-of-the-art technology utilizes two different encryption methods: symmetric and asymmetric encryption. 
+In the sampling process, an analogue signal is transformed into its digital representation.
-Symmetric encryption uses a single key for both the encryption and decryption process and is mainly used for securing data.
+This signal can be interpreted as any kind of waveform or motion that has not been processed by
-A common symmetric encryption algorithm is called "Advanced Encryption Standard (AES)". 
+a digital device yet.
-The security of data encrypted with algorithms like AES depends heavily on the length of the key used.
+For example, the sound of your voice or the tone of a guitar string is a suiting type of signal that we
-The longer the key, the more secure the encrypted data.
+want to digitize.
-Asymmetric encryption on the other hand uses pairs of keys --- a public and a private key --- to encrypt and decrypt information.
+However, a digital device like a computer or a phone cannot unterstand such an analogue signal, thus we have
-The principle behind asymmetric cryptography, as implemented by the "Rivest–Shamir–Adleman (RSA)" algorithm, stems from the complexity of factoring very large numbers into primes.
+to first convert it into some kind of electrical signal the device can unterstand.
-In summary, the security of symmetric and asymmetric encryption methods is based on the high computational effort required to break the encryption.
+We can achieve that by taking repeated "snapshots" of the current state of the analogue signal and saving
-While AES encryption with a long key requires trying a vast array of possible keys, RSA requires efficiently performing prime factorization on large numbers.
+the corresponding value.
-
+The resulting signal is now so called "time discreet", because we went from a continuous signal that has a value
-While symmetric and asymmetric encryption methods have proven effective in securing data, the continuous increase in performance of quantum computing could open up vulnerabilities in classical encryption algorithms.
+for every imaginable point in time to one where such values only exist at fixed, predefined points in time
-Quantum computers utilize a different approach to solve computational problems. 
+(i.e. every second).
-Instead of processing data in a binary format using ones and zeroes, quantum computers operate using qubits. 
+Going on, we now have a signal that consists of repeated snapshots of the originating signal where each value
-While qubits can represent two different values, like an ordinary bit, qubits are also capable of representing any value in between its two base states, for example zero and one. 
+can still be considered as continuous
-It is also important to note, that a qubit can, due to its physical properties, exist in multiple of these states at once. 
-This property allows a quantum computer to explore numerous possible solutions to a problem in parallel, significantly increasing the computation process. 
-Additionally, two qubits can also be created in such a way that their states depend on each other, making complex correlations between the two qubits possible.
-These two properties of qubits open up the possibility for quantum computers to solve the previously introduced numerical problems by encryption algorithms in an efficient way.
-
-As a result, quantum computers are able to solve the two problems making AES and RSA secure significantly faster than their classical counterparts. 
-To break the encryption of symmetric encryption algorithms like AES, "Grover's Algorithm" can be used.
-Grover's Algorithm is also commonly defined as the quantum search algorithm. 
-This means that Grover's Algorithm is capable of performing the task of _function inversion_.
-If a function is defined as $y = f(x)$, Gover's Algorithm is able to calculate the value of $x$ when given $y$.
-Comparing the operation of function inversion to the application of a symmetric encryption algorithm, $y$ can be seen as the encrypted data, while $x$ is the data to be encrypted by the algorithm $f()$.
-The notable difference between Grover's Algorithm and classical algorithms for the same task is the reduced number of steps required to find a solution. 
-Where classical algorithms would require $N$ steps to find a solution, Grover's Algorithm achieves the same result with $sqrt(N)$ steps.
-For example, brute-force searching a $128$-bit long key for AES encryption on a classical computer would require approximately $2^128$ trials, whereas Grover's algorithm could accomplish this in about $2^64$ trials.
-Another algorithm to break the classical encryption methods is "Shor's Algorithm", which is used to efficiently find the prime factors of an integer.
-As with Grover's Algorithm, Shor's Algorithm is able to find these prime factors faster than a classical algorithm. 
-The time complexity of the "General Number Field Sieve (GNFS)" Algorithm, which is considered the fastest classical integer factoring algorithm, is $O(2^N)$.
-In contrast, Shor's Algorithm has a time complexity of $O(log(N)^3)$.
-As a result, Shor's Algorithm reduces the complexity of finding the prime factors of an integer from exponential time to polynomial time, thus breaking the security of RSA, which depends on these prime factors.
-In conclusion, algorithms for quantum computers make it possible to speed up the process of breaking commonly used encryption methods.
-
-To address the vulnerabilities that quantum algorithms introduce, two solutions could be implemented.
-First, quantum-resistent algorithms could be implemented to undermine the efficiency of quantum computers. 
-
-
-Danach mögliche lösungen 
-
-DAnn zusammenfassung
 
+//To see how sampling works, we start with the sounds you make when you speak -- combinations of multiple sound waves at varying frequencies.
+/*For our purposes, however, we can simplify this complexity by modeling your voice as a single
+continuous sine wave, since this idealization does not affect the sampling process.
+Furthermore, we can think of this sine wave as the very first input into our communication pipeline.
+With the analogue signal established, we can go on and discuss the way our signal is transformed into a digital
+representation.
+*/
 
 Essay has a total of #total-words words.
 
 #pagebreak()
-
+#bibliography("bibliography.bib", style: "ieee", title: "References")
-#bibliography("./bibliography.bib", style: "ieee", title: "References")