First draft of problem solution essay

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 Marius Drechsler\
-Process Essay\
+Problem --- Solution Essay\
-May 17th, 2025
+July 5th, 2025
 
 #align(center, text(size: 17pt, weight: "bold")[
-  *Around the world in 133 ms*
+  *Essay Title*
 ])
 
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@@ -29,39 +42,57 @@ May 17th, 2025
 
 #show: word-count
 
-Have you ever wondered what really happens with your voice when you talking to someone on the phone?
+In an increasing digital world, securing information through encryption methods has become a necessity.
-From the instant the soundwaves leave your throat until they reach the ear of the person you are talking to,
+The rising trend of improvements in quantum computation poses a serious security vulnerability to information that is currently encrypted through classical encryption methods.
-a series of analog and digital processes collaborate to carry your message.
+This essay will explain the risk of quantum computers regarding cryptography and present possible solutions for it.
-In fact, this whole process can be broken down into three major steps -- sampling, quantisation and modulation.
+To properly understand the security vulnerability opened up by quantum computing, encryption methods in general will be investigated. 
-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.
 
-In the sampling process, an analogue signal is transformed into its digital representation.
+Current state-of-the-art technology utilizes two different encryption methods: symmetric and asymmetric encryption. 
-This signal can be interpreted as any kind of waveform or motion that has not been processed by
+Symmetric encryption uses a single key for both the encryption and decryption process and is mainly used for securing data.
-a digital device yet.
+A common symmetric encryption algorithm is called "Advanced Encryption Standard (AES)". 
-For example, the sound of your voice or the tone of a guitar string is a suiting type of signal that we
+The security of data encrypted with algorithms like AES depends heavily on the length of the key used.
-want to digitize.
+The longer the key, the more secure the encrypted data.
-However, a digital device like a computer or a phone cannot unterstand such an analogue signal, thus we have
+Asymmetric encryption on the other hand uses pairs of keys --- a public and a private key --- to encrypt and decrypt information.
-to first convert it into some kind of electrical signal the device can unterstand.
+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.
-We can achieve that by taking repeated "snapshots" of the current state of the analogue signal and saving
+In summary, the security of symmetric and asymmetric encryption methods is based on the high computational effort required to break the encryption.
-the corresponding value.
+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 resulting signal is now so called "time discreet", because we went from a continuous signal that has a value
+
-for every imaginable point in time to one where such values only exist at fixed, predefined points in time
+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.
-(i.e. every second).
+Quantum computers utilize a different approach to solve computational problems. 
-Going on, we now have a signal that consists of repeated snapshots of the originating signal where each value
+Instead of processing data in a binary format using ones and zeroes, quantum computers operate using qubits. 
-can still be considered as continuous
+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. 
+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.
 
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-#bibliography("bibliography.bib", style: "ieee", title: "References")
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+#bibliography("./bibliography.bib", style: "ieee", title: "References")