diff --git a/.gitignore b/.gitignore index cb6a416..dce8922 100644 --- a/.gitignore +++ b/.gitignore @@ -1,3 +1,4 @@ result* .direnv .envrc +main.pdf diff --git a/bibliography.bib b/bibliography.bib index e69de29..8b13789 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -0,0 +1 @@ + diff --git a/font/TypographerTextur-Bold.ttf b/font/TypographerTextur-Bold.ttf new file mode 100755 index 0000000..48c7f09 Binary files /dev/null and b/font/TypographerTextur-Bold.ttf differ diff --git a/font/TypographerTextur-Regular.ttf b/font/TypographerTextur-Regular.ttf new file mode 100755 index 0000000..4f33045 Binary files /dev/null and b/font/TypographerTextur-Regular.ttf differ diff --git a/font/TypographerTextur-Schatten.ttf b/font/TypographerTextur-Schatten.ttf new file mode 100755 index 0000000..2ea98f5 Binary files /dev/null and b/font/TypographerTextur-Schatten.ttf differ diff --git a/font/typographertextur.zip b/font/typographertextur.zip new file mode 100644 index 0000000..b297f4c Binary files /dev/null and b/font/typographertextur.zip differ diff --git a/main.pdf b/main.pdf deleted file mode 100644 index ccde7a6..0000000 Binary files a/main.pdf and /dev/null differ diff --git a/main.typ b/main.typ index 602bf52..690562c 100644 --- a/main.typ +++ b/main.typ @@ -2,21 +2,34 @@ #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\ -Process Essay\ -May 17th, 2025 +Problem --- Solution Essay\ +July 5th, 2025 #align(center, text(size: 17pt, weight: "bold")[ - *Around the world in 133 ms* + *Essay Title* ]) #set align(left) @@ -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? -From the instant the soundwaves leave your throat until they reach the ear of the person you are talking to, -a series of analog and digital processes collaborate to carry your message. -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. +In an increasing digital world, securing information through encryption methods has become a necessity. +The rising trend of improvements in quantum computation poses a serious security vulnerability to information that is currently encrypted through classical encryption methods. +This essay will explain the risk of quantum computers regarding cryptography and present possible solutions for it. +To properly understand the security vulnerability opened up by quantum computing, encryption methods in general will be investigated. -In the sampling process, an analogue signal is transformed into its digital representation. -This signal can be interpreted as any kind of waveform or motion that has not been processed by -a digital device yet. -For example, the sound of your voice or the tone of a guitar string is a suiting type of signal that we -want to digitize. -However, a digital device like a computer or a phone cannot unterstand such an analogue signal, thus we have -to first convert it into some kind of electrical signal the device can unterstand. -We can achieve that by taking repeated "snapshots" of the current state of the analogue signal and saving -the corresponding value. -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 -(i.e. every second). -Going on, we now have a signal that consists of repeated snapshots of the originating signal where each value -can still be considered as continuous +Current state-of-the-art technology utilizes two different encryption methods: symmetric and asymmetric encryption. +Symmetric encryption uses a single key for both the encryption and decryption process and is mainly used for securing data. +A common symmetric encryption algorithm is called "Advanced Encryption Standard (AES)". +The security of data encrypted with algorithms like AES depends heavily on the length of the key used. +The longer the key, the more secure the encrypted data. +Asymmetric encryption on the other hand uses pairs of keys --- a public and a private key --- to encrypt and decrypt information. +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. +In summary, the security of symmetric and asymmetric encryption methods is based on the high computational effort required to break the encryption. +While AES encryption with a long key requires trying a vast array of possible keys, RSA requires efficiently performing prime factorization on large numbers. + +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. +Quantum computers utilize a different approach to solve computational problems. +Instead of processing data in a binary format using ones and zeroes, quantum computers operate using qubits. +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. #pagebreak() -#bibliography("bibliography.bib", style: "ieee", title: "References") + +#bibliography("./bibliography.bib", style: "ieee", title: "References")