diff --git a/.gitignore b/.gitignore index dce8922..cb6a416 100644 --- a/.gitignore +++ b/.gitignore @@ -1,4 +1,3 @@ result* .direnv .envrc -main.pdf diff --git a/bibliography.bib b/bibliography.bib index 8b13789..e69de29 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -1 +0,0 @@ - diff --git a/font/TypographerTextur-Bold.ttf b/font/TypographerTextur-Bold.ttf deleted file mode 100755 index 48c7f09..0000000 Binary files a/font/TypographerTextur-Bold.ttf and /dev/null differ diff --git a/font/TypographerTextur-Regular.ttf b/font/TypographerTextur-Regular.ttf deleted file mode 100755 index 4f33045..0000000 Binary files a/font/TypographerTextur-Regular.ttf and /dev/null differ diff --git a/font/TypographerTextur-Schatten.ttf b/font/TypographerTextur-Schatten.ttf deleted file mode 100755 index 2ea98f5..0000000 Binary files a/font/TypographerTextur-Schatten.ttf and /dev/null differ diff --git a/font/typographertextur.zip b/font/typographertextur.zip deleted file mode 100644 index b297f4c..0000000 Binary files a/font/typographertextur.zip and /dev/null differ diff --git a/main.pdf b/main.pdf new file mode 100644 index 0000000..ccde7a6 Binary files /dev/null and b/main.pdf differ diff --git a/main.typ b/main.typ index 690562c..602bf52 100644 --- a/main.typ +++ b/main.typ @@ -2,34 +2,21 @@ #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\ -July 5th, 2025 +Process Essay\ +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. -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. +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. -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 +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 +//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")