diff --git a/main.typ b/main.typ index 8fc0565..690562c 100644 --- a/main.typ +++ b/main.typ @@ -24,9 +24,9 @@ size: 12pt, ) -Name\ -Type of essay\ -Date +Marius Drechsler\ +Problem --- Solution Essay\ +July 5th, 2025 #align(center, text(size: 17pt, weight: "bold")[ *Essay Title* @@ -42,6 +42,54 @@ Date #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. + +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 + Essay has a total of #total-words words.