108 lines
8.2 KiB
Typst
108 lines
8.2 KiB
Typst
#import "@preview/wordometer:0.1.4": word-count, total-words
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#set page(
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paper: "a4",
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//numbering: "1",
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margin: (top: 2.5cm, left: 2.5cm, right: 2.5cm, bottom: 2cm)
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)
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#set document(title: "The Challenge of Quantum Computing in Cryptography", author: "Marius Drechsler", date: auto)
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#if (context here().page()) != 1 [
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#set page(
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numbering: "1"
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)
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]
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#set page(
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footer: context {
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if here().page() > 1 {
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align(center)[#counter(page).display()]
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}
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}
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)
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#set text(
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size: 12pt,
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)
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Marius Drechsler\
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Problem --- Solution Essay\
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July 6th, 2025
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#align(center, text(size: 17pt, weight: "bold")[
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*The Challenge of Quantum Computing in Cryptography*
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])
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#set align(left)
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#set par(
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justify: true,
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leading: 2em,
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spacing: 2em,
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first-line-indent: (amount: 3em, all: true)
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)
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#show: word-count
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In today's increasingly digital landscape, the protection of sensitive information through encryption has become essential.
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However, the rapid advancements in quantum computing present a formidable challenge to the security of data encrypted using classical methods.
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Quantum computers, with their unique ability to perform complex calculations at unprecedented speeds, threaten to undermine the very foundations of traditional cryptography @steane1998quantum.
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This essay will explain the risk of quantum computers regarding cryptography and present possible solutions for it.
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To properly understand the security vulnerability opened up by quantum computing, encryption methods in general will now be investigated.
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Current state-of-the-art technology utilizes two different encryption methods: symmetric and asymmetric encryption.
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Symmetric encryption uses a single key for both the encryption and decryption process and is mainly used for securing data.
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A common symmetric encryption algorithm is called "Advanced Encryption Standard (AES)" @abdullah2017advanced.
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The security of data encrypted with algorithms like AES depends heavily on the length of the key used.
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The longer the key, the more secure the encrypted data.
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Asymmetric encryption on the other hand uses pairs of keys --- a public and a private key --- to encrypt and decrypt information.
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The principle behind asymmetric cryptography, as implemented by the "Rivest–Shamir–Adleman (RSA)" algorithm @milanov2009rsa, stems from the complexity of factoring very large numbers into primes.
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In summary, the security of symmetric and asymmetric encryption methods is based on the high computational effort required to break the encryption.
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While AES encryption with a long key requires trying a vast array of possible keys, RSA requires efficiently performing prime factorization on large numbers.
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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.
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Quantum computers utilize a different approach to solve computational problems @steane1998quantum.
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Instead of processing data in a binary format using ones and zeroes, quantum computers operate using qubits @ozhigov1998quantum.
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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.
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It is also important to note, that a qubit can, due to its physical properties, exist in multiple of these states at once.
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This property allows a quantum computer to explore numerous possible solutions to a problem in parallel, significantly increasing the computation process.
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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.
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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 @ozhigov1998quantum.
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As a result, quantum computers are able to solve the two problems making AES and RSA secure significantly faster than their classical counterparts @ugwuishiwu2020overview.
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To break the encryption of symmetric encryption algorithms like AES, "Grover's Algorithm" @jozsa1999searching can be used.
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Grover's Algorithm is also commonly defined as the quantum search algorithm.
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This means that Grover's Algorithm is capable of performing the task of _function inversion_.
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If a function is defined as $y = f(x)$, Gover's Algorithm is able to calculate the value of $x$ when given $y$.
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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 $f()$.
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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.
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Where classical algorithms would require $N$ steps to find a solution, Grover's Algorithm achieves the same result with $sqrt(N)$ steps.
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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.
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Another algorithm to break the classical encryption methods is "Shor's Algorithm" @ugwuishiwu2020overview, which is used to efficiently find the prime factors of an integer.
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As with Grover's Algorithm, Shor's Algorithm is able to find these prime factors faster than a classical algorithm.
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The time complexity of the "General Number Field Sieve (GNFS)" Algorithm, which is considered the fastest classical integer factoring algorithm, is $O(2^N)$.
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In contrast, Shor's Algorithm has a time complexity of $O(log(N)^3)$.
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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.
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In conclusion, algorithms for quantum computers make it possible to speed up the process of breaking commonly used encryption methods.
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To address the vulnerabilities that quantum algorithms introduce, two solutions could be implemented.
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First, quantum-resistant encryption algorithms could be implemented @balamurugan2021post.
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These algorithms are designed in such a way that the speed advantages of quantum computers cannot provide any decisive benefit.
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For example, Lattice-Based cryptography introduces methods to develop new encryption algorithms which are resistant to attacks using Shor's Algorithm.
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The NTRU @hoffstein1999ntru encryption algorithm is a relatively new, lattice-based alternative to asymmetric encryption algorithms like RSA.
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Lattice-based algorithms are currently not known to be breakable using quantum computers.
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To account for symmetric encryption algorithms, significantly increasing the key length makes algorithms like AES already resistant to attacks by a quantum computer.
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Second, instead of designing classical encryption algorithms that would also require excessive computational effort for quantum computers, quantum-based encryption algorithms be used @balamurugan2021post.
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Algorithms like E91 or BB84 @begimbayeva2022research are engineered such that the quantum physical properties of qubits make it physically impossible to derive the used key.
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However, these quantum-based algorithms would require new, quantum-specific communication infrastructure that is not yet widely adopted @lewis2022secure.
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Therefore, quantum-resistant algorithms appear to be the more sensible of the two solutions, as they do not require new communication infrastructure.
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In conclusion, the emergence of quantum computing represents a profound challenge to the security of current encryption methods such as AES and RSA.
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As quantum algorithms like Grover's or Shor's algorithm threaten to compromise the security of encrypted data, solutions to protect sensitive information have to be found.
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The adoption of quantum-resistant encryption algorithms, such as the lattice-based NTRU, along with the implementation of security enhancement measures for symmetric cryptography, represent crucial steps to secure communication in the context of quantum computing.
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//Essay has a total of #total-words words.
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#pagebreak()
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#bibliography("./bibliography.bib", style: "ieee", title: "References")
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