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#set document(title: "The Challenge of Quantum Computing in Cryptography", author: "Marius Drechsler", date: auto)
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Marius Drechsler\
Problem --- Solution Essay\
July 5th, 2025
July 6th, 2025
#align(center, text(size: 17pt, weight: "bold")[
*Essay Title*
*The Challenge of Quantum Computing in Cryptography*
])
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#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.
In today's increasingly digital landscape, the protection of sensitive information through encryption has become essential.
However, the rapid advancements in quantum computing present a formidable challenge to the security of data encrypted using classical methods.
Quantum computers, with their unique ability to perform complex calculations at unprecedented speeds, threaten to undermine the very foundations of traditional cryptography @steane1998quantum.
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.
To properly understand the security vulnerability opened up by quantum computing, encryption methods in general will now 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)".
A common symmetric encryption algorithm is called "Advanced Encryption Standard (AES)" @abdullah2017advanced.
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.
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.
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.
Quantum computers utilize a different approach to solve computational problems @steane1998quantum.
Instead of processing data in a binary format using ones and zeroes, quantum computers operate using qubits @ozhigov1998quantum.
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.
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.
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.
As a result, quantum computers are able to solve the two problems making AES and RSA secure significantly faster than their classical counterparts @ugwuishiwu2020overview.
To break the encryption of symmetric encryption algorithms like AES, "Grover's Algorithm" @jozsa1999searching 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 $y$.
@ -75,7 +78,7 @@ Comparing the operation of function inversion to the application of a symmetric
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.
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.
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 $O(log(N)^3)$.
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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.
First, quantum-resistant encryption algorithms could be implemented @balamurugan2021post.
These algorithms are designed in such a way that the speed advantages of quantum computers cannot provide any decisive benefit.
For example, Lattice-Based cryptography introduces methods to develop new encryption algorithms which are resistant to attacks using Shor's Algorithm.
The NTRU @hoffstein1999ntru encryption algorithm is a relatively new, lattice-based alternative to asymmetric encryption algorithms like RSA.
Lattice-based algorithms are currently not known to be breakable using quantum computers.
To account for symmetric encryption algorithms, significantly increasing the key length makes algorithms like AES already resistant to attacks by a quantum computer.
Second, instead of designing classical encryption algorithms that would also require excessive computational effort for quantum computers, quantum-based encryption algorithms be used @balamurugan2021post.
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.
However, these quantum-based algorithms would require new, quantum-specific communication infrastructure that is not yet widely adopted @lewis2022secure.
Therefore, quantum-resistant algorithms appear to be the more sensible of the two solutions, as they do not require new communication infrastructure.
In conclusion, the emergence of quantum computing represents a profound challenge to the security of current encryption methods such as AES and RSA.
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.
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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DAnn zusammenfassung
Essay has a total of #total-words words.
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