Quantum computing, an interdisciplinary field of computer science and quantum mechanics, offers the promise of solving complex problems far beyond the reach of classical computers, opening the door to computational possibilities that were once considered purely theoretical. Quantum computing, particularly through algorithms like Shor's, presents both significant challenges and groundbreaking opportunities in modern cryptography.

## Table of Contents

- The Quantum Leap into Computing
- Superposition: From Qubits to Quantum Registers
- The Transformative Power of Quantum Computing
- Challenges to Symmetric Cryptography
- Breaking Asymmetric Cryptography
- The Quantum-Secure Horizon: Preparing for the Q-Day
- The NIST Standardization
- Quantum Cryptography
- Quantum Computing in 2023
- Conclusion
- Support My Work
- Further Reading
- Comments

## The Quantum Leap into Computing

In the first half of the 20th century, a pair of groundbreaking scientific revolutions unfolded. On one hand, visionaries like Konrad Zuse, Alan Turing, and John von Neumann pioneered the creation of the first functional computers. On the other, the classical worldview of physics, largely unaltered since Newton's era, underwent a dramatic transformation. A radically new theory was forged: Quantum mechanics. The new theory didn't just add to our knowledge; it completely redefined our understanding of reality.

These groundbreaking scientific movements rapidly paved the way for technological progress. The extent to which computers shape our society, worldview, and perception of humanity is evident to all. However, many are less aware of how quantum mechanics similarly influences our daily lives. It's the quantum mechanical understanding of the atom that has allowed the invention of semiconductors and lasers, leading to inventions like transistor radios, CD players, and the very heart of modern computing hardware.

In recent decades, these two sciences have converged, giving rise to a new interdisciplinary field known as Quantum Computing. The aim is to build quantum computers, develop quantum algorithms, and explore their implications on information transmission. Quantum Computers operate on principles that defy conventional computing logic. They are fundamentally different. With these machines, it is possible to solve problems that conventional computers cannot solve, regardless of their increasing computational power. These include absolute secure message transmission, teleportation of information, generating true random numbers, and breaking classical encryption methods.

Predicting the full impact of quantum computing is like trying to map the stars in an undiscovered galaxy, but the signs point towards a new scientific revolution. Under laboratory conditions, quantum computers have already become a reality. Quantum supremacy became a global race with tech giants like Amazon, Google, and IBM at the forefront. Even intelligence agencies like the NSA recognize the strategic importance of this technology, evidenced by their significant investments.

## Superposition: From Qubits to Quantum Registers

While the classic computer operates with bits set to either 0 or 1, quantum computing introduces a game-changing concept: the quantum bit, or qubit. Unlike its classical counterpart, a qubit can exist in a state of superposition - essentially embodying both 0 and 1 simultaneously.

The power of superposition becomes more evident as we scale up: In classical computing, a machine with n bits can represent 2^n different numbers but can only store or process one of these numbers at a time. In contrast, a quantum computer with n qubits can represent and process 2^n numbers simultaneously. This means that a quantum algorithm can operate on all possible inputs in a single computation, leading to a superposition of all possible outcomes.

It is crucial to understand that quantum computers are not advanced parallel processors. The parallel processors in use today, equipped with multiple CPUs, perform several calculations at the same time. However, Quantum Computers operate on a different principle.

Firstly, accessing the mentioned superposition of solutions is not straightforward. Attempting to read a qubit generally influences its state. This requires what is known as the measurement of a qubit. Upon measurement, the superposition is destroyed and we receive one of the computation results "more or less randomly". Skillful techniques are required to extract the desired information.

To illustrate the essence of quantum superposition and measurement, consider the famous thought experiment of Schrödinger's cat. In this scenario, a cat in a sealed box has a 50% chance of being killed by a toxic gas, depending on the random decay of an atom. According to quantum theory, until the box is opened, the cat is both dead and alive - it exists in a superposed state of both possibilities. Only when the box is opened, and the cat is observed, does this superposition collapse into a definitive state of either alive or dead.

When qubits are combined, they form a quantum register. As expected, the state of a quantum register of n qubits is a superposition of the 2^n classical states.

## The Transformative Power of Quantum Computing

One of the most astonishing capabilities of quantum computing is the so-called teleportation of information. While teleportation is often portrayed in science fiction as a means to transfer physical objects across space, quantum teleportation operates quite differently. It "only" transfers quantum information. Intriguingly, in quantum teleportation, the sender doesn't need to know the specific quantum state being transferred. Furthermore, the recipient's location may remain undisclosed.

While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from one location to the next, quantum teleportation only transfers quantum information. The sender does not have to know the particular quantum state being transferred. Moreover, the location of the recipient can be unknown, but to complete the quantum teleportation, classical information needs to be sent from sender to receiver. Because classical information needs to be sent, quantum teleportation cannot occur faster than the speed of light.

The first successful teleportations were carried out in 1997 by Anton Zeilinger's research group in Innsbruck. By 2012, teleportation over a distance of 97 km had been achieved.

A highly significant algorithm was introduced in 1996 by Lov Grover working for Bell Labs at Lucent Technologies. He demonstrated that quantum computers have the potential to vastly outpace classical computers in search efficiency. To put this into perspective: Imagine searching for a specific phone number in an unsorted phone book with a million entries. A classical computer might need to check, on average, 500,000 entries, or in the worst case, all million. In contrast, a quantum computer using Grover's algorithm would only require about 1000 accesses. This represents a quadratic speedup, making quantum computing exceptionally powerful for database searching and data mining tasks.

The potential applications of quantum computing extend far beyond these examples. In pharmaceuticals, quantum computing could revolutionize drug discovery by simulating molecular interactions at an unprecedented scale. In finance, it could optimize complex risk assessments and portfolio management. Additionally, quantum computing could solve intricate optimization problems in logistics, improve climate modeling in environmental science, and even unlock new mysteries in fundamental physics.

## Challenges to Symmetric Cryptography

Symmetric ciphers use the same key for both encryption and decryption. The security of these algorithms relies heavily on the secrecy of the key. Common examples include AES (Advanced Encryption Standard) and DES (Data Encryption Standard).

The classical approach to breaking a symmetric cryptography involves brute-force attacks, where every possible key is tried until the correct one is found. The time it takes to break the encryption increases exponentially with the length of the key. For example, a 128-bit key would require 2^128 different combinations to be tried in a brute-force attack.

As described above, "searching" all possible keys using Grover's Algorithm takes a quantum computer only the square root of the time it would take a classical computer. For the example of a 128-bit key, a quantum computer can potentially do a complete brute-force attack in 2^64 operations.

In general, this means that in the context of symmetric cryptography, the time for a brute force search is reduced from 2^n to 2^(n/2) for an n bit key using a quantum computer. This implies that as of today, a 256-bit AES key is still considered secure, even in the age of quantum computing, as it effectively provides a 128-bit level of security against quantum attacks. However, shorter keys might be lengthened or replaced to maintain robust security against quantum computing attacks. Nevertheless, for symmetric encryption methods like AES, quantum computers pose a relatively small threat.

## Breaking Asymmetric Cryptography

Shor's algorithm, introduced in 1994 by mathematician Peter Shor, is arguably the most famous quantum algorithm. When it was published, it represented the first algorithm to offer an efficient solution for a problem that classical computers struggle with: factoring large integers. This capability is particularly crucial because it directly challenges the security of RSA cryptography, a public key cryptographic system and one of the pillars of modern digital security.

In asymmetric systems, there are two keys involved: a public key, which senders use to encrypt messages, and a private key, which the recipient must keep secret because it is used to decrypt messages that have been encrypted with the public key. Asymmetric cryptography is one of the foundations to secure digital communication as symmetric cryptography is not sufficient primarily because it requires both parties to securely share and maintain a secret key beforehand.

The RSA cryptosystem's security relies on the mathematical challenge of factoring large integers into prime numbers, a task that is "difficult" and "too time consuming" for classical computers. However, Shor's algorithm can factor these large numbers exponentially faster than the best-known algorithms running on classical computers. This means that, in theory, a sufficiently powerful quantum computer could break RSA encryption, decrypting data and messages thought to be secure.

The implications of Shor's algorithm extend beyond RSA: Other popular asymmetric cryptosystems like Diffie-Hellman, ElGamal, and those based on elliptic curves are threatened by quantum computing as well. They rely on the difficulty of certain mathematical problems (discrete logarithm problem and the elliptic curve discrete logarithm problem) which could be potentially solved by similar quantum algorithms to Shor's algorithm.

## The Quantum-Secure Horizon: Preparing for the Q-Day

As of 2023, quantum computers lack the processing power to break widely used cryptographic algorithms. However, the cryptographic community is anticipating the Q-Day, the day when existing algorithms will be vulnerable to quantum computing attacks.

Daniel Bernstein introduced the term post-quantum cryptography (PQC) which refers to the subfield of cryptography that deals with cryptographic primitives which are practically impossible to decrypt even with the use of quantum computers. Bernstein was also involved in organizing the first specialist conference, PQCrypto, on this topic in 2006.

The goal is to create systems that are practically indecipherable by quantum computers. Several promising approaches are being explored in the development of quantum-resistant encryption algorithms. One such approach is based on mathematical lattices, which form the foundation of algorithms like Ring-LWE and NTRUEncrypt. Additionally, research is being conducted in various other domains, including:

- Multivariate polynomials, with methods like the Unbalanced Oil and Vinegar scheme.
- Cryptographic hash functions, exemplified by the Merkle Signature Scheme and the Lamport One-Time Signature method.
- Error-correcting codes, such as the McEliece Cryptosystem.
- Supersingular elliptic curve isogeny cryptography, as seen in the SIKE protocol.
- Lattice-based cryptography, with algorithms like CRYSTALS DILITHIUM emerging as frontrunners.

## The NIST Standardization

The National Institute of Standards and Technology (NIST) played a crucial role in anticipating the quantum computing era through its PQC Standardization Process. Its call for submissions in 2016 marked the beginning of a process which led to a multi-round evaluation of the submissions to select a few that showed promise for eventual standardization. The focus was on ensuring that these algorithms could protect sensitive information, especially for U.S. Government communications, even post the Q-Day.

By 2023 NIST has selected four algorithms for standardization. These include CRYSTALS-Kyber, a public key encapsulation mechanism, and three digital signature schemes: CRYSTALS-Dilithium, FALCON, and SPHINCS+.

So far CRYSTALS-Kyber is the only quantum-safe key exchange algorithm approved by NIST. It is important to understand that standardization does not equate to the absolute security of these algorithms. The recent discovery of vulnerabilities in algorithms like CRYSTALS-Kyber through advanced methods like AI-assisted side-channel attacks, underscores this point.

## Quantum Cryptography

Post-Quantum-Cryptography (PQC) as explained above focuses on developing cryptographic algorithms that are "quantum resistant" while still operating on classical computers. This is distinct from quantum cryptography which uses the principles of quantum mechanics itself for secure communication.

Quantum cryptography is the science of exploiting quantum mechanical properties to perform cryptographic tasks. One of its key applications is quantum key distribution (QKD), which offers a secure way to exchange cryptographic keys using quantum particles, primarily photons. This method ensures a high level of security, as any attempt at eavesdropping can be detected due to the fundamental principles of quantum mechanics.

## Quantum Computing in 2023

As of 2023, the state of quantum computing has seen considerable progress, with Big Tech companies actively working towards increasing the number of qubits to be used in quantum computing.

One of the leaders in this field is IBM, which took the record for the largest quantum computing system "Osprey" with a processor that contained 433 qubits. GlobalData predicts that full-scale commercial quantum computing will likely begin in 2027.

Big Tech companies have begun offering quantum-as-a-service (QaaS), notably Microsoft's Azure Quantum, which provides users access to hardware from other quantum companies. This market is expected to grow as more companies will invest in quantum.

However, it is important to note that the number of qubits alone does not entirely determine the computational power of a quantum computer. Factors like qubit coherence, error rates, and quantum volume (a measure of quantum computational power that considers the number of qubits, connectivity, and error rates) are crucial.

Current quantum computers are generally considered to be in the Noisy Intermediate-Scale Quantum (NISQ) stage of development, meaning they still face significant challenges such as qubit isolation and noise management, which affect their reliability and computational capabilities.

## Conclusion

As continuous progress in the development of quantum computing technology can be observed, with leading companies such as IBM, Google, and Amazon, who are expanding the boundaries of qubit capabilities, the quantum supremacy is moving closer.

The onset of quantum computing presents notable difficulties for the field of cryptography. Symmetric algorithms like AES require careful key management. Asymmetric systems, on the other hand, are more vulnerable, with algorithms like Shor's posing a real threat to their security. This has led to the research into post-quantum cryptography (PQC), aiming to develop cryptographic methods resilient agaist quantum attacks. The NIST PQC Standardization Process has been evaluating and selecting algorithms that can withstand the quantum thread.

However, the journey towards secure quantum-resistand cryptography is ongoing, as part of a continuous process in ensuring the security of our digital world in tha face of advancing technologies.

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## Further Reading

- NIST: Post-Quantum Cryptography
- The Quantum Insider, Matt Swayne: NIST Releases Four PQC Algorithms For Standardization
- Peter W. Shor: Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
- Communications of ACM: Post-Quantum Cryptography
- Matthias Homeister: Quantum Computing verstehen
- Nature, Forty thousand kilometers under quantum protection
- phys.org: New technology developed for quantum cryptography applications
- MIT Technology Review: IBM wants to build a 100,000-qubit quantum computer
- Verdict: The state of quantum computing in 2023
- Security Week: AI Helps Crack NIST-Recommended Post-Quantum Encryption Algorithm

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