Author: Mohamed Aziz Chebil.

## Introduction

Cryptography is the art of using mathematical algorithms and the algebraic structure in various protocols to secure information against potential adversaries who might gain unauthentic access to the confidential information, were it not for the mathematical assurances of the cryptographic systems.

Perhaps it is important to lay the emphasis on the fact that the core intuition of cryptography is not to incorporate infinite hardness in the way the information is encrypted, since that might lead to it never being decryptable even by the intended party, but rather make the decryption “hard enough” for any adversary to be able to break into and unlock with ease, and only the intended recipient be able to do so (sort of like a “trapdoor” only for the receiver and an eternity for any other third-party adversary, as the computers (classical computers) may as well take indefinitely forever to break the encryption!).

Trapdoor permutations are the mathematical assumptions that are:

- Invertible
- Efficient to Compute
- Difficult to Invert
- Easy to Invert Given Some Auxiliary Information

These trapdoor permutations, which are one of the ways to build PKE in a complexity-theoretic approach, are obtained from the so-called “RSA Assumption.”

The two most common classical cryptographic techniques include:

: The sender and the receiver parties share a common secret key that is used for both encryption and decryption.__Symmetric-Key Encryption (SKE)__: A pair of keys is used, one – called the public key – for encryption by the sender, and the other – called the private key – for decryption by the receiver.__Asymmetric/Public-Key Encryption (PKE)__

** NOTE**: The terms “private key” and “secret key” are used interchangeably very often.

## Classical Cryptography

Classical cryptography’s encryption methods are based purely on the complexity of the mathematics involved; more specifically, the computational difficulty of factoring large numbers is where the security of encryption in classical cryptography lies. Quantum cryptography on the other hand, is a more recently developed method of encryption such that two parties can establish for themselves a secure communication entirely reliant on the immutable laws of quantum mechanics, whose properties are utilized in order to transmit data through theoretically unhackable means. Both forms of cryptography have their own approaches and solutions to the key exchange problem (which says that, to build a secure communication channel where no one else may access a copy of the key(s)/data, it is necessary to share any keys or other information).

One of the arguably strongest classical cryptographic algorithms is RSA. It entirely revolves around logarithmic functions to stay computationally convoluted enough to resist brute-force attacks and yet remain streamlined enough to be swift post-deployment. For the encryption and decryption of general data, RSA switches the order in which the key sets are used; to encrypt and decrypt the data, the recipient’s public and private keys are used respectively, thereby eliminating any need for key exchange in scenarios where RSA is employed.

#### RSA Encryption Algorithm:

One of the famous and most widely used Public-Key Encryption (PKE) algorithms is the RSA Encryption, invented in 1977 by Ron ** R**ivest, Adi

**hamir, and Leonard**

__S__**dleman. This algorithm uses a pair of keys, a public key that can be freely distributed among everybody, and a private key that is kept secret by a owner, and it promises security based on the difficulty of factoring large numbers into their prime factors.**

__A__The algorithm works as follows:

: This first step of generating a pair of keys involves selecting two large prime numbers, p and q, and computing their product, n = p x q. The totient of n, given by the Euler’s totient function φ(n), is computed as φ(n) = (p-1) (q-1). The public key is the pair (e, n), e being a small, odd integer that is relatively prime to φ(n). The private key is the pair (d, n), d being the multiplicative inverse of e modulo φ(n).__Key Generation__: To encrypt a message m, the sender uses the recipient’s public key (e, n) to compute the ciphertext c = m__Encryption__^{e }mod n.: To decrypt a ciphertext c, the recipient uses their private key (d, n) to compute the plaintext m = c__Decryption__^{d }mod n, recovering the original message.

Besides RSA, there are also many other strong cryptographic techniques/protocols such as the Advanced Encryption Standard (AES), widely used in online banking, file encryption, and VPNs; Elliptic Curve Cryptography (EEC), a PKE algorithm that uses elliptic curves to generate keys, widely used in mobile devices and embedded systems; the Secure Hash Algorithm (SHA), a family of hash functions that are used for digital signatures and message authentication codes; and many more.

## Quantum Cryptography

Quantum cryptography, on the other hand, uses the principles of quantum mechanics to secure the quantum information and communication. The key idea behind this quantum-mechanical analogue of classical cryptography is that it is impossible to measure a quantum state without disturbing it. What this implies is that if an eavesdropper tries to intercept a message, the act of measuring the state (or interacting with the quantum information) will alter it, thereby revealing the presence of the intruder.

As a result, quantum cryptography provides an unbreakable form of communication that is resistant to hacking!

** Advantages of quantum cryptography over its classical counterpart**:

: The so-called “factoring assumption” in the classical cryptography, which is based on the assumption that it is insanely difficult to factor large numbers, is broken by quantum computers (with the advent of quantum algorithms such as the Shor’s Algorithm), which have the potential to solve such mathematically hard problems much faster than classical computers.*Security*: Any attempt to intercept a message will not go undetected, in contrast to the classical cryptography where an eavesdropper can intercept a message without being detected.*Ability to detect eavesdroppers*

__Drawbacks of Quantum Cryptography:__

*Complexity and Cost**:*Quantum Cryptography requires specialized equipment, such as single-photon detectors and Quantum Key Distribution systems, which are expensive and difficult to implement on a large scale.

Despite the drawbacks, this is still a budding area of research with many ongoing efforts to develop more efficient and practical quantum encryption methods.

** Quantum Key Distribution (QKD)**:

One of the most powerful and commonly used quantum protocols is the Quantum Key Distribution (QKD), which enables two parties to share a secret key over a quantum channel, such as a fiber-optic cable or free space. QKD is based on the principle of quantum entanglement, which ensures that any attempt to intercept the key will disturb the quantum state, alerting the parties to the presence of an eavesdropper. QKD is a fundamental building block of many quantum-cryptographic systems.

Two important quantum cryptographic protocols that are used for QKD are BB84 and E91.

: It is one of the earliest and most widely used QKD protocols, introduced by Charles Bennett and Giles Brassard in 1984. The protocol uses the properties of quantum mechanics, such as the polarization of photon, to establish a shared secret key between two parties. BB84 runs the following way:__BB84 Protocol__

: Alice generates a string of random bits (the key) and encodes them onto individual photons, which are sent to Bob. She chooses randomly between four bases (two orthogonal bases: vertical/horizontal polarization, and two diagonal bases: +45° and -45° polarization) to encode each bit.__Key Generation__: Alice sends the photons to Bob through a quantum channel. This channel can be a fiber-optic cable or free space (vacuum).__Transmission__Bob receives the photons and measures them using one of the four bases randomly. He records the results but does not tell Alice what bases he used to measure.__Measurement:__Alice and Bob publicly compare the bases they used for encoding and measuring. They discard any bits for which they used different bases.__Public Discussion:__Alice and Bob use classical communication to identify and correct errors in the remaining bits. They discard any bits that could not be corrected.__Error Correction:__Alice and Bob use a hashing function to extract a shorter key from the remaining bits. This key is guaranteed to be secret and known to only Alice and Bob (thus ensuring the security of the key exchange).__Privacy Amplification:__

__ __

: Also called the Ekert Protocol, it is also a QKD protocol, like BB84, that uses the properties of entangled particles to establish a shared secret key between two parties. It was introduced by the British-Polish professor of quantum physics at the Mathematical Institute, University of Oxford, Artur Ekert in 1991.__E91 Protocol__

** NOTE**: This is the Ekert’s paper on “Quantum Cryptography based on Bell’s Theorem” published in 1991

This protocol is based on the principle that measuring the state of one particle in an entangled pair will *instantaneously *affect the state of the other particle regardless of the distance between them, which ensures that any attempt to intercept the key will be detected instantaneously. The E91 protocol runs as follows:

: Alice generates a pair of entangled photons and sends one to Bob.__Entanglement Generation__: Alice randomly chooses one of three bases (two orthogonal bases: vertical/horizontal polarization, and one diagonal basis: diagonal polarization), and Bob chooses one of two basses (vertical/horizontal polarization) to measure his photon.__Measurement Choice__: Alice and Bob communicate their measurement choices through a public channel.__Transmission__: Bob measures his photon and records the result. Alice measures her photon in the same basis as Bob or a complementary basis, depending on their measurement choices.__Measurement__: Alice and Bob publicly compare their measurement choices for the photons they shared. If their bases are the same, they keep the result, and if they are different, they discard the result.__Public Discussion__: Alice and Bob randomly select a subset of the results they kept and compare them to estimate error rate due to noise or eavesdropping.__Error Estimation__: Alice and Bob use classical communication to perform a process called “sifting” to extract a smaller subset of bits that are known to be secure. They then use a hashing function to extract a shorter key from this subset.__Privacy Amplification__

## Post-Quantum Cryptography

Post quantum cryptography aims to build strong encryption methods that algorithms cannot break, and which obstructs undetected eavesdropping. Theoretically, it is impossible to hack quantum cryptography, but its practical uses are limited. Existing protocols might have to be modified to handle larger signatures or key sizes. Current challenges are mostly surrounded by transmission rates and application limitations. These are hard challenges to overcome as there is a need for transmittable quantum states in long distances, sustainable communication networks, and understanding the utilization of emerging technologies and public adaptation.

Up until now, many of the challenges in secure quantum information have been solved in regard to post quantum cryptography. Within its cost and complexity, quantum migration takes time. It’s crucial to begin planning the next steps of this replacement for a safe actualization.

Post-quantum cryptography is, more specifically, a type of cryptography that is designed to be resistant to attacks by quantum computers. This is important because quantum computers have the potential to break many of the cryptographic algorithms that are currently in use by classical computers.

One of the main issues associated with quantum computing is that it can easily factor large numbers, which is the basis of many cryptographic algorithms. Post-quantum cryptography uses mathematical problems that are believed to be resistant to quantum computers, such as lattice-based cryptography and code-based cryptography.

Photonic quantum technologies have shown great promise through their high-speed transmission and low-noise specialized photons. Photon-based quantum technologies can be explained as providing secure communication by encoding a variety of the BB84 protocol discussed earlier. The use of single-photon sources is important in these protocols, as it enables the creation of a secure key and is caused by the unavailability of single photon sources since today’s cryptography systems are relying on photons from laser pulses. Another alternative might be quantum cryptography based on entangled photon pairs with low power consumptions and small device footprints. Optical communications could have a visible effect on reducing power consumption.

There is no direct relation between quantum-proof encryption and distance limitation. Quantum-proof encryption (also known as post-quantum cryptography) is designed to be resistant to attacks by quantum computers, whereas distance limitation refers to the maximum distance over which a quantum communication channel can reliably transmit quantum information. However, it is worth noting that some post-quantum cryptography schemes may require a quantum communication channel to securely distribute the encryption keys, in which case, the distance limitation of the quantum communication channel may become a limiting facet.

## Conclusion

In conclusion, quantum cryptography offers a new and exciting paradigm for secure communication that has the potential to revolutionize the way we think about cryptography. Unlike classical cryptography, quantum cryptography uses the principles of quantum mechanics to provide a level of security that is theoretically unbreakable. While there are still some challenges to be overcome, such as scalability and cost, recent advances in quantum technologies have brought us closer than ever to realizing the full potential of quantum cryptography.

A particularly promising application of quantum cryptography is of course the use of quantum key distribution (QKD) to encrypt VPNs specifically. This technology offers a new level of security that is resistant to attacks from even the most powerful adversaries, making it an attractive option for organizations that require the highest level of security for their communications (such as with the military and government; for communications, they can use quantum cryptography to securely transmit sensitive info without worrying about interception from third parties; in the medical and healthcare field as well, this can be used to send patient records, healthcare data, and medical research). With the ongoing development of quantum technologies, it likely that we will see even more powerful and efficient applications of quantum cryptography soon.

Overall, while there are still many challenges to be overcome, quantum cryptography represents an exciting new frontier in the field of cryptography and has the potential to transform the way we communicate and secure our information.

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