# Intelligent Bio-Latticed Cryptography: A Quantum-Proof Efficient Proposal

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## Abstract

**:**

## 1. Introduction

- Normalising data by Gaussian normal distribution. Gaussian distribution is a normal unique probability distribution that is preferred for describing random systems because numerous random processes that occur in nature act as normal distributions. The central limit theorem declares that for any distribution, the sum of random variables tends to fall within normal distribution under moderate circumstances. Thus, normal distribution has flexible mathematical attributes [7];
- Calculating a covariance matrix;
- Deriving eigenvalue and eigenvectors based on the covariance matrix;
- Picking eigenvector dimension k;
- Computing the Karhunen–Loève transformation into k.

- While humanity is benefitted by the information technology revolution, which derives its power from information to transfer knowledge and share resources, many exploit this revolution for malicious purposes such as attacking, eavesdropping, fraud, etc. According to PwC’s survey 2022 [17], cybercrimes have topped the list of external fraud threats faced by businesses worldwide. The survey included 1296 chief executive officers belonging to 53 countries in the world, and nearly half of these corporations ($46\%$) admitted that they had been subjected to cyber-attack, fraud, or financial crimes due to the high rate of cybercrimes and fraud around the world since the emergence of COVID-19. Moreover, Ref. [17] stated that cyber-attacks pose more risk to organisations than before, as fraud and cyber-attacks have become more sophisticated. One in five global businesses, whose revenues exceed $10 billion, have been exposed to a fraud case that cost more than $50 million. More than a third of corporations ($38\%$) with revenues of less than USD 100 million reported having experienced some form of cybercrime and $22\%$ of these corporations were affected by more than USD 1 million. Consequently, to counter this dangerous phenomenon, we propose intelligent performance-efficient lattice-driven cryptography using biometrics.
- The main benefit of combining lattice theory and biometrics is that doing so eliminates the need to save or send biometric templates, private keys, or any secret information, which solves some public key infrastructure problems, such as public keys distribution challenges and key expiration issues, thereby preserving privacy, improving cybersecurity in a post-quantum era, and minimising the risk of information leakage online or offline.
- The proposed cryptography resists quantum attacks such as Shor’s quantum algorithm. At the same time, it inherits neither the shortcomings of the quantum computer, such the large gap between the implementation of real devices and physical quantum theory, nor the defects of quantum cryptography, such as a vulnerability to side-channel attacks, source flaws, laser damage, Trojan horses, injection-locking lasers, and timing attacks. Since the first quantum cryptosystems—represented by quantum key distribution systems—were made available, many adversaries have attempted to hack them with unsettling success. Fierce attacks have focused on exploiting flaws in the equipment used to transmit quantum information. Consequently, adversaries have demonstrated that the equipment is not perfect, even though the laws of quantum physics imply perfect security and privacy. Furthermore, one of the most significant drawbacks of quantum computing and quantum cryptography is the limited distance that must be considered for transmitting photons, which often should not exceed tens of kilometres. This is due to the probability that the polarisation of photons may change or even disappear completely as a result of consecutive collisions with other particles while travelling long distances. However, this problem can be solved by adding spaced quantum repeaters at uniform intervals that amplify optical signals and maintain quantum randomness for thousands of kilometres.
- Enhancement of cybersecurity allows the private keys created from biometric encryption to be stronger, more complex, and less vulnerable to cybersecurity attacks. Traditional/classical biometric systems are susceptible to various attacks, such as manipulations, impersonation attacks, stolen verifier attacks, device compromise attacks, replay attacks, denial-of-service (DoS) attacks, distributed denial-of-service (DDoS) attacks, integrity threats, privacy threats, confidentiality concerns, and insider attacks. The use of the proposed advanced algorithm eliminates these vulnerabilities. It also enhances accuracy and performance by using artificial intelligence (AI), such as machine learning (artificial neural networks) and genetic algorithms.

## 2. Biometrics

- An enrolment module acquires the data related to biometrics;
- A feature-extraction module extracts the required set of characteristics from the collected biometric data;
- A matching module compares the extracted features with the features in existing data;
- A decision-making module checks whether the identity of the user exists and whether it is accepted or rejected.

- Universality: each person must have it;
- Uniqueness: there should be sufficient and significant differences between the characteristics of any two persons;
- Longevity: it must be adequately invariant over a certain period.

## 3. Merits of Combining Biometrics and Asymmetric Encryption

#### 3.1. Management of Public and Private Keys

#### 3.2. No Storage of Biometric Data

#### 3.3. Cancellation and Revocation in Biometric Systems

#### 3.4. Security against Known Vulnerabilities in Biometric Systems

#### 3.5. Security and Privacy of Personal Data

#### 3.6. Public Acceptance Based on Embedded Privacy and Security

#### 3.7. Making Biometric Systems Scalable

## 4. Lightweight Intelligent Bio-Latticed Cryptography

^{n}for any integer n and prime p.

- Key generation:$${S}_{1}=shif{t}_{\alpha}\left(shuffl{e}_{H}(h\left(corrected\phantom{\rule{3.33333pt}{0ex}}facial\phantom{\rule{3.33333pt}{0ex}}features\right)\oplus \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathsf{{\rm Y}})\right)$$
_{H}is a Henon shuffling map.Because biometric data is naturally changeable, while the symmetrical cryptography process requires exact data to operate properly, the representation of biometrics must be corrected symmetrically before it can be employed. To stabilise the biometric matrix, error-correction principles are symmetrically applied [30,40,41]. Further detailed information concerning the principles of the process of data correction can be found in [30,40,41].$${S}_{2}=\mathsf{\Lambda}\oplus \mathsf{\Gamma}\oplus {S}_{1}$$Secret key (private key): (S_{1}, S_{2})$$PK={S}_{1}^{-1}\xb7{S}_{2}^{T}$$Public key: (β, PK) - Encryption:$$En{c}_{o}=Shuffle\left[Shif{t}_{\beta}[Msg\left(mod\beta \right)\oplus \beta ]\right]$$$$En{c}_{1}=En{c}_{o}\xb7PK$$$$Enc={T}_{s}\parallel En{c}_{1}$$
_{s}is the current time of the transmitter’s device (the sender picks up timestamp T_{s}) and ∥ denotes a concatenation operation. - Decryption:$${M}_{o}=[\left[(Enc\oplus \beta )mod\beta \right]\xb7{S}_{1}]\xb7[{\left({S}_{2}^{T}{S}_{2}\right)}^{-1}{S}_{2}^{T}]$$$$Msg={Shift}_{\beta}^{-1}\left({Shuffle}^{-1}\left({M}_{o}\right)\right)$$When the receiver obtains the encrypted message at time T
_{r}, this message is decrypted via the secret key ($S1,{S}_{2}$) and then tested for the timestamp freshness T_{s}. If (T_{r}− T_{s}) > $\Delta $T, the receiver will reject this message, since it is expired; i.e., it is significantly vulnerable to reply attack, where $\Delta $T indicates the expected time interval for the communication delay in the wireless networks. Conversely, if (T_{r}− T_{s}) ≤ $\Delta $T, the receiver will accept this message.

**Figure 3.**Square lattice (right-isosceles-triangular) [42].

## 5. Implementation-Based Evaluation Results

- An 80 km
^{2}classical attocell network topology in an urban macrocell scenario such as a smart city; - Five cells; each cell (${R}_{cell}$) is 2 km;
- 24 Base Stations (BSs). ${R}_{cell}$ is a hexagon bordered by BSs placed at the vertices, with another BS placed at the centre;
- The number of devices is 250, and these are positioned randomly in 2-dimensional space;
- In traditional cellular communications, all the devices transmit uplink/downlink requests at the same time to the BS. In other words, all transmissions between cellular devices must be accomplished through the BS without considering if Device-to-Device (D2D) communication is within two cellular devices’ reach;
- The initial energy for a cell device ($Eo$) is 5000–10,000 Joules.

## 6. Security Proof of the Proposed Lattice-Driven Encryption Scheme

**Lemma**

**1.**

**Proof.**

_{1}, v

_{2}, v

_{3}, …, v

_{n}in the Euclidean norm. In the closest vector problem (CVP), the goal is to find the point closest to the given vector. For example, if v

_{1}, v

_{2}, v

_{3}, …, v

_{n}were the given basis vectors and v was a given target vector, our objective would be to determine the closest lattice point (non-zero vector) within the Euclidean norm closest to the given target vector v.

**Lemma**

**2.**

**Proof.**

_{s}and T

_{r}are used in the proposed lattice-driven cryptosystem to abort any attempted replay attacks. If an attacker exploits the communication to replay the old packet, the attacker cannot succeed in the proof process where (T

_{r}− T

_{s}) > $\Delta $T; thus, this packet is aborted because it is expired. $\Delta $T denotes the expected time interval for the communication delay in the wireless network, T

_{s}refers to the current time of the sender’s node, and T

_{r}refers to the message reception time at the receiver’s device.

**Lemma**

**3.**

**Biometric templates blended with encryption keys can improve online cybersecurity while simultaneously preserving users’ privacy.****Proof.**In the lightweight intelligent bio-latticed cryptosystem, the one-way hash function of a corrected facial image is XORed with $\mathsf{{\rm Y}}$ matrix $\in {\mathbb{F}}_{{P}^{n}}^{i\times j}$2-dimension good Galois polynomial entropy to regenerate the private key (S_{1}, S_{2}) on-the-fly without transmitting or holding any secret data that may be compromised and breach the privacy of users. In biometric encryption, there is no requirement to memorise either facial images or templates of these images, which adresses the classic flaw of biometric methods. Additionally, an adversary cannot retrieve an encryption parameter $\mathsf{{\rm Y}}$, a facial image, or a facial template after binding them and then discarding them. This leads to the dramatic enhancement of security features, such as generating a robust biometric-based lattice private key (S_{1}, S_{2}) on-the-fly and at the same time maintaining the performance of restricted-resource devices, since this private key has more security than typical passwords and less storage space than biometric facial data. Additionally, the key generating process is characterised by its low computational requirements and lightweight operations.**Lightweight intelligent bio-latticed cryptography and Galois field ${\mathbb{F}}_{{P}^{n}}^{i\times j}$.****Proof.**A finite field, which is commonly referred to as a Galois field, is a set of numbers to perform mathematical operations such as addition, multiplication, subtraction, and division that always produces a result contained within the same set of numbers. Cryptography benefits from this, since a restricted set of very large numbers can be used [68]. The proposed bio-lattice cryptography uses Galois field theory, which has many applications in cryptography. Some of the main reasons for this are that it is possible for arithmetic operations to scramble data quickly and efficaciously when the data is represented as a vector in a Galois Field, and subtraction and multiplication in a Galois Field need extra operations/steps, unlike in Euclidean space [69].In the proposed bio-lattice cryptography, ${\mathbb{F}}_{{2}^{63}}^{i\times j}$ is used, since manipulating the bytes is required. ${\mathbb{F}}_{{2}^{63}}^{i\times j}$ has an array of elements that together represent all of the various potential values that may be assigned to a byte. Because the Galois field’s addition and multiplication operations are closed, it is easy to perform arithmetic operations on any two bytes to yield a new byte belonging to the array of that field, making it ideal for manipulating bytes [70]. Furthermore, multiplications in ${\mathbb{F}}_{{2}^{63}}$ can be optimised securely for applications in cryptography when the ${P}^{n}$ is smaller than the bits of the device (i.e., ${P}^{n}$ < 64, on standard desktops or smartphones) [71].The National Institute of Standards and Technology (NIST) has issued a request for standardization of Post-Quantum Cryptography (PQC) [72] because of the growing awareness of the need for PQC in light of the impending arrival of quantum computing. According to Danger et al. [71], code-based encryption, along with multivariate and lattice-based cryptosystems, is one of the primary competitors for this challenge, because of its inherent resistance to quantum cyberattacks. Despite being nearly as age-old as RSA and Diffie–Hellman, the original McEliece cryptography has never been widely employed, mostly because of its large key sizes [71]. There are numerous cryptosystems defined on ${\mathbb{F}}_{{2}^{N}}$ that were recognised as candidates for the first round of the NIST PQC competition [71].The carry-less feature of addition in ${\mathbb{F}}_{{2}^{N}}$ makes arithmetic operations in this setting notably desirable. Consequently, many cryptographic approaches use it, since it provides efficiency in both hardware and software implementations because there is no carry and, thus, there are no lengthy delays [71]. In addition, Danger et al. [71] present a case study in which they assess several implementations of ${\mathbb{F}}_{{2}^{N}}$ multiplication with regard to both their level of safety and how well they perform. They claim in their conclusion that their findings are applicable to accelerate and secure implementations of the other PQC outlined in their research, in addition to symmetric cyphers such as AES that operate on finite fields ${\mathbb{F}}_{{2}^{N}}$.Moreover, in the implementation of any cryptographic application, the size of the employed finite fields and the conditions imposed on the field parameters are determined by security concerns [73]. Therefore, the proposed intelligent bio-latticed cryptography devotes a square lattice (right-isosceles-triangular) $\mathcal{L}$ over Galois field ${\mathbb{F}}_{{P}^{n}}^{i\times j}$ such that $\mathcal{L}$⊂${\mathbb{F}}_{{P}^{n}}^{i\times j}$ good prime unique Galois polynomial entropies with dimensions i and j and order p^{n}for any integer n and prime p.**Entropic randomness, shifting, shuffling, XOR, and proposed lattice-driven cryptosystem.****Proof.**Random entropies are essential for assuring the security of sensitive information stored electronically [74,75]. Furthermore, a MATLAB-based shuffling package was developed to enhance a cryptosystem in [76]. This research paper included suggestions to enhance cryptography and make it invulnerable to data leaks using random shuffling. Hence, in the proposed cryptosystem, entropic randomness distribution, shifting, shuffling, and XORing are all used to make it difficult for an adversary without the appropriate private key to extrapolate anything valuable about the message (plaintext) from the encrypted message (corresponding ciphertext), strengthening proposed cryptosystem’s ability to resist data leaks and preserve privacy and making it more secure.Reyzin summarised essential entropy concepts used to study cryptographic architectures in [77], since the capability of assigning a random variable’s value in a single try is often used as a significant metric of its quality, especially in applications related to cybersecurity. Moreover, he defined this capability as follows:A random variable A has min-entropy b, indicated by ${H}_{\infty}\left(A\right)=b$, if ${max}_{a}Pr[A=a]={2}^{-b}$.Extractors of randomness have been expressed in terms of their compatibility with any distribution having a min-entropy [77,78]. Furthermore, the outputs from robust extractors are almost uniform, regardless of the seed, and tend to maximal min-entropy, as these extractors are able to generate outputs with a high possibility over the seed selection [77].Similar to cryptography literature employing shuffling to prevent information leaks from encoded correspondences [76], Henon shuffling maps are used in the proposed lattice-driven cryptosystem. Using a random shuffling package, the researcher in [76] improved the security and efficacy of the Goldreich–Goldwasser–Halevi (GGH) public-key scheme. She proposed enhanced functions of GGH encryption and decryption principally relying on MATLAB- based shuffling to prevent sensitive information from leaking in images. In [32], public-key cryptography established on the closest vector problem was presented by Goldreich, Goldwasser, and Halevi to be an NP-hard lattice problem at the Crypto ’97 conference. Unfortunately, later at the Crypto ’99 conference, in [79], Phong Nguyen analysed the GGH cryptography and demonstrated that there are serious shortcomings including: any encrypted message can leak sensitive data concerning the plain message, and the difficulty of decryption can be reduced to a particular closest vector problem, which will significantly be easier than the general problem.

#### Reducing a Vector Module to a Lattice-Based Problem

Algorithm 1: Reduction of the proposed lattice-based scheme |

INPUT: Basis $H\in {\mathbb{F}}_{{P}^{n}}^{i\times j}$ of Hermite Normal Form, $v\in {\mathbb{F}}_{{P}^{n}}^{i\times j}$ OUTPUT: $b\in {\mathbb{F}}_{{P}^{n}}^{i\times j}$ Start $b\leftarrow v$ for $y\leftarrow j$ to 1 do for $x\leftarrow i$ to 1 do ${c}_{x,y}\leftarrow {b}_{x,y}\xf7{h}_{x,y}$ $b\leftarrow b-{c}_{x,y}\times {b}_{x,y}$ end end return b End |

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PKI | Public Key Infrastructure |

6G | Sixth-generation Networks |

NB-IoT | Narrowband-Internet of Things |

IoT | Internet of Things |

5G PPP | European Infrastructure Public Private Partnership |

LEACH | Low Energy Adaptive Clustering Hierarchy |

BS | Base Station |

Msg | Message |

AI | Artificial Intelligence |

DoS | Denial-of-Service attack |

DDoS | Distributed Denial-of-Service attack |

PCA | Principal Component Analysis |

iIoT | Industrial Internet of Things |

NP | Nondeterministic Polynomial-type |

NP-hardness | Nondeterministic Polynomial-time hardness |

SVP | Shortest Vector Problem |

CVP | Closest Vector Problem |

RP | Random Polynomial time |

LLL | Lenstra–Lenstra–Lovász |

BKZ | Block Korkine–Zolotarev |

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**Figure 4.**Secure heterogeneous cellular network in traditional mode with quantum-resistant intelligent bio-latticed cryptography.

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**MDPI and ACS Style**

Althobaiti, O.S.; Mahmoodi, T.; Dohler, M.
Intelligent Bio-Latticed Cryptography: A Quantum-Proof Efficient Proposal. *Symmetry* **2022**, *14*, 2351.
https://doi.org/10.3390/sym14112351

**AMA Style**

Althobaiti OS, Mahmoodi T, Dohler M.
Intelligent Bio-Latticed Cryptography: A Quantum-Proof Efficient Proposal. *Symmetry*. 2022; 14(11):2351.
https://doi.org/10.3390/sym14112351

**Chicago/Turabian Style**

Althobaiti, Ohood Saud, Toktam Mahmoodi, and Mischa Dohler.
2022. "Intelligent Bio-Latticed Cryptography: A Quantum-Proof Efficient Proposal" *Symmetry* 14, no. 11: 2351.
https://doi.org/10.3390/sym14112351