Elliptic Curve Research Articles

SECURE and energy-efficient routing protocol based on micro-segmentation and batch authentication

An effective technique in designing routing algorithms for Wireless Sensor Networks (WSNs) is named clustering nodes, which augments the network lifespan. More tasks are performed by the Cluster Heads (CHs) of the clustered WSNs, thereby consuming more energy. Hence, a secure energy-efficient Routing Protocol (RP) centered on micro-segmentation and batch authentication is proposed here. The proposed method begins by initializing the WSN nodes. After that, a random key is engendered for every node centered on their Node ID. The nodes undergo a Micro-segmentation process; in addition, they are securely grouped utilizing the Supremum Distance based K-Prototype (SD-KP) algorithm. Now, a fog layer with a number of fog nodes is initialized; also, every single micro-segmented group is assigned with an optimal fog node. Then, by using the Quasi Deterministic Sequence-Black Widow Optimization (QDS-BWO) algorithm, the optimal fog nodes are selected. By using the Root Squared- Diffie Hellman (R2-DH) technique, a key agreement is created for each group centered on their Node IDs. By using the Bitwise Cyclic Shift –BLAKE 512 (BCS-BLAKE-512) algorithm, the created keys are converted into hashcode. Now, for the purpose of performing batch authentication, the hashcode of each group is sent to the trusted authority. Subsequent to verification, data sensing occurs. Lastly, by using the Elliptic Curve Cryptography (ECC) algorithm, the data is securely encrypted and it is transmitted via the routes selected utilizing the Learned Gradient Weight Initialization-based Geography and Energy Aware Routing (LGWI-GEAR) algorithm. The proposed framework's efficacy is proved by the experimental outcomes.

Constructing abelian varieties from rank 2 Galois representations

Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let ${\mathbb {L}}$ be a rank $2$, geometrically irreducible lisse $\overline {{\mathbb {Q}}}_\ell$-sheaf on $U$ with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field $E\subset \overline {\mathbb {Q}}_{\ell }$, and has bad, infinite reduction at some closed point $x$ of $X\setminus U$. We show that ${\mathbb {L}}$ occurs as a summand of the cohomology of a family of abelian varieties over $U$. The argument follows the structure of the proof of a recent theorem of Snowden and Tsimerman, who show that when $E=\mathbb {Q}$, then ${\mathbb {L}}$ is isomorphic to the cohomology of an elliptic curve $E_U\rightarrow U$.

An image encryption method based on modified elliptic curve Diffie-Hellman key exchange protocol and Hill Cipher

Abstract Digital image protection is crucial since images often contain private and sensitive information in business, medical, and military. One of the best techniques for securing the content of these images is encryption. This article introduces a cryptosystem known as the elliptic curve Diffie-Hellman Hill Cipher (ECDHHC) that uses the modified eliptic curve Diffie-Hellman (ECDH) key exchange protocol to generate the shared secret key integrated with the Hill Cipher. An elliptic curve point-based secret shared key matrix using the ECDHHC, which will be used for encryption and decryption, is generated. Thereafter, the input image is split into a set of 8 × 8 submatrices and then changes the values of these matrices by multiplying each block with the secret shared key matrix. The encrypted image is constructed by merging all encrypted blocks. With this combination, the correlation between adjacent pixels in the ciphered image is effectively removed, and the level of unpredictability and uncertainty for the ciphered image is also enhanced. The suggested approach used the key space, entropy, histogram, antinoise attack, differential attack, and correlation coefficient to evaluate the performance of the encryption method. According to simulation findings, the proposed method offers a high level of security and efficiency, and resists attackers.

Comparison of focusing property and radiation force between autofocusing Bessel beams and focused Gaussian beams.

As abruptly autofocusing beams, autofocusing Bessel beams (ABBs) have been proven to be a class solution for the Helmholtz equation [Opt. Express31, 33228 (2023)10.1364/OE.500383]. In this paper, we use the Fresnel number as the basic parameter and accurately compare the focusing property and radiation force of ABBs versus focused Gaussian beams (FGBs) under the same Fresnel number. Unlike FGBs, ABBs can achieve autofocusing without the need for an initial focusing phase. Our analysis of the beam width defined by power in the bucket, revealed that FGBs exhibit uniform focusing along the straight line, whereas ABBs demonstrate accelerated focusing along the elliptic curve. At the same Fresnel number, FGBs exhibit a higher peak intensity in the focal plane, yet ABBs excel in gradient force on particles. In comparison to FGBs, ABBs exhibit smaller potential well widths, allowing for stable and precise trapping of high refractive index particles at the focal point. While FGBs are considered suitable for laser processing and ablation due to their high peak power density, ABBs possess significant advantages in optical manipulation due to their great gradient force. Furthermore, we conduct a comparative analysis between ABBs and circular Airy beams (CABs). The peak intensity and gradient force exhibited by CABs are slightly lesser than those of ABBs. CABs are appropriate for multi-point trapping along the axis, whereas ABBs are more suited for precise single-point trapping.

Smoothed particle hydrodynamics implementation of the standard viscous–plastic sea-ice model and validation in simple idealized experiments

Abstract. The viscous–plastic (VP) rheology with an elliptical yield curve and normal flow rule is implemented in a Lagrangian modelling framework using the smoothed particle hydrodynamics (SPH) meshfree method. Results show, from a perturbation analysis of SPH sea-ice dynamic equations, that the classical SPH particle density formulation expressed as a function of sea-ice concentration and mean ice thickness leads to incorrect plastic wave speed. We propose a new formulation for particle density that gives a plastic wave speed in line with theory. In all cases, the plastic wave in the SPH framework is dispersive and depends on the smoothing length (i.e., the spatial resolution) and on the SPH kernel employed in contrast to its finite-difference method (FDM) implementation counterpart. The steady-state solution for the simple 1D ridging experiment is in agreement with the analytical solution within an error of 1 %. SPH is also able to simulate a stable upstream ice arch in an idealized domain representing the Nares Strait in a low-wind regime (5.3 m s−1) with an ellipse aspect ratio of 2, an average thickness of 1 m and free-slip boundary conditions in opposition to the FDM implementation that requires higher shear strength to simulate it. In higher-wind regimes (7.5 m s−1) no stable ice arches are simulated – unless the thickness is increased – and the ice arch formation showed no dependence on the size of particles, in contrast to what is observed in the discrete-element framework. Finally, the SPH framework is explicit, can take full advantage of parallel processing capabilities and shows potential for pan-Arctic climate simulations.

Secure heart disease prediction model using ESVO-based Swish Bessel CNN classifier

Heart disease is a critical issue that affects people, causes serious sickness, and is the main cause of mortality worldwide. Early diagnosis of disease plays a significant role in heart disease prediction and is attained by various automation techniques. The availability of automation techniques initiates the necessity for medical data and the storage of medical data becomes a research problem due to its high sensitivity. The emergence of IoT networks formed a promising solution for data storage through the cloud server and preventing the data from various threats is a challenging problem. A secure heart disease prediction system is developed by the utility of the ESVO-based Swish Bessel CNN classifier (Emperor Spheniscidae Vampire Optimization-based Swish Bessel Convolutional Neural Network), and the important significance of the research depends on the ESVO optimization that helps in gaining a deeper insight of the classifier as well as helps in preventing the threatening of data. The security of the cloud server is enhanced by the EDH-ECC (Entropy Diffie Hellman – Elliptic Curve Cryptography) which promotes the information exchange even in unsecured channels. Similarly, the authentication and authorization of the cloud server are carried out using the EAN-13 and salt-based digital signature that initiates strong credentials and enhance data security. Finally, the heart disease is diagnosed using the ESVO-based Swish Bessel CNN classifier. Assessing the accuracy, sensitivity, specificity, and F1-measure, which provided values of 94.877 %, 95.464 %, 93.293 %, and 95.14 % shows the effectiveness of the research.

A Provably Secure Lightweight Authentication Based on Elliptic Curve Signcryption for Vehicle-to-Vehicle Communication in VANETs

A Provably Secure Lightweight Authentication Based on Elliptic Curve Signcryption for Vehicle-to-Vehicle Communication in VANETs

Securing patient data in the healthcare industry: A blockchain-driven protocol with advanced encryption.

Ensuring the security and privacy of patient data is a critical concern in the healthcare industry. The growing utilization of electronic data transmission and storage in medical records has amplified apprehensions about data security. However, due to varying stakeholder interests, not all data can be freely shared, necessitating the development of secure protocols. This study presents a highly secure protocol that integrates blockchain technology, patient biometric information, and robust cryptographic algorithms (elliptic curve cryptography (ECC) and advanced encryption algorithm (AEC)) to facilitate data encryption and decryption. The protocol encompasses secure login, secure key sharing, and data sharing mechanisms among miners, offering comprehensive security measures. To validate the effectiveness of the proposed protocol, both informal and formal security analyses are conducted. The security protocol description language in Scyther is utilized to evaluate the protocol's resilience against attacks. The culmination of this research is a secure protocol that leverages blockchain technology and ECC for the secure storage and sharing of medical records. The protocol covers all stages, including system setup, user registration, login mechanisms, key exchange between users and blockchain, communication between blockchains, and interaction with other miners, with a steadfast emphasis on security. Furthermore, the protocol's communication and computation costs are assessed, with a comparison to existing blockchain-based schemes. Informal proofs establish the protocol's security against common attacks faced by medical institutions. Formal simulation of the protocol using the Scyther tool provides definitive evidence of its resistance to attacks. As a result, this protocol presents a viable real-time implementation solution for safeguarding patient data within the healthcare domain, representing a significant contribution to data security.

Occlusion Avoidance in Image-Based Control of Multiple Spacecraft

This study develops a novel image-based control scheme to address the on-orbit service mission that multiple spacecraft track a tumbling uncooperative target under a visibility constraint. The chaser spacecraft are commanded to hover above and track stably the feature points attached to the target as well as to avoid the mutual visual occlusion caused by interspacecraft motion. To this end, a new expression of image-based visual servoing dynamics for spacecraft is derived, where the physical meaning of each term is analyzed in detail so that the exact upper bounds of noncooperative terms can be estimated. After that, a novel occlusion description model is established by the projective theory of quadrics. The mathematical elements for describing the imaging elliptic curve and occlusion discrimination function are proposed, whose association with interspacecraft motion is given to guide occlusion avoidance controller design. Combining the artificial potential function with the backstepping control method, a robust control algorithm is designed. The proposed control scheme can achieve the six-degree-of-freedom control objective based on image space, and motion trajectory will not influence others’ visibility. Finally, the validity of the control scheme is verified by simulation for a given multispacecraft observation scenario.

MATHEMATICAL MODEL OF DIGITAL SIGNATURE BASED ON ECDSA AND SCHMIDT-SAMOA CRYPTOSYSTEM

Digital Signature Technology is replacing paper-based work for customers and employees in various industries and e-commercial environment. Digital Signature provides cryptographic services like authentication, non-repudiation, and integrity for the digital data. With the development of internet, Digital Signature becomes increasingly important for security because of its integrity and authenticity. It is an electronic signature that can be used to authenticate the identity of the sender. Digital Signature does not provide confidentiality until an encryption algorithm is applied. In this study, a new model of Digital Signature is introduced using the Elliptic Curve Digital Signature Algorithm (ECDSA)with an encryption technique Schmidt Samoa Cryptosystem. This model provides double layer security with encryption as well as signing protocol. Proposed model provides features like confidentiality, non-repudiation, and authenticity.

A Comprehensive Survey on Exploring Covert Timing Channels in Network Security

Covert timing channels play pivotal roles in spying, hacking, and data theft, enabling secret communication and surreptitious data exchange. They also find utility in secure conversations, message concealment, security testing, and privacy protection, underscoring their versatile application across diverse domains. However, the accuracy of covert timing channels may be compromised by network traffic patterns and time delays, potentially affecting data integrity. This survey paper primarily provides an brief overview of identifying and mitigating these influencing factors of existing covert timing channels and presents a suggested system that integrates Reversible Covert Timing Channel (CTC) steganography and Elliptic Curve Cryptography (ECC) to ensure secure communication and data encryption. By leveraging this combination, the system aims to enhance the resilience of covert communication channels against adversarial interference while safeguarding the confidentiality and integrity of transmitted data.

Hyperelliptic curve based authentication for the internet of drones

Drones provide an alternative progression in protection submissions since they are capable of conducting autonomous seismic investigations. Recent advancement in unmanned aerial vehicle (UAV) communication is an internet of a drone combined with 5G networks. Because of the quick utilization of rapidly progressed registering frameworks besides 5G officialdoms, the information from the user is consistently refreshed and pooled. Thus, safety or confidentiality is vital among clients, and a proficient substantiation methodology utilizing a vigorous sanctuary key. Conventional procedures ensure a few restrictions however taking care of the assault arrangements in information transmission over the internet of drones (IOD) environmental frameworks. A unique hyper elliptical curve (HEC) cryptographically based validation system is proposed to provide protected data facilities among drones. The proposed method has been compared with the existing methods in terms of packet loss rate, computational cost, and delay and thereby provides better insight into efficient and secure communication. Finally, the simulation results show that our strategy is efficient in both computation and communication.

Modeling arithmetic systems of elliptic curve cryptography using Microsoft Excel VBA

Purpose. This study aims to develop a new teaching module to illustrate the arithmetic systems of Elliptic Curve Cryptography, a powerful yet simple algorithm for information security, by exploring the capability of the Visual Basic Applications of Microsoft Excel in user friendly way. Methodology. The research is performed using research and development approach, which is divided into five steps utilizing VBA features of Microsoft Excel. It starts with modeling arithmetic in Microsoft Excel spreadsheet, then testing the validity through calculation and setup of the actual arithmetic of Elliptic Curve Cryptography using VBA Excel, before performing the test of the VBA application and finally visualizes the results in graphical mode. Findings. Novel teaching software based on of Microsoft Excel Visual Basic Applications is produced that is able to simulate arithmetic system behind Elliptic Curve Cryptography in an easy way for students. Originality. To the best of the authors’ knowledge, this is the first simulation based on Excel VBA to illustrate the arithmetic systems of Elliptic Curve Cryptography for teaching purposes. Practical value. In general, mastering cryptography will need a steep learning curve; however, using Microsoft Excel as a simulation platform will accelerate learning. The main practical value is the ease of Microsoft Excel, which will turn cryptography learning which was commonly very difficult for student to become easier and user friendly.

Enhanced Scalar Multiplication Algorithm over Prime Field Using Elliptic Net

Scalar multiplication in elliptic curve cryptography is the most expensive and time-consuming operation. The elliptic curve cryptography attracted interest due to the development of modern technology since it could offer the equivalent high level of security with a reduced length of key. Therefore, improving elliptic curve scalar multiplication performance has always been the primary goal of cryptography. In this paper, a novel scalar multiplication algorithm based on the modified double and double add via elliptic net with Karatsuba method was proposed in order to enhance the efficiency of scalar multiplication. In the experimental results, the elliptic net equivalence sequence was applied to the Twisted Edwards curve together with safe curves of numsp384t1 and numsp512t1. At the point operational level, the proposed method reduced the cost of multiplication by 46.15% and 42.30% for double and double add, respectively, when compared to elliptic net using eight blocks method. The proposed double lowered the multiplication cost by 12.5% and the squaring cost by 20% when compared to elliptic net using ten temporary variables method. Following this, proposed double add cost reductions of 6.25% and 20% were obtained to multiplication and squaring. At the field operational level, in comparison to the binary method, the eight-block elliptic net method, and the elliptic net method with ten temporary variables for the 384 bits scenario, the developed scalar multiplication algorithm obtained cost reductions of 57.6%, 31.3%, and 13.2%, respectively. On 512 bits with similar comparison, the designed algorithm exhibited better performance by averages 59.2%, 31.0% and 13.2%. The results signified that the designed algorithm over prime field performed better at the point and field operational levels with larger scalar bit size.

Common valuations of division polynomials

In this note, we prove a formula for the cancellation exponent $k_{v,n}$ between division polynomials $\psi _n$ and $\phi _n$ associated with a sequence $\{nP\}_{n\in \mathbb {N}}$ of points on an elliptic curve $E$ defined over a discrete valuation field $K$ . The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue fields.

A note on Wedderburn and Hasse theorems

Wedderburn and Hasse theorems have always fascinated many algebraists as there are just very few versions of their proofs and that are somehow not easy to follow because they contain other results inside. In this note, we aim to look at their proofs again but nowwith more simplified versions that include sufficient details in order to let them be understandable for most readers in the subjects of finite fields and elliptic curves. In the proof of Wedderburn’s theorem, we succeed to show how it is equivalent to define either a stabilizer or centralizer of non-central elements and that helps to consider the equation on the order of group by using interchangeably the index of its subgroup and the order of the orbit. As for Hasse’s theorem, we provide a short proof in just two paragraphs with a recall of the most important results that have been used, and then, we state three of its version with an explanation of how this all leads in the end to the need of an algorithm like the one of René Schoof in 1985.

RETRACTED ARTICLE: Efficient identity security authentication method based on improved R-LWE algorithm in IoT environment

In recent years, various smart devices based on IoT technology, such as smart homes, healthcare, detection, and logistics systems, have emerged. However, as the number of IoT-connected devices increases, securing the IoT is becoming increasingly challenging. To tackle the increasing security challenges caused by the proliferation of IoT devices, this research proposes an innovative method for IoT identity authentication. The method is based on an improved ring-learning with errors (R-LWE) algorithm, which encrypts and decrypts communication between devices and servers effectively using polynomial modular multiplication and modular addition operations. The main innovation of this study is the improvement of the traditional R-LWE algorithm, enhancing its efficiency and security. Experimental results demonstrated that, when compared to number theory-based algorithms and elliptic curve cryptography algorithms at a 256-bit security level, the enhanced algorithm achieves significant advantages. The improved algorithm encrypted 20 data points with an average runtime of only 3.6 ms, compared to 7.3 ms and 7.7 ms for the other algorithms. Similarly, decrypting the same amount of data had an average runtime of 2.9 ms, as opposed to 7.3 ms and 8 ms for the other algorithms. Additionally, the improved R-LWE algorithm had significant advantages in terms of communication and storage costs. Compared to the number theory-based algorithm, the R-LWE algorithm reduced communication and storage costs by 3 °C each, and compared to elliptic curve cryptography, it reduced them by 4 °C each. This achievement not only enhances the efficiency of encryption and decryption but also lowers the overall operational costs of the algorithm. The research has made significant strides in improving the security and efficiency of IoT device identity authentication by enhancing the R-LWE algorithm. This study provides theoretical and practical foundations for the development and application of related technologies, as well as new solutions for IoT security.

Transformation of Elliptic Curve Discrete Logarithm Problem to QUBO Using Direct Method in Quantum Annealing Applications

This paper investigates how to reduce the elliptic curve discrete logarithm problem over prime fields to the quadratic unconstrained binary optimization (QUBO) problem in order to obtain as few logical qubits as possible. In the best case scenario, if n is the bitlength of a characteristic of prime field Fp, approximately 3n³ logical qubits are required for such a reduction in the Edwards curve case. We present a practical attack on an elliptic curve discrete logarithm problem over the 3-bit prime field F7 for an elliptic curve with the subgroup of order 8. We solved this problem using the D-Wave Advantage QPU. To the best of the authors' knowledge, no one has made, so far, a practical attack on the elliptic curve discrete logarithm over a prime field using the direct quantum method.

Deformation and Quantisation Condition of the $$\mathcal {Q}$$-Top Recursion

AbstractWe consider a deformation of a family of hyperelliptic refined spectral curves and investigate how deformation effects appear in the hyperelliptic refined topological recursion as well as the $$\mathcal {Q}$$ Q -top recursion. We then show a coincidence between a deformation condition and a quantisation condition in terms of the $$\mathcal {Q}$$ Q -top recursion on a degenerate elliptic curve. We also discuss a relation to the corresponding Nekrasov–Shatashivili effective twisted superpotential.

Statistics for iwasawa invariants of elliptic curves, II

Statistics for iwasawa invariants of elliptic curves, II