Elliptic Curve Research Articles

Studying Hilbert’s 10th problem via explicit elliptic curves

Studying Hilbert’s 10th problem via explicit elliptic curves

Realization of Authenticated One-Pass Key Establishment on RISC-V Micro-Controller for IoT Applications

Internet-of-things networks consist of multiple sensor devices spread over a wide area. In order to protect the data from unauthorized access and tampering, it is essential to ensure secure communication between the sensor devices and the central server. This security measure aims to guarantee authenticity, confidentiality, and data integrity. Unlike traditional computing systems, sensor node devices are often limited regarding memory and computing power. Lightweight communication protocols, such as LoRaWAN, were introduced to overcome these limitations. However, despite the lightweight feature, the protocol is vulnerable to different types of attacks. This proposal presents a highly secure key establishment protocol that combines two cryptography schemes: Elliptic Curve Qu–Vanstone and signcryption key encapsulation. The protocol provides a method to establish a secure channel that inherits the security properties of the two schemes. Also, it allows for fast rekeying with only one exchange message, significantly reducing the handshake complexity in low-bandwidth communication. In addition, the selected schemes complement each other and share the same mathematical operations in elliptic curve cryptography. Moreover, with the rise of a community-friendly platform like RISC-V, we implemented the protocol on a RISC-V system to evaluate its overheads regarding the cycle count and execution time.

Moore matrices and Ulrich bundles on an elliptic curve

Moore matrices and Ulrich bundles on an elliptic curve

Numerical simulation of optimizing the swing curve of a 3DOF biomimetic pectoral fin in drag mode

In response to the optimization problem of the motion law of the pectoral fins of biomimetic robotic fish, this paper uses computational fluid dynamics (CFD) methods to achieve the CFD optimization of two types of rigid biomimetic pectoral fins with three degrees of freedom (DOF): elliptical and 8-shaped swing curves. By analyzing the hydrodynamic characteristic curve of the pectoral fins, pressure distribution upstream and downstream of the fin surface, velocity vector of the surrounding flow field during pectoral fin swing, and three-dimensional vortex structure, it is found that when the pectoral fins are pushed forward in an elliptical swing curve, the resistance peak value decreases after two optimizations. However, at the same time, a certain amount of thrust is lost. When the pectoral fin advances in an 8-shaped swinging curve, after two optimizations, the pectoral fins resistance decreases while maintaining an enormous thrust, comprehensively improving the propulsive effect of the pectoral fin on the robotic fish. By substituting the optimized 8-shaped pattern into the robotic fish model, it is found that the flapping angle of 35° has better drag reduction performance, and the average swimming speed of the robotic fish is the fastest at this angle.

Improved RSA dynamic cryptographic accumulator-based anonymous batch authentication scheme for Internet of Vehicles

The Internet of Vehicles (IoV) provides various services to vehicles through information sharing to ensure road safety and improve traffic efficiency. However, malicious attackers can challenge the privacy and security systems of the IoV by conducting security attacks. Currently, many schemes use bilinear pairing or elliptic curves to construct authentication systems, ensuring anonymity of vehicle identities and message security. However, these schemes suffer drawbacks such as low computational efficiency and high communication overhead in highly dynamic IoV. Therefore, we propose a new efficient certificateless message authentication scheme. The proposed scheme constructs a lightweight security authentication protocol by improving the RSA dynamic accumulator and combining it with non-interactive discrete logarithm zero-knowledge proofs. In addition, our scheme eliminates the need for bilinear pairing operations, thereby reducing computational overhead. Simultaneously, it enables real-time tracking and revocation of malicious vehicle identities. Security analysis shows that our scheme can be reduced to a secure S-RSA assumption under the random oracle model, meeting various security requirements of the IoV. Performance analysis shows that our scheme diminishes not only the computational overhead but also the communication overhead.

CryptoCard: Blockchain-Powered Library Card Management System

Abstract: Institutions such as libraries often face challenges in managing and verifying the authenticity of issued library cards. Traditional methods of card issuance and verification may be susceptible to fraud and manipulation. To address these challenges, we present a Blockchain-Enabled Library Card Management System. Our system leverages blockchain technology to securely store and manage library card data. Each library card issuance is recorded as a block on the blockchain, ensuring transparency, immutability, and tamper-resistance. Digital signatures generated using the Elliptic Curve Digital Signature Algorithm (ECDSA) are employed to authenticate the integrity of each card. The system allows users to easily issue library cards by providing their personal information and a photo. The generated library card includes user details and a unique identifier, along with a digital signature to validate its authenticity. Additionally, users can verify the validity of library cards by querying the blockchain and verifying the digital signatures. By employing blockchain technology, our system enhances the security and trustworthiness of library card management, reducing the risk of fraud and ensuring the integrity of issued cards. It provides a robust and efficient solution for libraries to manage their card issuance process whilemaintaining data integrity and security.

Secured V2X communication using optimized prime field ECC architecture

The objective of this research is to ensure a high level of security in various forms of vehicular communication (V2V, V2I, and V2N) and to support vehicles in safely receiving all keys and messages from Roadside Unit (RSU), other vehicles, or the network with the support of simple cryptographic techniques. Elliptic Curve Cryptography (ECC) is one of the many cryptographic techniques that offer a desirable response to this requirement. The Koblitz curve secp256k1 is supported by the designed ECC processor for 256-bit point multiplication and point addition. A powerful illustration of the "Divide and Conquer" strategy's ability to accelerate multiplication asymptotically is given by the Karatsuba algorithm. The ECC processor's multiplication operation is accelerated further by the incorporation of pipelining. With a maximum clock frequency of 238.40MHz, the proposed Karatsuba-based ECC processor executes 256-bit single point multiplication in 0.937ms, providing 273.21kbps throughput, and taking up 8.42k slices in a Virtex-7 FPGA. Scalar multiplication is extended by incorporating a pipeline by increasing the highest clock frequency by up to 7.97%, which decreases time consumption by 9.90% and boosts throughput by 10.99%. In terms of area, operating frequency, area-delay product, and throughput, the suggested pipelined Karatsuba multiplier based ECC processor performs better than the existing designs.

Analysis of Isomorphism Classes of a Family of Elliptic Curves Over Finite Fields

Doche et al. constructed a family of elliptic curves (DIK elliptic curves) and proposed more efficient tripling formulas leading to a fast scalar multiplication algorithm. In this paper we present a direct method to compute the number of \(\bar{F}\)q-isomorphism classes (isomorphism over \(\bar{F}\)q ) and \(\bar{F}\)q isomorphism classes of DIK family of elliptic curves defined over a finite field \(\bar{F}\)q. We give the explicit formulae for the number of \(\bar{F}\)q-isomorphism and an estimate formulae for the number of isomorphism classes. These result can be used in the elliptic curve cryptosystems.

GIS and cellular automata based slope rainwater movement process model and its application

Rainfall serves as a significant factor contributing to slope stability challenges in mountainous areas, and simulating the process of slope rainwater movement is a crucial approach for analyzing the stability of slopes triggered by rainfall. By combining computer numerical simulation technology with traditional hydraulic and hydrological calculation theories, it is possible to create an efficient and precise rainwater movement model that can simulate and analyze the process of rainwater movement on slopes. Utilizing natural slopes as the focal point of our research, the cellular automaton method was applied to simulate rainfall runoff on slopes, and a Cellular Automata (CA) based model for rainwater movement process was developed. This model modified the Green-Ampt (G-A) infiltration model by adopting an elliptical water content curve and introducing a coefficient that quantifies the ratio of saturated to unsaturated depth. Additionally, we refined the rules governing runoff generation and convergence within the slope and on its surface, enabling a comprehensive simulation of the entire rainwater movement process on the slope. Furthermore, the effectiveness of the model was validated through analytical solutions derived from simplified assumptions, laboratory experiments on infiltration and runoff in the flume, and a case study of a natural slope. The results show that the infiltration calculation results of the rainwater movement model are closer to the experimental values, and their overall values are slightly higher than the measured values, which are basically consistent with the model test results; The runoff calculation results show a phenomenon of initially increasing and gradually approaching the measured values compared to the measured values. When applying the model to an actual slope, it was found that the model comprehensively accounts for the influence of slope seepage, infiltration and runoff process, has better performance compared to G-A modified model. The model can be used to describe the spatial distribution and temporal variation of infiltration and runoff processes.

Key Agreement Scheme for Authorization and Authentication of WSN in IoT-5G Using Elliptic Curve Cryptography

Key Agreement Scheme for Authorization and Authentication of WSN in IoT-5G Using Elliptic Curve Cryptography

A function related to the Mordell-Weil rank of elliptic curves

A function related to the Mordell-Weil rank of elliptic curves

Elliptic curves with a rational 2-torsion point ordered by conductor and the boundedness of average rank

In this paper we refine recent work due to A. Shankar, A. N. Shankar, and X. Wang on counting elliptic curves by conductor to the case of elliptic curves with a rational 2-torsion point. This family is a small family, as opposed to the large families considered by the aforementioned authors. We prove the analogous counting theorem for elliptic curves with so-called square-free index as well as for curves with suitably bounded Szpiro ratios. We note that the assumptions we impose on the size of the Szpiro ratios are less stringent than would be expected by the naive generalization of their approach; this requires an application of a theorem of Browning and Heath-Brown. Our proof also relies on linear programming techniques.

Torsion primes for elliptic curves over degree 8 number fields

Let d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 1$$\\end{document} be an integer and let p be a rational prime. Recall that p is a torsion prime of degree d if there exists an elliptic curve E over a degree d number field K such that E has a K-rational point of order p. Derickx et al. (Algebra Number Theory 17(2):267–308, 2023) have computed the torsion primes of degrees 4, 5, 6 and 7. We verify that the techniques used in Derickx et al. (Algebra Number Theory 17(2):267–308, 2023) can be extended to determine the torsion primes of degree 8.

An efficient provably secure authentication and key agreement protocol for satellite communication networks

AbstractThe leverage of satellite resources to establish a communication network offers a wide range of benefits, including the capability to support news gathering, broadcasting, and aeronautical and maritime tracking. The unique aspects of the satellite‐linked terrestrial network provide high‐speed services, dependable and consistent network quality, and comprehensive geographic coverage of remote regions. Unfortunately, these structural features also render the network vulnerable to unauthorized intrusion, potentially leading to significant disruptions. Consequently, the implementation of authentication measures presents an appealing solution for enhancing the overall service quality of this network. Recently, Kumar et al. presented an authentication and key agreement scheme for satellite communications. Strikingly, their scheme proves inadequate in safeguarding against several malicious attacks and reveals certain design weaknesses. In this article, we present a reliable and secure authentication protocol that takes advantage of the complexities inherent in the elliptic curve discrete logarithm problem. We assess the protocol's effectiveness against various types of attacks using formal proof, Burrows–Abadi–Needham logic, informal descriptive proof analysis, and the verification tool SCYTHER. Furthermore, we compare the computational, communication, and storage overhead of our proposed protocol to an existing one, demonstrating its efficiency and superiority.

Towards a classification of isolated 𝑗-invariants

We develop an algorithm to test whether a non-complex multiplication elliptic curve E / Q E/\mathbf {Q} gives rise to an isolated point of any degree on any modular curve of the form X 1 ( N ) X_1(N) . This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to E E . Running this algorithm on all elliptic curves presently in the L L -functions and Modular Forms Database and the Stein–Watkins Database gives strong evidence for the conjecture that E E gives rise to an isolated point on X 1 ( N ) X_1(N) if and only if j ( E ) = − 140625 / 8 , − 9317 j(E)=-140625/8, -9317 , 351 / 4 351/4 , or − 162677523113838677 -162677523113838677 .

WILES DEFECT OF HECKE ALGEBRAS VIA LOCAL-GLOBAL ARGUMENTS

Abstract In his work on modularity of elliptic curves and Fermat’s last theorem, A. Wiles introduced two measures of congruences between Galois representations and between modular forms. One measure is related to the order of a Selmer group associated to a newform $f \in S_2(\Gamma _0(N))$ (and closely linked to deformations of the Galois representation $\rho _f$ associated to f), whilst the other measure is related to the congruence module associated to f (and is closely linked to Hecke rings and congruences between f and other newforms in $S_2(\Gamma _0(N))$ ). The equality of these two measures led to isomorphisms $R={\mathbf T}$ between deformation rings and Hecke rings (via a numerical criterion for isomorphisms that Wiles proved) and showed these rings to be complete intersections. We continue our study begun in [BKM21] of the Wiles defect of deformation rings and Hecke rings (at a newform f) acting on the cohomology of Shimura curves over ${\mathbf Q}$ : It is defined to be the difference between these two measures of congruences. The Wiles defect thus arises from the failure of the Wiles numerical criterion at an augmentation $\lambda _f:{\mathbf T} \to {\mathcal O}$ . In situations we study here, the Taylor–Wiles–Kisin patching method gives an isomorphism $ R={\mathbf T}$ without the rings being complete intersections. Using novel arguments in commutative algebra and patching, we generalize significantly and give different proofs of the results in [BKM21] that compute the Wiles defect at $\lambda _f: R={\mathbf T} \to {\mathcal O}$ , and explain in an a priori manner why the answer in [BKM21] is a sum of local defects. As a curious application of our work we give a new and more robust approach to the result of Ribet–Takahashi that computes change of degrees of optimal parametrizations of elliptic curves over ${\mathbf Q}$ by Shimura curves as we vary the Shimura curve. The results we prove are not attainable using only the methods of Ribet–Takahashi.

Torsion points of elliptic curves over multi-quadratic number fields

We compute the Mordell–Weil groups of the modular Jacobian varieties of hyperelliptic modular curves X1(M,MN) over every composite field of some quadratic number fields. Also we prove criteria for the existence of elliptic curves over such number fields with prescribed torsion points generalizing the results for quadratic number fields of Kamienny and Najman.

Secure ECDSA SRAM-PUF Based on Universal Single/Double Scalar Multiplication Architecture.

Physically unclonable functions (PUFs) are crucial for enhancing cybersecurity by providing unique, intrinsic identifiers for electronic devices, thus ensuring their authenticity and preventing unauthorized cloning. The SRAM-PUF, characterized by its simple structure and ease of implementation in various scenarios, has gained widespread usage. The soft-decision Reed-Muller (RM) code, an error correction code, is commonly employed in these designs. This paper introduces the design of an RM code soft-decision attack algorithm to reveal its potential security risks. To address this problem, we propose a soft-decision SRAM-PUF structure based on the elliptic curve digital signature algorithm (ECDSA). To improve the processing speed of the proposed secure SRAM-PUF, we propose a custom ECDSA scheme. Further, we also propose a universal architecture for the critical operations in ECDSA, elliptic curve scalar multiplication (ECSM), and elliptic curve double scalar multiplication (ECDSM) based on the differential addition chain (DAC). For ECSMs, iterations can be performed directly; for ECDSMs, a two-dimensional DAC is constructed through precomputation, followed by iterations. Moreover, due to the high similarity of ECSM and ECDSM data paths, this universal architecture saves hardware resources. Our design is implemented on a field-programmable gate array (FPGA) and an application-specific integrated circuit (ASIC) using a Xilinx Virtex-7 and an TSMC 40 nm process. Compared to existing research, our design exhibits a lower bit error rate (2.7×10-10) and better area-time performance (3902 slices, 6.615 μs ECDSM latency).

A secure data hiding system in biomedical images using grain 128a algorithm, logistic mapping and elliptical curve cryptography

A secure data hiding system in biomedical images using grain 128a algorithm, logistic mapping and elliptical curve cryptography

A hybrid scheme using hyper-chaotic system and elliptic curve cryptography for image encryption

A hybrid scheme using hyper-chaotic system and elliptic curve cryptography for image encryption