Elliptic Curve Research Articles

PYTHON-МОДЕЛЬ ПРОТОКОЛУ УЗГОДЖЕННЯ СЕКРЕТНОГО КЛЮЧА У ГРУПІ З ДОВІЛЬНОЇ КІЛЬКОСТІ УЧАСНИКІВ

The article is devoted to the problem of common secret key agreement in a group of an arbitrary number of participants. Data is exchanged between participants through open data channels. The problem of sharing a secret key over open data channels arose due to the need for a secure exchange of information between two or more parties that may be remote from each other and do not have a common confidential communication channel. Reliable methods of secret key exchange, such as transferring the key in person or using a secure channel, are not practical in remote or scalable scenarios. In the process of developing and modelling cryptographic systems, in which there is a need of cryptographic keys agreement in a group of two or more participants, it is very convenient to have a model that implements these algorithms. The agreement protocol is based on the Diffie-Hellman protocol on elliptic curves (ECDH). The paper contains theoretical justifications, a flow chart of the algorithm, and a Python implementation of the algorithm that performs the secret key agreement in a group of an arbitrary number of participants. To implement cryptographic operations on elliptic curves, the Python library Cryptography is used, in particular, the X25519 algorithms that use the elliptic curve Curve25519. The results of the work are shown on an example for a group of four participants, which demonstrate the correct operation of the model and the same secret key obtained as a result of agreement process. The paper also contains the link to a GitHub repository with the full program. The multi_participant_ecdh.py file contains a secret key agreement program for N participants written in Python. The file two_participant_ecdh.py demonstrates a typical two-participant Diffie-Hellman protocol. Both programs use the X25519 algorithm implemented in the Cryptography Python library. Thus, this work makes it possible to better understand the principles of secret key exchange algorithms between two and an arbitrary number of participants, to compare the results with other implementations, to apply the developed model for educational and demonstration purposes, and may be useful for a number of other scientific and engineering tasks.

Efficient attribute-based strong designated verifier signature scheme based on elliptic curve cryptography.

In an attribute-based strong designated verifier signature, a signer who satisfies the access structure signs the message and assigns it to a verifier who satisfies the access structure to verify it, which enables fine-grained access control for signers and verifiers. Such signatures are used in scenarios where the identity of the signer needs to be protected, or where the public verifiability of the signature is avoided and only the designated recipient can verify the validity of the signature. To address the problem that the overall overhead of the traditional attribute-based strong designated verifier signature scheme is relatively large, an efficient attribute-based strong designated verifier signature scheme based on elliptic curve cryptography is proposed, as well as a security analysis of the new scheme given in the standard model under the difficulty of the elliptic curve discrete logarithm problem (ECDLP). On the one hand, the proposed scheme is based on elliptic curve cryptography and uses scalar multiplication on elliptic curves, which is computationally lighter, instead of bilinear pairing, which has a higher computational overhead in traditional attribute-based signature schemes. This reduces the computational overhead of signing and verification in the system, improves the efficiency of the system, and makes the scheme more suitable for resource-constrained cloud end-user scenarios. On the other hand, the proposed scheme uses LSSS (Linear Secret Sharing Schemes) access structure with stronger access policy expression, which is more efficient than the "And" gate or access tree access structure, making the computational efficiency of the proposed scheme meet the needs of resource-constrained cloud end-users.

Lightweight Certificateless Signcryption Scheme Using Type-3 Pairing on Elliptic Curve

Lightweight Certificateless Signcryption Scheme Using Type-3 Pairing on Elliptic Curve

Two-loop non-planar four-point topology with massive internal loop

We study a set of two-loop non-planar master integrals needed for the NNLO QCD corrections to diphoton and dijet production at hadron colliders. The top-sector topology contains an internal massive fermion loop and is known to contain elliptic curves. Leveraging the method of differential equations, we provide a comprehensive discussion for deriving an ϵ-factorized differential equation related to the most intricate sector within the Feynman integral family. Despite the dependence on multiple scales and the presence of two elliptic sectors, we demonstrate how to leverage the properties of their maximal cuts and the factorization of the Picard-Fuchs operator to deal with the complexity of the analytic computation. In particular, we construct a transformation matrix that brings the differential equations into a format enabling the convenient expression of analytic results in terms of Chen’s iterated integrals.

Compatible Poisson Brackets Associated with Elliptic Curves in $G(2,5)$

We prove that a pair of Feigin-Odesskii Poisson brackets on ${\mathbb P}^4$ associated with elliptic curves given as linear sections of the Grassmannian $G(2,5)$ are compatible if and only if this pair of elliptic curves is contained in a del Pezzo surface obtained as a linear section of $G(2,5)$.

Picard sheaves, local Brauer groups, and topological modular forms

AbstractWe develop tools to analyze and compare the Brauer groups of spectra such as periodic complex and real ‐theory and topological modular forms, as well as the derived moduli stack of elliptic curves. In particular, we prove that the Brauer group of is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Our main computational focus is on the subgroup of the Brauer group consisting of elements trivialized by some étale extension, which we call the local Brauer group. Essential information about this group can be accessed by a thorough understanding of the Picard sheaf and its cohomology. We deduce enough information about the Picard sheaf of and the (derived) moduli stack of elliptic curves to determine the structure of their local Brauer groups away from the prime 2. At 2, we show that they are both infinitely generated and agree up to a potential error term that is a finite 2‐torsion group.

Composite continuum robots: Accurate modeling and model reduction

Models of continuum robots often trade off accuracy and speed. The constant curvature assumption model can perform fast calculations but with poor accuracy. Static models can obtain good calculation accuracy but with slower computation speed. In this paper, a precise and efficient method for establishing a non-constant curvature kinematic model for composite continuum robots is proposed. First, an algorithm based on the Beam Constraint Model is proposed to establish the static model of the continuum robot. The situation in which friction changes gradually in continuum robots is considered in the static model. Then, the Kepler elliptic curve is used to simplify the static model to build a non-constant curvature kinematic model of the continuum robot. Then, complex inverse kinematics solutions are transformed into explicit sets of equations by mathematical transformation. Finally, simulations and experiments are designed to demonstrate the algorithm proposed in this paper. The simulation results show that the proposed algorithm is 1572 times faster than the Levenberg–Marquardt algorithm. The experimental results show that the average error of the static model is 0.065 mm, accounting for 0.13% of its length. The maximum error is 0.309 mm, accounting for 0.618% of its length. The positioning accuracy of the non-constant curvature model proposed in this paper is 2.02 times that of the constant curvature model. The results show that the simplified kinematic model algorithm based on the static model has high accuracy and computational efficiency. This research is crucial for the practical application of continuum robots in engineering.

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

Multi-dimension-precision chaotic encryption mechanism for Internet of Things

The massive amount of information processed in the Internet of Things(IoT) environment requires effective encryption schemes to ensure secure transmission. However, traditional encryption methods such as Elliptic Curve Cryptography (ECC) or Data Encryption Standard (DES) are not suitable for IoT environments where large amounts of data need to be quickly encrypted. In this paper, a multi-precision encryption mechanism based on multi-dimensional chaotic system is proposed. Firstly, the proposed scheme uses floating-point, 32-bit, 24-bit, and 16-bit data accuracy respectively, which can effectively reduce the complexity through fixed-point calculation. Secondly, aiming at the problem that traditional chaotic dynamic analysis methods such as the Lyapunov index cannot be applied, an analysis scheme based on complexity measure is proposed. Finally, a free dimension encryption mechanism based on Logistic, Tent, and PWLCM is proposed. By adding random disturbance to a one-dimensional chaotic map, the weak randomness of the single chaotic sequence is overcome effectively. Finally, the security analysis of the encryption scheme is carried out, and the result proves that the scheme can effectively resist the common attack means. In the 3D chaotic system, the encryption effect of 24-bit data accuracy is close to floating point accuracy, which is suitable for the IoT environment with large data volume, high encryption efficiency, and vulnerability.

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.