Elliptic Curve Cryptosystem Research Articles

A New Septupling Point 7P Arithmetic Formula for LD Coordinate and Affine Over Binary Elliptic Curve Cryptosystem

The Elliptic Curve Cryptography (ECC) is one of the most prominent Asymmetric-based cryptosystems as it affords a higher level of security with small keys. According to National Institute of Standards and Technology (NIST), ECC gains the smallest secure key over the binary curve. In literature, the best field over binary curves is Lopez-Dahab (LD) and Affine coordinates, and it considered fit for lightweight cryptography in resource-constrained devices such as Internet of Things (IoTs). ECC consists of three operational levels; scalar multiplication, point arithmetic and field arithmetic. This research focuses on point arithmetic precomputation useful in scalar multiplication and then for field arithmetic. There is no existing formula for Septupling point over binary curves in LD and Affine coordinates. A new precomputed Septupling point is introduced in this paper using LD and non-supersingular affine coordinate over the binary field . This paper uses the form , consisting of Doubling point, Tripling point and the point addition. Also, the form means the Sixtupling point is also proposed. Results show that the is characterized by a cost of , while the cost of Sixtupling point is . The point is mathematically proved as valid. The proposed point can be implemented for different scalar multiplication such as for , and multi-based scalar such as {2, 3, 7}-based scalar multiplication method.

Internet of things encryption technology combining elliptic curve cryptosystem, hash function, and RFID-based authentication

As an important means of connecting the physical world and the digital world, the reliability and security of the Internet of Things network are key issues. To raise the security of Internet of Things data, a research proposes an encryption technique that combines elliptic curve cryptosystem and hash functions. In this process, the radio frequency identification system is used as the central element for data encryption, with the elliptic curve cryptosystem employed to secure the transmitted data. Non-adjacent scalar representations are used to reduce the expected running time of scalar multiplication, and a bidirectional authentication protocol for Internet of Things encryption is designed using Hash functions. The experimental results showed that in the communication overhead test, the research method had a communication overhead of 242 bits when the Hash function output length was 40 bits in server communication. When analyzing the success rate of intercepting abnormal data access behavior, the research method achieved a success rate of 99.1% when the host file size was 100 Kb. In the analysis of scalar multiplication operation time, the research method only took 18 ms when the output length of the Hash function reached 340bits in a local area network environment. This illustrates that the raised method has a good encryption effect on the Internet of Things and can effectively ensure the security of Internet of Things communication. The research is expected to provide certain technical support for the development of the Internet of Things.

Application of Lenstra–Lenstra–Lovasz on Elliptic Curve Cryptosystem Using IOT Sensor Nodes

The Internet of Things (IoT) model is presented in this paper with multi-layer security based on the Lenstra-Lenstra-Lovasz (LLL) algorithm. End nodes for the Internet of Things include inexpensive gadgets like the Raspberry Pi and Arduino boards. It is not practical to run rigorous algorithms on them, as opposed to computer systems. Therefore, a cryptography procedure is required that could function on this IOT equipment. Bitcoins and Ethereum are examples of cryptocurrency and Ripple employs techniques such as elliptic curve digital signature, Elliptic-Curve Diffie-Hellman (ECDH), and algorithm to sign any cryptocurrency on SECP256k1 elliptic curves transactions. By using Lenstra-Lenstra-Lovasz on a real-world Bitcoin blockchain and applying it to multiple dimensions, such as nonce leakage and weak nonces across several elliptic curves with different bit sizes on a Raspberry Pi, we can demonstrate the security of elliptic curve cryptosystems. Public key encryption techniques are seriously threatened by the development of quantum computing. Therefore, employing lattice encryption with Nth Degree Truncated Polynomial Ring Units (NTRU-NTH) on the Bitcoin blockchain will increase the resistance of Bitcoin blocks to quantum computing assaults. The execution time taken on SECP256k1 is 131.7 Milli seconds comparatively faster than NIST-224P and NIST-384P.

A Digital Signature on Cubic Pell Cryptosystem CP256-1299

Elliptic curves have proven to be a suitable foundation for cryptosystems, with Elliptic Curve Cryptosystems (ECC) offering strong security with smaller key sizes. Recent advancements in ECC design aim to create more efficient and secure curves. In this paper, we introduce a new digital signature scheme, named CP256-1299. It is a 256-bit scheme based on a cubic Pell curve where the arithmetic operations are efficient and straightforward. In previous works, cubic Pell curves have been used to design public key cryptosystems. Our main motivation in proposing the new digital signature algorithm is to exploit the effectiveness of the arithmetic of cubic Pell curves, while maintaining reasonable keys and high security. We compare our new scheme to three widely-used digital signature algorithms based on ECC, namely ED25519, SECP256K1 and SECP256R1. It turns out that our cubic Pell curve based digital signature algorithm is designed to operate with a larger periodic order while maintaining at least similar computational requirements to most popular elliptic curve cryptosystems. Our new scheme is also suitable to support a central bank digital currency.

AN ADVANCED ENCRYPTION SCHEME WITH BI-PERIODIC FIBONACCI MATRICES ON ECC

In this study, we present a new Elliptic Curve Cryptosystem using the bi-periodic Fibonacci matrix sequence. We propose a new cryptographic method on elliptic curves over a finite field. We map a plaintext matrix constructed from points corresponding to letters on an elliptic curve. The 2m*2m message matrix M is divided into 2*2 block matrices, Bi, for i=1,...,m^2, and each block is multiplied by the corresponding elements of the bi-periodic Fibonacci matrix sequence. Finally, the resulting matrix is multiplied by a permutation matrix to increase the complexity of the encryption algorithm.

The use of chaotic pseudo random number and elliptic curve cryptosystem in an efficient OTP-based authentication scheme for online learning system

The use of chaotic pseudo random number and elliptic curve cryptosystem in an efficient OTP-based authentication scheme for online learning system

Advancing Security: A Novel Approach with Linear Bézier Curve in Menezes-Vanstone Elliptic Curve Cryptosystem

Advancing Security: A Novel Approach with Linear Bézier Curve in Menezes-Vanstone Elliptic Curve Cryptosystem

A secure transaction and smart contract architecture for internet of vehicles.

Recently, with the popularity of Internet of vehicles (IoV), there is an increasing market demand for integrating blockchain technology and smart contracts in electric vehicles. By utilizing distributed blockchain and smart contract technology, it provides data integrity preservation, traceability, and prevention of forgery. These features can be effectively used in the automated charging system. As a result, IoV can utilize blockchain and smart contract technologies to facilitate seamless payment for charging fees, tolls, parking fees, online shopping and other associated expenses without the need for intermediaries. However, since blockchain and smart contracts operate in an open internet environment, there is a risk of tampering with transaction records. Therefore, enhancing security becomes a crucial concern. The main aim of this study is to emphasize the growing significance of securing transactions and smart contracts in an open transmission environment to prevent potential damage or alteration.In order to achieve the implementation of secure smart contracts within a resource-constrained IoV, this study investigates various cryptographic mechanisms and realizes the elliptic curve cryptosystem (ECC) is widely recognized for providing strong security with shorter key lengths compared to other public key cryptography methods. This makes it ideal for environments with limited resources. Furthermore, in order to enhance the performance of smart contracts, this study proposes a cloud server architecture for parallel storage of smart contracts to improve access speed. Additionally, we introduce an automated trading framework for smart contracts that operates without human intervention.Finally, this study uses ECC to ensure the secure transmission of transaction data among the vehicle, base station, and cloud server, as well as to provide mutual verification of identities across all entities involved in the operation. As a result, this study proposes a secure architecture for blockchain and smart contracts that can autonomously initiate transactions while ensuring secure transmission and protection in the IoV.

Addition chain heuristics in application to elliptic curve cryptosystems

Addition chain heuristics in application to elliptic curve cryptosystems

Secure pairing-free certificateless aggregate signcryption scheme for IoT

Secure pairing-free certificateless aggregate signcryption scheme for IoT

A Privacy-Preserving Three-Factor Authentication System for IoT-Enabled Wireless Sensor Networks

A Privacy-Preserving Three-Factor Authentication System for IoT-Enabled Wireless Sensor Networks

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.

Cryptography Based on Elliptic Curve and Special Matrix with Linear System Algorithm

“Elliptic Curve Cryptosystem, which offers a high level of security with a reduced key size, has emerged as the best option for public key encryption. Elliptic curve cryptosystem has proven to be the best solution for public key encryption, where it provides a good level of security with smaller key size. In this paper, we attempt to develop an enhanced image encryption algorithms based on ECC by use two keys as circulant matrix in one of them, while generation another key from simple linear system modulo to purpose of image cryptography. The results of the suggested algorithm comparison with recent research are used to assess it where results were better than previous works”

Advanced Authentication and Energy-Efficient Routing Protocol for Wireless Body Area Networks

Recently, wireless body area network (WBAN) becomes a hot research topic in the advanced healthcare system. The WBAN plays a vital role in monitoring the physiological parameters of the human body with sensors. The sensors are small in size, and it has a small-sized battery with limited life. Hence, the energy is limited in the multi-hop routing process. The patient data is collected by the sensor, and the data are transmitted with high energy consumption. It causes failure in the data transmission path. To avoid this, the data transmission process should be optimized. This paper presents an advanced authentication and energy-efficient routing protocol (AAERP) for optimal routing paths in WBAN. Patients’ data are aggregated from the WBAN through the IoMT devices in the initial stage. To secure the patient’s private data, a hybrid mechanism of the elliptic curve cryptosystem (ECC) and Paillier cryptosystem is proposed for the data encryption process. Data security is improved by authenticating the data before transmission using an encryption algorithm. Before the routing process, the data encryption approach converts the original plain text data into ciphertext data. This encryption approach assists in avoiding intrusions in the network system. The encrypted data are optimally routed with the help of the teamwork optimization algorithm (TOA) approach. The optimal path selection using this optimization technique improves the effectiveness and robustness of the system. The experimental setup is performed by using Python software. The efficacy of the proposed model is evaluated by solving parameters like network lifetime, network throughput, residual energy, success rate, number of packets received, number of packets sent, and number of packets dropped. The performance of the proposed model is measured by comparing the obtained results with several existing models.

Dynamic Privacy-Preserving Anonymous Authentication Scheme for Condition-Matching in Fog-Cloud-Based VANETs.

Secure group communication in Vehicle Ad hoc Networks (VANETs) over open channels remains a challenging task. To enable secure group communications with conditional privacy, it is necessary to establish a secure session using Authenticated Key Agreement (AKA). However, existing AKAs suffer from problems such as cross-domain dynamic group session key negotiation and heavy computational burdens on the Trusted Authority (TA) and vehicles. To address these challenges, we propose a dynamic privacy-preserving anonymous authentication scheme for condition matching in fog-cloud-based VANETs. The scheme employs general Elliptic Curve Cryptosystem (ECC) technology and fog-cloud computing methods to decrease computational overhead for On-Board Units (OBUs) and supports multiple TAs for improved service quality and robustness. Furthermore, certificateless technology alleviates TAs of key management burdens. The security analysis indicates that our solution satisfies the communication security and privacy requirements. Experimental simulations verify that our method achieves optimal overall performance with lower computational costs and smaller communication overhead compared to state-of-the-art solutions.

Remote sensing image encryption algorithm based on novel hyperchaos and an elliptic curve cryptosystem

Remote sensing images carry crucial ground information, often involving the spatial distribution and spatiotemporal changes of surface elements. To safeguard this sensitive data, image encryption technology is essential. In this paper, a novel Fibonacci sine exponential map is designed, the hyperchaotic performance of which is particularly suitable for image encryption algorithms. An encryption algorithm tailored for handling the multi-band attributes of remote sensing images is proposed. The algorithm combines a three-dimensional synchronized scrambled diffusion operation with chaos to efficiently encrypt multiple images. Moreover, the keys are processed using an elliptic curve cryptosystem, eliminating the need for an additional channel to transmit the keys, thus enhancing security. Experimental results and algorithm analysis demonstrate that the algorithm offers strong security and high efficiency, making it suitable for remote sensing image encryption tasks.

Construction of Big Data Information Security Protection System in Industrial Internet Environment

Abstract With the continuous development and integration of information technology and industrialization-related technologies, industrial Internet control system security attacks occur frequently, and it is more and more important to build an information security protection system. This study focuses on the research improvement from two aspects of access control and intrusion prevention and designs an industrial Internet security access control strategy based on the homomorphic encryption algorithm of the Hyper Elliptic Curve Cryptosystem (HCC) and the key splitting algorithm based on threshold. Meanwhile, the convolutional neural network, two-way gating loop unit, and multi-head attention mechanism are integrated to construct the CMAG intrusion detection model. The encryption algorithm and CMAG model are applied and analyzed. The encryption and decryption times of this paper’s algorithm are both relatively smooth, with an average time consumption of about 1.93ms and 0.46ms, respectively, and significantly better than other algorithms with the increase in the number of bits. The throughput of this paper’s encryption algorithm is 13.68 KB/s, which is approximately 2 times, 19 times, and 29 times higher than the throughput of GM, ElGamal, and Paillier algorithms, respectively. The other algorithms cannot match its throughput rate during decryption. The CMAG model has an accuracy of 99.14%, which is better than that of the other models, and its average checking accuracy, average recall, and average F1-Score are 0.9889, 0.9783, and 0.9834, respectively, which are 1.25%-5.16%, 4.31%-7.19%, and 3.32%, respectively, compared with that of the other three algorithms. 7.19% and 3.32%-6.76%, respectively. This paper is of great practical significance for the construction and optimization of a big data information security protection system in an industrial Internet environment.

Optimization of Smart Campus Cybersecurity and Student Privacy Protection Paths Based on Markov Models

Abstract This paper starts with the application of hyper-convergence technology, builds the framework of a university smart campus based on it, and gives the framework description of the smart campus. In order to analyze the network security for the smart campus, the Markov model is used as the basis combined with the reinforced Q learning algorithm for network node security detection, and a specific simulation analysis is given. The encryption performance and defense performance of the elliptic curve cryptosystem are analyzed through the elliptic curve cryptosystem to formulate the encryption scheme for students’ private data in the smart campus. The results indicate that the Markov model node detection combined with reinforcement Q-learning in this paper takes a maximum time of about 5.75s when the network node size reaches 150. When the number of nodes in the smart campus network is 30, under brute force attack, the whole network is captured only when the number of malicious nodes increases to more than 22, while under random attack, it takes as many as 30 malicious nodes to join before the network completely falls. This illustrates that the use of the Markov model can be realized to analyze the network security of the smart campus and that student privacy protection needs to further improve the awareness of student data privacy protection and develop the habit of assessing the privacy risk beforehand in their daily network behavior.

Implementation of Elliptic Curve Cryptosystem with Bitcoin Curves on SECP256k1, NIST256p, NIST521p, and LLL

Very recent attacks like ladder leaks demonstrated the feasibility of recovering private keys with side-channel attacks using just one bit of secret nonce. ECDSA nonce bias can be exploited in many ways. Some attacks on ECDSA involve complicated Fourier analysis and lattice mathematics. This paper will enable cryptographers to identify efficient ways in which ECDSA can be cracked on curves NIST256p, SECP256k1, NIST521p, and weak nonce, kind of attacks that can crack ECDSA and how to protect yourself. Initially, we begin with an ECDSA signature to sign a message using the private key and validate the generated signature using the shared public key. Then we use a nonce or a random value to randomize the generated signature. Every time we sign, a new verifiable random nonce value is created, and a way in which the intruder can discover the private key if the signer leaks any one of the nonce values. Then we use Lenstra–Lenstra–Lovasz (LLL) method as a black box, we will try to attack signatures generated from bad nonce or bad random number generator (RAG) on NIST256p, SECP256k1 curves. The combination of nonce generation, post-message signing, and validation in ECDSA helps achieve Uniqueness, Authentication, Integrity, and Non-Repudiation. The analysis is performed by considering all three curves for the implementation of the Elliptic Curve Digital Signature Algorithm (ECDSA). The comparative analysis for each of the selected curves in terms of computational time is done with the leak of nonce and with the Lenstra–Lenstra–Lovasz method to crack ECDSA. The average computational costs to break ECDSA with curves NIST256p, NIST521p, and SECP256k1 are 0.016, 0.34,0.46 respectively which is almost zero depicting the strength of the algorithm. The average computational costs to break ECDSA with curves SECP256K1 and NIST256p using LLL are 2.9 and 3.4 respectively

General analysis on essential mathematical principles of elliptic curve cryptography

Prevalent is the practical application of Elliptic Curve Cryptography (ECC) in the modern public-key cryptosystem, especially the implementation of ECC algorithm in Bitcoin source code. With the thorough introduction of discrete logarithm and Diffie-Hellman key exchange, ECC has gradually progressed to be sophisticated and efficient simultaneously. Therefore, it currently has been widely regarded as the successor of RSA algorithm in terms of inheritance for its shorter lengths of keys, faster speed and higher safety under the same encryption strength. Due to the potential safety and complexity of Elliptic Curve Cryptosystem, it is apparently noticed that there is included a large volume of Maths principles related to the establishment of ECC algorithm. As a consequence, this paper will mainly focus on qualitative research and exemplary analysis to specifically elucidate the general knowledge on essential mathematical principles of ECC, including the Law of Addition, the Elliptic Curve Discrete Logarithm Problems (ECDLP) and the Elliptic Curve ElGamal (EC ElGamal), together with the corresponding applications combined with their deprivation processes.