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Elliptic Curve Cryptography (ECC) is one of the strongest and most efficient cryptographic techniques in modern cryptography. This paper gives the following introduction: The introduction of cryptography’s development; the introduction of the elliptic curve; the principle of ECC; the horizontal comparison between ECC and other types of cryptography; the modern breakthrough of ECC; the applications of ECC; by using a method of literature review. The study’s findings indicate that this factor is responsible for the rapid historical development of cryptography, from the classical password to the leap to modern cryptography. Elliptic Curve Cryptography (ECC), as one of the most important modern cryptographies, is stronger than most other cryptographies both in terms of security and strength, because it uses an elliptic curve to construct and, at the same time, uses mathematical operations to encrypt and generate keys. At the same time, elliptic curve cryptography can continue to improve the speed and intensity with the improvement of accelerators, scalar multiplication, and the speed of order operation. The applications of the elliptic curve in ECDSA and SM2 are very efficient, which further illustrates the importance of elliptic curve cryptography.

With the swift evolution of wireless technologies, the demand for the Internet of Things (IoT) security is rising immensely. Elliptic curve cryptography (ECC) provides an attractive solution to fulfill this demand. In recent years, Edwards curves have gained widespread acceptance in digital signatures and ECC due to their faster group operations and higher resistance against side-channel attacks (SCAs) than that of the Weierstrass form of elliptic curves. In this paper, we propose a high-speed, low-area, simple power analysis (SPA)-resistant field-programmable gate array (FPGA) implementation of ECC processor with unified point addition on a twisted Edwards curve, namely Edwards25519. Efficient hardware architectures for modular multiplication, modular inversion, unified point addition, and elliptic curve point multiplication (ECPM) are proposed. To reduce the computational complexity of ECPM, the ECPM scheme is designed in projective coordinates instead of affine coordinates. The proposed ECC processor performs 256-bit point multiplication over a prime field in 198,715 clock cycles and takes 1.9 ms with a throughput of 134.5 kbps, occupying only 6543 slices on Xilinx Virtex-7 FPGA platform. It supports high-speed public-key generation using fewer hardware resources without compromising the security level, which is a challenging requirement for IoT security.

This paper shows that properties of projective modules over a group ring

This research paper presents an efficient data collection scheme for Wireless Sensor Networks (WSNs) that simultaneously compresses and encrypts sensor data to extend network lifespan. To address WSN resource limitations, the scheme combines Compressive Sensing (CS) with Elliptic Curve Cryptography (ECC) and Elliptic Curve Diffie–Hellman (ECDH) key exchange. Sensor data is securely compressed and encrypted using ECC-based public key mechanisms, mitigating CS-related attacks during aggregation and transmission. The measurement matrix seed serves as a private key that is exchangeed between sensor nodes and the base station, enhancing both security and efficiency. A prime-number-based Tree Path Identifier (TPID) routing and Cluster Head (CH) selection strategy is employed to optimize communication. Seven CS algorithms—including Orthogonal Matching Pursuit (OMP), Binary Compressive Sensing (BCS), Subspace Pursuits (SP), Approximate Message Passing (AMP), Split Bregman Iterative (SBI), Basis Pursuit (BP) and Compressive Sampling Matching Pursuit (CoSaMP) algorithms—are evaluated across various data sparsity levels. Results show that SP, AMP, and SBI algorithms outperform others in preserving energy, extending network life, and delaying the First Dead Node (FDN) appearance. Performance metrics include residual energy, network lifetime, total energy dissipation, and throughput. Energy savings confirm the superiority of the proposed hybrid scheme over traditional CS algorithms.

In recent decades there has been an increasing interest in Elliptic curve cryptography (ECC) and, especially, the Elliptic Curve Digital Signature Algorithm (ECDSA) in practice. The rather recent developments of emergent technologies, such as blockchain and the Internet of Things (IoT), have motivated researchers and developers to construct new cryptographic hardware accelerators for ECDSA. Different types of optimizations (either platform dependent or algorithmic) were presented in the literature. In this context, we turn our attention to ECC and propose a new method for generating ECDSA moduli with a predetermined portion that allows one to double the speed of Barrett’s algorithm. Moreover, we take advantage of the advancements in the Artificial Intelligence (AI) field and bring forward an AI-based approach that enhances Schoof’s algorithm for finding the number of points on an elliptic curve in terms of implementation efficiency. Our results represent algorithmic speed-ups exceeding the current paradigm as we are also preoccupied by other particular security environments meeting the needs of governmental organizations.

Image encryption based on elliptic curves (ECs) is emerging as a new trend in cryptography because it provides high security with a relatively smaller key size when compared with well-known cryptosystems. Recently, it has been shown that the cryptosystems based on ECs over finite rings may provide better security because they require the computational cost for solving the factorization problem and the discrete logarithm problem. Motivated by this fact, we proposed a novel image encryption scheme based on ECs over finite rings. There are three main steps in our scheme, where, in the first step, we mask the plain image using points of an EC over a finite ring. In step two, we create diffusion in the masked image with a mapping from the EC over the finite ring to the EC over the finite field. To create high confusion in the plain text, we generated a substitution box (S-box) based on the ordered EC, which is then used to permute the pixels of the diffused image to obtain a cipher image. With computational experiments, we showed that the proposed cryptosystem has higher security against linear, differential, and statistical attacks than the existing cryptosystems. Furthermore, the average encryption time for color images is lower than other existing schemes.

Recent studies have shown that existing elliptic curve‐based cryptographic standards provide backdoors for manipulation and hence compromise the security. In this regard, two new elliptic curves known as Curve448 and Curve25519 are recently recommended by IETF for transport layer security future generations. Hence, cryptosystems built over these elliptic curves are expected to play a vital role in the near future for secure communications. A high‐speed elliptic curve cryptographic processor (ECCP) for the Curve448 is proposed in this study. The area of the ECCP is optimised by performing different modular operations required for the elliptic curve Diffie–Hellman protocol through a unified architecture. The critical path delay of the proposed ECCP is optimised by adopting the redundant‐signed‐digit technique for arithmetic operations. The segmentation approach is introduced to reduce the required number of clock cycles for the ECCP. The proposed ECCP is developed using look‐up‐tables (LUTs) only, and hence it can be ported to any field‐programmable gate array family or standard ASIC libraries. The authors' ECCP design offers higher speed without any significant area overhead to recent designs reported in the literature.

Ciphertext-policy attribute-based encryption (CP-ABE) is a suitable solution for the protection of data privacy and security in cloud storage services. In a CP-ABE scheme which provides an access structure with a set of attributes, users can decrypt messages only if they receive a key with the desired attributes. As the number of attributes increases, the security measures are strengthened proportionately, and they can be applied to longer messages as well. The decryption of these ciphertexts also requires a large decryption key which may increase the decryption time. In this paper, we proposed a new method for improving the access time to the CP using a new elliptic curve that enables a short key size to be distributed to the users that allows them to use the defined attributes for encryption and decryption. Each user has a specially created key which uses the defined attributes for encryption and decryption based on the Diffie-Hellman method. After the implement, the results show that this system saves nearly half of the execution time for encryption and decryption compared to previous methods. This proposed system provides guaranteed security by means of the elliptic curve discrete logarithmic problem.

The rapid proliferation of Internet of Things (IoT) devices and blockchain technology requires robust and efficient cryptographic solutions to ensure secure communication and data integrity. This paper presents an optimized Edward-Elgamal signature algorithm that uses Edward curved signature storage mechanism to improve performance in IoT and blockchain environments to overcome the limitations of schemes, especially Elliptic Curve Digital Signature Algorithm (ECDSA) and Hyper-ECDSA. The proposed method achieved faster signature generation times, with a 33% improvement over ECDSA and a 25% improvement over Hyper-ECDSA, making it more suitable for faster applications. The validation times of the optimized systems were relatively low, with a 32% improvement over ECDSA and a 24% improvement over Hyper-ECDSA. The efficiency of the proposed system was significantly higher, showing a 51% improvement over ECDSA and a 35% improvement over Hyper-ECDSA. The latency was reduced by 33% compared to ECDSA and 26% compared to Hyper-ECDSA, indicating the effectiveness of the optimized system in terms of time-related performance. This paper contributes to the advancement of cryptographic techniques and provides suitable solutions for secure IoT and blockchain applications. The proposed system achieves a highest improvement rate in key performance parameters over existing methods, resulting in a robust solution for modern cryptography requirements.