Catalogue Search | MBRL
Are you sure you want to remove the book from the shelf?
{{itemTitle}}
1,589
result(s) for
"Curves, Elliptic"
Sort by:
Iwasawa Theory, projective modules, and modular representations
This paper shows that properties of projective modules over a group ring
eBook
The Overview of Elliptic Curve Cryptography (ECC)
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.
Journal Article
Design and Implementation of High-Performance ECC Processor with Unified Point Addition on Twisted Edwards Curve
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.
Journal Article
Hashing to Elliptic Curves Through Cipolla–Lehmer–Müller’s Square Root Algorithm
The present article provides a novel hash function
H
to any elliptic curve of
j
-invariant
≠
0
,
1728
over a finite field
F
q
of large characteristic. The unique bottleneck of
H
consists of extracting a square root in
F
q
as well as for most hash functions. However,
H
is designed in such a way that the root can be found by (Cipolla–Lehmer–)Müller’s algorithm in constant time. Violation of this security condition is known to be the only obstacle to applying the given algorithm in the cryptographic context. When the field
F
q
is highly 2-adic and
q
≡
1
(
mod
3
)
, the new batching technique is the state-of-the-art hashing solution except for some sporadic curves. Indeed, Müller’s algorithm costs
≈
2
log
2
(
q
)
multiplications in
F
q
. In turn, original Tonelli–Shanks’s square root algorithm and all of its subsequent modifications have the algebraic complexity
Θ
(
log
(
q
)
+
g
(
ν
)
)
, where
ν
is the 2-adicity of
F
q
and a function
g
(
ν
)
≠
O
(
ν
)
. As an example, it is shown that Müller’s algorithm actually needs several times fewer multiplications in the field
F
q
(whose
ν
=
96
) of the standardized curve NIST P-224.
Journal Article
Compressive sensing techniques based on secure data aggregation in WSNs
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.
Journal Article
Speeding-Up Elliptic Curve Cryptography 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.
Journal Article
Vector bundles on degenerations of elliptic curves and Yang–Baxter equations
In this paper the authors introduce the notion of a geometric associative $r$-matrix attached to a genus one fibration with a section and irreducible fibres. It allows them to study degenerations of solutions of the classical Yang-Baxter equation using the approach of Polishchuk. They also calculate certain solutions of the classical, quantum and associative Yang-Baxter equations obtained from moduli spaces of (semi-)stable vector bundles on Weierstrass cubic curves.
eBook
Cryptanalysis and Improved Image Encryption Scheme Using Elliptic Curve and Affine Hill Cipher
In the present era of digital communication, secure data transfer is a challenging task in the case of open networks. Low-key-strength encryption techniques incur enormous security threats. Therefore, efficient cryptosystems are highly necessary for the fast and secure transmission of multimedia data. In this article, cryptanalysis is performed on an existing encryption scheme designed using elliptic curve cryptography (ECC) and a Hill cipher. The work shows that the scheme is vulnerable to brute force attacks and lacks both Shannon’s primitive operations of cryptography and Kerckchoff’s principle. To circumvent these limitations, an efficient modification to the existing scheme is proposed using an affine Hill cipher in combination with ECC and a 3D chaotic map. The efficiency of the modified scheme is demonstrated through experimental results and numerical simulations.
Journal Article