Elliptic curves having non-trivial p-part of Shafarevich-Tate groups and satisfying the Birch and Swinnerton-Dyer conjecture modulo p

An explicit bound for Siegel zeros and the torsion of elliptic curves with complex multiplication

ABSTRACT Cognitive radio for vehicular ad hoc networks (CR-VANET) plays a key role in managing the spectrum dynamically and provides data communication in smart transportation. However, the security aspect is threatened by Sybil attacks where the adversary creates multiple fake identities in the network to hinder performance, damage topology, and mount denial of service attacks. To address these challenges, we present Vision-Augmented Split-Attention Neural Architectures for Sybil Resilience via Chaos-Driven Secure Elliptic Key Synthesis to Assured Data Exchange in CR-VANETs (Fuzz-CViAt_DuBe), a new approach. The proposed comprises of (i) Cluster Head selection with the Sooty Tern Maximizer for efficient communication; (ii) Sybil attack detection using Convolutional Neural Networks Augmented by Vision Transformers with split attention enhanced by the Dung Beetle Adaptive Optimizer; and (iii) Cryptographic security with the Fuzz-Resilient Chaotic Elliptic Curve Cryptographic Infrastructure. In simulations it is seen that there is a very much enhancement in the network performance. The proposed system increases the packet delivery ratio by 97.4%, improves throughput by 95.8%, and reduces latency by 88.3%. Additionally, the security rate is enhanced by 98.5%, while encryption time for 100KB of data is reduced to 15.2 s, demonstrating its superior performance over existing models. These findings highlight the benefits of the Fuzz-CViAt_DuBe framework in protecting CR-VANETs against Sybil attacks and enabling safe communication, providing solid grounds for improved future intelligent transportation systems.

Abstract We investigate limit linear series on chains of elliptic curves, giving a simple proof of a conjecture of Farkas stating the existence of curves with a theta-characteristic with a given number of sections for the expected range of genera. Using the additional structure afforded by considering limit linear series on chains of elliptic curves, we find examples of reducible Brill–Noether loci, admitting at least two components, with and without a theta-characteristic respectively. This allows us to display reducible Hilbert schemes for $$r\ge 3$$ r ≥ 3 and $$d=g-1$$ d = g - 1 . We also give examples of Brill–Noether loci with three components. On the positive side, we provide optimal bounds on the degree under which Brill–Noether loci are irreducible when $$r=2$$ r = 2 .

Data security has become one of the primary concerns, particularly for images, as conventional encryption methods such as the Vigenere cipher or ring [Formula: see text] -based methods are no longer robust against modern types of attacks. To resolve the issue, this paper introduces a novel hybrid image encryption technique that combines elliptic curve cryptography (ECC), finite field extensions integrating the AJ chaotic map for dynamic parameters generation and image-dependent key adaptation. First, the method depends on the pseudorandom selection of an elliptic curve over the field [Formula: see text] designed based on an irreducible polynomial of degree 8. Second, two substitution tables are designed based on discrete logarithm and exponentiation operations to improve the impact of the confusion and diffusion. Numerous in-depth security experiments, including entropy, NPCR, UACI, correlation coefficients, and the NIST statistical test, have been performed, indicating high cryptographic robustness against known attacks. The findings affirm the performance of the proposed method in achieving an average entropy of 7.9998 bits per pixel, a correlation coefficient of adjacent pixels less than 0.001, NPCR of 99.79%, and UACI of 33.59% for the Peppers color image. In addition, our technique achieves a competitive execution time, confirming both its security and efficiency. Therefore, it is evident that the employment of ECC over a finite body along with a dynamically designed SBox is an ideal high-performance method for privacy protection, secure data storage, and trustworthy transmission of sensitive data.

The workshop focused on various directions of arithmetic statistics in algebra and number theory. These include statistical problems for random polynomials and varieties, probabilistic Galois theory, and counting and distribution problems for algebraic functions, algebraic number fields, elliptic curves, L -functions, as well as arithmetic problems in non-abelian settings (eg, arithmetic statistics for algebraic groups).

ABSTRACT Recently, certificate management has been considered one of the crucial tasks of public key infrastructure (PKI), which also provides crucial privacy and security over the blockchain. Moreover, the recent discovery of multiple promising approaches provides less security, threats, incremental deployment challenges, and inefficiency, which strongly impact blockchain. On account of these issues, this research established a new strategy for certificate management, Hybrid Certificate Verification (Hy‐Cert) for Public Key Infrastructure. The contribution of this work is deployed in two phases, including the verification phase and key generation phase, which contemplate secure communication for certificate management. In this approach, the trapdoor, as well as proposed efficiency factor‐based verification, is implemented as a two‐step verification, which highly scrutinizes the authenticity of users and allows superior and safe communication. On the other hand, Hybrid public key generation (Hy‐Key) using a hybrid public key infrastructure is developed, which resembles the traditional key generation mechanism for encryption and decryption of data over the information‐centric network. In Hy‐Key, more powerful techniques such as Rivest, Shamir, Adleman (RSA) and Elliptic Curve Cryptography (ECC) algorithms are associated and undergo enormous security measures for better communication during certificate management. The system validation was performed on Students mark sheet and COVID‐19 datasets by varying users (500–2500) and transactions (1000–4000). For 2500 users, Hy‐Cert achieved 0.59 ms certification delay, 1.09 ms delay, 0.91 genuine user rate, 254.60 KB memory, 3.29 ms responsiveness, and 5.50 ms time. Similarly, for 4000 transactions, the corresponding values were 0.50 ms, 1.26 ms, 0.89, 257.93 KB, 3.75 ms, and 5.95 ms, respectively.

As contemporary power grids are becoming more complex with the integration of renewable energy sources, distributed generation, and smart grid technologies. Conventional contingency analysis techniques, based on centralized architectures and static rule-based evaluations, tend to be inadequate in real-time fault detection, automated response, and cybersecurity. This paper suggests a hybrid approach that combines machine learning algorithms with blockchain technology to improve both predictive intelligence and security of contingency analysis. For the IEEE 30-bus test case, different line outage and generator failure cases were simulated. Different machine learning models, such as random forest (RF), support vector machine (SVM), and gradient boosting (GB), were trained to classify and predict these contingencies. In parallel, cryptographic primitives like advanced encryption standard (AES), Rivest–Shamir–Adleman (RSA), and elliptic curve cryptography (ECC) were tested in a blockchain setting to provide security for event data and enable automatic recovery steps through smart contracts. Outcomes illustrate that the GB showed the maximum fault classification rate (93.4%), and ECC ensured light yet robust data protection for blockchain activities. Against the conventional system, the designed model enhanced the response time in case of faults, accuracy, and system fault tolerance. This two-layer mechanism presents a scalable, proactive, and cyber-safe mechanism for the power grid in the future.

This paper presents a novel modified seagull monarch butterfly optimization (MSMBO) algorithm, with a multi-objective focus on privacy and personalization in the fitness recommender system using a refined three-tier deep learning structure. The method is divided into three phases. In the first phase, fitness data from wearable devices undergoes preprocessing to eliminate noise and standardize features. The second phase incorporates improved elliptic curve cryptography (IECC) alongside the MSMBO to encrypt user data securely, ensuring privacy in cloud storage. This phase also enhances neural network performance by optimizing weights and hyperparameters through feature selection, effectively reducing data complexity while boosting accuracy. In the third phase, ConvCaps extracts spatial data features, while Bi-LSTM identifies temporal dependencies. The proposed system balances multiple objectives like novelty, accuracy, and precision, while safeguarding user data through robust encryption. With the experimental findings, our suggested method performs better than current existing models, especially in heart rate prediction and fitness pattern identification. The overall outcome makes the system ideal for privacyconscious, personalized fitness recommendations. The model’s shows significant improvement in mean squared error (MSE), normalized mean squared error (NMSE), and mean absolute percentage error (MAPE), thus verifying its effectiveness in secure, real-time fitness tracking.

Odd and Even Elliptic Curves with Complex Multiplication

The blistering growth of the Mobile Cloud Computing (MCC) has enabled the smooth usage of omnipresent services, but it poses a great security issue because of the intrinsic resource limitations of mobile devices. Conventional authentication systems based on either large key sizes or heavy modular exponentiation are expensive in terms of computation and energy usage for a mobile device. In this paper, a lightweight mutual authentication protocol that can be applied in a mobile cloud environment is proposed, based on Elliptic Curve Cryptography (ECC) and 1-way hash functions that can be used to provide high levels of security at minimum overhead. In order to determine the effectiveness of the proposed scheme, an extensive statistical and performance study was done. Comparative results indicate that the protocol reduces computational latency by approximately 35-40% compared to standard RSA-based frameworks. In particular, processing time on the mobile client side will be kept at less than 15ms, keeping battery life intact. From a communication perspective, the protocol minimizes the exchange to three messages, reducing total bit-overhead by 28%. Performance evaluations using the BETH and Multi-Cloud Kaggle datasets show that the proposed protocol decreases computational delay by more than 90% compared to traditional RSA-2048 frameworks, while achieving a 98.5% throughput success rate even under high traffic conditions. The protocol is also checked through formal security verification with BAN Logic and AVISPA simulation tools and found to be resistant to common attack vectors, such as Man-in-the-Middle (MitM), replay attacks, and impersonation. More statistical testing additionally shows that the keys created by the generated session have high levels of entropy and guarantee Perfect Forward Secrecy (PFS). The results indicate that the suggested protocol offers the best tradeoff ratio between the security level and operation efficiency and is, therefore, very appropriate in real-time mobile cloud applications in which the latency and the life of the device matter.

With the growing use of unmanned aerial vehicles (UAVs) or drones, ensuring secure communication in open-access environments has become essential. While authentication systems based on Elliptic Curve Cryptography (ECC) are widely adopted due to their efficiency and reduced key sizes, further improvements are needed to enhance communication security and computational performance. To address this, we introduce LAPHECC, a lightweight authentication protocol that replaces ECC with Hyperelliptic Curve Cryptography (HECC). HECC is particularly well-suited for resource-limited devices, as it provides stronger security per bit and requires even smaller key sizes than ECC. The protocol incorporates a preregistration stage and employs a Schnorr-based zero-knowledge session key update mechanism to guarantee both forward and backward secrecy. Security evaluations show that LAPHECC delivers minimal computational and communication overhead while ensuring core security properties such as confidentiality, mutual authentication, and resistance to replay and impersonation attacks.

For every group [Formula: see text], there exists an intermediate modular curve [Formula: see text]. In this paper, we determine all curves [Formula: see text] with infinitely many points of degree [Formula: see text] over [Formula: see text]. To do that, we developed a method to compute possible degrees of rational morphisms from [Formula: see text] to an elliptic curve.

Abstract The compactly supported $\mathbb {A}^1$ -Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an analogue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties $\mathrm {K}_0(\mathrm {Var}_k)$ taking values in the Grothendieck-Witt ring $\mathrm {GW}(k)$ of the base field k . The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, the first author and Pál construct a power structure on $\mathrm {GW}(k)$ and show that the compactly supported $\mathbb {A}^1$ -Euler characteristic respects these two power structures for $0$ -dimensional varieties, or equivalently étale k -algebras. In this paper, we define the class $\mathrm {Sym}_k$ of symmetrisable varieties to be those varieties for which the compactly supported $\mathbb {A}^1$ -Euler characteristic respects the power structures and study the algebraic properties of the subring $\mathrm {K}_0(\mathrm {Sym}_k)$ of symmetrisable varieties. We show that it includes all cellular varieties, and even linear varieties as introduced by Totaro. Moreover, we show that it includes non-linear varieties such as elliptic curves. As an application of our main result, we compute the compactly supported $\mathbb {A}^1$ -Euler characteristics of symmetric powers of Grassmannians and certain del Pezzo surfaces.

The fast expansion of the Internet of Things (IoT) has increased the need for strong security measures to protect the enormous network of interconnected devices. This paper proposes a unique approach that combines optimization, intuitive design principles, and Least Weighted Elliptic Curve Cryptography (LWECC) to improve IoT device security while reducing power consumption. The proposed optimization strategy focuses on lowering computational overhead, which is critical for IoT devices with limited energy and processing power. The proposed method significantly reduces the amount of energy required for cryptographic operations by carefully selecting appropriate elliptic curves and optimizing cryptographic algorithms, ensuring that IoT devices may continue to function without compromising security. Furthermore, by selecting elliptic curves with minimal attack vulnerability, the use of LWECC provides an additional layer of protection. This technique ensures that, even in the face of emerging threats, IoT devices remain highly resilient, reducing the chance of security breaches while preserving functionality without using excessive power. Experimental results show a power consumption of only 0.156 W and 0.25 W for memory and router topologies, respectively, with an error margin of 0.01. The stated error margin pertains to the simulation-based evaluation of transmission-level data handling within the LWECC-enabled memory/router pipeline, rather than the risk of physical memory-cell failure or fabrication yield. The value shows the maximum amount of packet/data-stream loss detected during encrypted data transfer, rather than hardware memory reliability.

A spinor definition of matter is extended to simplest cycles in interval [0,1] and to permutations of quartic-cubic roots in elliptic curves. Minkowski spacetime and Mandelstam plane are linked with wave vectors due to rotation on real interval. Metrical geometry is discussed by rational triangles which are connected with modular invariants. An adiabatic solution in terms of cyclotomic units for constant elliptic invariants is derived and extended to iterated invariants. The pseudo-congruences allow to formulate coupling constants. Iterated invariants connect a scale factor with diagram expansions in different eras in bifurcating k-components.

Abstract Let $p$ be an odd prime, and let $E_1$ and $E_2$ be two elliptic curves defined over a number field $K$ , with good ordinary reduction at $p$ . We compare the $\Lambda$ -ranks and (generalized) Iwasawa invariants of the Pontryagin duals of the Selmer groups of $E_1$ and $E_2$ over ${\mathbb{Z}}_p^d$ -extensions $\mathbb{L}_\infty$ of $K$ for general $d \ge 1$ under the hypothesis that $E_1[p^i] \cong E_2[p^i]$ as Galois modules for a sufficiently large $i$ . This generalizes and complements previous work over ${\mathbb{Z}}_p$ -extensions. The comparison of generalized Iwasawa invariants is related via an up-down approach to the comparison of the variation of classical Iwasawa invariants over the ${\mathbb{Z}}_p$ -extensions of $K$ which are contained in $\mathbb{L}_\infty$ .

Abstract We present , a zkSNARK solution for large-scale matrix multiplication. Classical zkSNARK protocols typically underperform in data analytic contexts, hampered by the large size of datasets and the superlinear nature of matrix multiplication. excels in its scalability. The prover time of scales linearly with respect to the number of non-zero elements in the input matrices. For $$n \times n$$ n × n matrix multiplication with N non-zero elements across three input matrices, employs a structured reference string (SRS) of size O ( n ), and achieves RAM usage of $$O(N+n)$$ O ( N + n ) , transcript size of $$O(\log n)$$ O ( log n ) , prover time of $$O(N+n)$$ O ( N + n ) , and verifier time of $$O(\log n)$$ O ( log n ) . The prover time, notably at $$O(N+n)$$ O ( N + n ) and surpassing all existing protocols, includes $$O(N+n)$$ O ( N + n ) field multiplications and O ( n ) exponentiations and pairings within bilinear groups. These efficiencies make effective for linear algebra on large matrices common in real-world applications. We evaluated with $$2^{15} \times 2^{15}$$ 2 15 × 2 15 input matrices each containing 1 G non-zero integers, which necessitate 32 T integer multiplications in naive matrix multiplication. recorded prover and verifier times of 150.84s and 0.56s, respectively. When applied to $$1M \times 1M$$ 1 M × 1 M sparse matrices each containing 1 G non-zero integers, it demonstrated prover and verifier times of 1, 384.45s and 0.67s. Our approach outperforms current zkSNARK solutions by successfully handling the large matrix multiplication task in experiment. We extend matrix operations from field matrices to group matrices, formalizing group matrix algebra. This mathematical advancement brings notable symmetries beneficial for high-dimensional elliptic curve cryptography. By leveraging the bilinear properties of our group matrix algebra in the context of the two-tier commitment scheme, achieves efficiency gains over previous matrix multiplication arguments. To accomplish this, we extend and enhance Bulletproofs to construct an inner product argument featuring a transparent setup and logarithmic verifier time.

A Multi-Population Genetic Algorithm for Secure and Efficient Elliptic Curve Parameter Generation

ABSTRACT In the Internet of Vehicles (IoV), secure and low-latency authentication is challenging due to high mobility and computational constraints. This paper proposes a certificateless conditional privacy-preserving authentication (CPPA) protocol based on elliptic curve cryptography, eliminating bilinear pairing and certificate management overhead. A split-key mechanism mitigates key escrow, while a seed-based pseudonym approach enables privacy-preserving and conditionally traceable identity management. The scheme also supports batch verification to improve efficiency under high traffic loads. Security and performance analyses demonstrate that the protocol achieves mutual authentication, message integrity, conditional privacy, and traceability with reduced computational and communication overhead compared to existing approaches.