Truncated Polynomial Rings Research Articles

A model for the canonical algebras of bimodules type (1, 4) over truncated polynomial rings

Let C((ε))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}(\\!(\\varepsilon )\\!)$$\\end{document} be the field of complex Laurent series. We use Galois descent techniques to show that the simple regular representations of the species of type (1,4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1,\\, 4)$$\\end{document} over C((ε))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}(\\!(\\varepsilon )\\!)$$\\end{document} are naturally parametrized by the closed points of Spec(C((ε))[x])∪˙{1,2}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Spec}(\\mathbb {C}(\\!(\\varepsilon )\\!)[x]){\\dot{\\cup }}\\{1,\\,2\\}$$\\end{document}. Moreover, we provide weak normal forms for those representations. We use our representatives of the simple regular representations to describe the canonical algebras associated to the species of type (1, 4) over C((ε))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}(\\!(\\varepsilon )\\!)$$\\end{document}. This suggests a model of those algebras in the sense of the work of Geiss et al. (Invent Math 209(1):61–158, 2017; Math Z 295:1245–1277, 2020).

Secure and Scalable Internet of Things Model Using Post-Quantum MACsec

For the secure deployment of network platforms tailored for IoT devices, the encryption of data transmission is equally as crucial as the process of authentication. In this context, we introduce the Secure and Scalable IoT network (SSI) network platform, designed to accommodate a diverse range of IoT devices. It provides scalability and implements effective many-to-many and end-to-end encryption across extensive regions. With the emergence of quantum computing, secure public key exchange mechanisms have become important. Among the various post-quantum cryptography (PQC) algorithms assessed, Nth Degree Truncated Polynomial Ring Units (NTRUs) have emerged as an optimally suited PQC algorithm for IoT devices constrained by limited computational capabilities. We have integrated NTRUs with SSI as a lightweight PQC solution. Moreover, SSI-PQM (Post-Quantum MACsec) enhances the SSI’s initial authentication structure to minimize PQC-TLS session attempts and protect the SSI’s important configuration information. When applying TLS with PQC for secret key exchange purposes, it was verified that this approach ensures stable performance in IoT environments. Upon the implementation of our proposed SSI-PQM on Raspberry Pi 3B+ based IoT devices, SSI-PQM exhibited acceptable performance at security levels from 80 to 128 and achieved a minimum speed improvement of 161% over RSA at security levels above 160. It can be concluded that SSI-PQM stands out as an effective Zero Trust-based IoT network platform, demonstrating its viability and efficiency in safeguarding data transmission against potential quantum computing threats.

Enhanced Security in Public Key Cryptography: A Novel Approach Combining Gaussian Graceful Labeling and NTRU Public Key Cryptosystem

This research explores an encryption system that combines the Nth-degree Truncated Polynomial Ring Unit (NTRU) public key cryptosystem with Gaussian Graceful Labeling. This process assigns distinct labels to a graph's vertices, resulting in successive Gaussian integers. The NTRU method offers enhanced security and efficient key exchange. The communication encryption process depends on integers P, a, and b, with P being the largest prime number in the vertex labeling. The original receivers are the vertex labeling with the largest prime number coefficient, while all other receivers receive messages from the sender. A polynomial algebraic mixing system and a clustering principle based on the abecedarian probability proposition are used in NTRU encryption and decryption. The choice of relatively prime integers p and q in NTRU plays a role in the construction of polynomial rings used for encryption and decryption, with specific choices and properties designed to ensure scheme security.

Lightweight Computational Complexity Stepping Up the NTRU Post-Quantum Algorithm Using Parallel Computing

The Nth-degree Truncated polynomial Ring Unit (NTRU) is one of the famous post-quantum cryptographic algorithms. Researchers consider NTRU to be the most important parameterized family of lattice-based public key cryptosystems that has been established to the IEEE P1363 standards. Lattice-based protocols necessitate operations on large vectors, which makes parallel computing one of the appropriate solutions to speed it up. NTRUEncrypt operations contain a large amount of data that requires many repetitive arithmetic operations. These operations make it a strong candidate to take advantage of the high degree of parallelism. The main costly operation that is repeated in all NTRU algorithm steps is polynomial multiplication. In this work, a Parallel Post-Quantum NTRUEncrypt algorithm called PPQNTRUEncrypt is proposed. This algorithm exploits the capabilities of parallel computing to accelerate the NTRUEncrypt algorithm. Both analytical and Apache Spark simulation models are used. The proposed algorithm enhanced the NTRUEncrypt algorithm by approximately 49.5%, 74.5%, 87.6%, 92.5%, 93.4%, and 94.5%, assuming that the number of processing elements is 2, 4, 8, 12, 16, and 20 respectively.

Hardness of Module-LWE with Semiuniform Seeds from Module-NTRU

The module learning with errors (MLWE) problem has attracted significant attention and has been widely used in building a multitude of lattice-based cryptographic primitives. The hardness of the MLWE problem has been established for several variants, but most of the known results require the seed distribution (i.e., the distribution of matrix A ) to be the uniform distribution. In this paper, we show that under the Module-N-th degree Truncated polynomial Ring Units (NTRU) (MNTRU) assumption, the search MLWE problem can still be hard for some distributions that are not (even computationally indistinguishable from) the uniform distribution. Specifically, we show that if the seed distribution is a semiuniform distribution (namely, the seed distribution can be publicly derived from and has a “small difference” to the uniform distribution), then for appropriate settings of parameters, the search MLWE problem is hard under the MNTRU assumption. Moreover, we also show that under the appropriate settings of parameters, the search learning with errors over rings problem with semiuniform seeds can still be hard under the NTRU assumption due to our results for the search MLWE problem with semiuniform seeds being rank-preserving.

A Comparative Study of Post-Quantum Cryptographic Algorithm Implementations for Secure and Efficient Energy Systems Monitoring

The Internet of Things (IoT) has assumed a pivotal role in the advancement of communication technology and in our daily lives. However, an IoT system such as a smart grid with poorly designed topology and weak security protocols might be vulnerable to cybercrimes. Exploits may arise from sensor data interception en route to the intended consumer within an IoT system. The increasing integration of electronic devices interconnected via the internet has galvanized the acceptance of this technology. Nonetheless, as the number of users of this technology surges, there must be an aligned concern to ensure that security measures are diligently enforced within IoT communication systems, such as in smart homes, smart cities, smart factories, smart hospitals, and smart grids. This research addresses security lacunae in the topology and configuration of IoT energy monitoring systems using post-quantum cryptographic techniques. We propose tailored implementations of the Rivest–Shamir–Adleman (RSA), N-th degree Truncated Polynomial Ring Units (NTRU), and a suite of cryptographic primitives based on Module Learning With Rounding (Saber) as post-quantum cryptographic candidate algorithms for IoT devices. These aim to secure publisher–subscriber end-to-end communication in energy system monitoring. Additionally, we offer a comparative analysis of these tailored implementations on low-resource devices, such as the Raspberry Pi, during data transmission using the Message Queuing Telemetry Transport (MQTT) protocol. Results indicate that the customized implementation of NTRU outperforms both SABER and RSA in terms of CPU and memory usage, while Light SABER emerges as the front-runner when considering encryption and decryption delays.

Comparative Analysis between Elgamal and NTRU Algorithms and their implementation of Digital Signature for Electronic Certificat

The emergence of electronic certificates, which are official documents in the form of digital files transmitted via the internet, facilitates the exchange of information. However, internet use has risks, such as data theft for fabricating and modifying information. This problem can be solved by applying a digital signature. This problem can be solved by applying a digital signature. The main concern in this research is how to perform a comparative analysis between asymmetric cryptographic Elgamal and NTRU (Nth-Degree Truncated Polynomial Ring) algorithms and their implementation of a digital signature as an effort to improve information security in electronic certificates. The stages of the research method are divided into the key generation process, signing, and verification. In the signing and verification process, the SHA-512 hash function is also used for hashing messages to be encrypted-decrypted and QR Code as the signature. Comparison of performance of NTRU with Elgamal algorithms required running at a pdf extension with security levels 80,128,192, 256 bits will be obtained from the templates.office.com website. The results obtained that the El Gamal algorithm is better than the NTRU algorithm, but at a higher security level, the NTRU algorithm is better than the Elgamal algorithm. In the verification experiment that has been carried out, it can be concluded that by using SHA-512 as a hash function, the N parameter used for NTRU must be greater than or equal to 512 to avoid error results from verification.

Resolutions for truncated Ore extensions

We extend recent results in order to construct projective resolutions for modules over twisted tensor products of truncated polynomial rings. We begin by taking note of the conditions necessary to think of these algebras as a type of Ore extension. We then use this parallel with Ore extensions to develop a method for constructing projective resolutions. Finally, we use the new construction to compute a resolution for a family of examples.

Smart contract token-based privacy-preserving access control system for industrial Internet of Things

Due to mobile Internet technology's rapid popularization, the Industrial Internet of Things (IIoT) can be seen everywhere in our daily lives. While IIoT brings us much convenience, a series of security and scalability issues related to permission operations rise to the surface during device communications. Hence, at present, a reliable and dynamic access control management system for IIoT is in urgent need. Up till now, numerous access control architectures have been proposed for IIoT. However, owing to centralized models and heterogeneous devices, security and scalability requirements still cannot be met. In this paper, we offer a smart contract token-based solution for decentralized access control in IIoT systems. Specifically, there are three smart contracts in our system, including the Token Issue Contract (TIC), User Register Contract (URC), and Manage Contract (MC). These three contracts collaboratively supervise and manage various events in IIoT environments. We also utilize the lightweight and post-quantum encryption algorithm-Nth-degree Truncated Polynomial Ring Units (NTRU) to preserve user privacy during the registration process. Subsequently, to evaluate our proposed architecture's performance, we build a prototype platform that connects to the local blockchain. Finally, experiment results show that our scheme has achieved secure and dynamic access control for the IIoT system compared with related research.

Security Analysis of DBTRU Cryptosystem

DBTRU was proposed by Thang and Binh in 2015. As a variant of NTRU, the integer polynomial ring is replaced by two binary truncated polynomial rings . DBTRU has some advantages over NTRU in terms of security and performance. In this paper, we propose a polynomial-time linear algebra attack against the DBTRU cryptosystem, which can break DBTRU for all recommended parameter choices. The paper shows that the plaintext can be achieved in less than 1 s via the linear algebra attack on a single PC.

Motivic cohomology of fat points in Milnor range via formal and rigid geometries

We present a formal scheme based cycle model for the motivic cohomology of the fat points defined by the truncated polynomial rings $$k[t]/(t^m)$$ with $$m \ge 2$$ , in one variable over a field k. We compute their Milnor range cycle class groups when the field has sufficiently many elements. With some aids from rigid analytic geometry and the Gersten conjecture for the Milnor K-theory resolved by M. Kerz, we prove that the resulting cycle class groups are isomorphic to the Milnor K-groups of the truncated polynomial rings, generalizing a theorem of Nesterenko-Suslin and Totaro.

Review on Cryptography and Network Security Zero Knowledge Technique in Blockchain Technology

A huge amount of data has been generated every day across the world by organizations with the emergence of the World Wide Web, social networks, along with e-commerce applications. To safeguard the network along with data transmission that occurs over a wireless network, cryptography along with network security (NS) is utilized. In data transmission over a wireless untrustworthy network, securing data is a major concern. Providing security to the wireless sensor networks is highly significant; NS provides security not only to the end system but also to all over the network system. Consequently, NS, cryptography, present progress in NS, linear cryptanalysis (LC), together with differential cryptanalysis (DC) are explicated in this work. The outcomes of the encryption time (ET) versus file size for data encryption standard (DES), advanced encryption standard (AES), 3DES, Rivest-Shamir-Adleman (RSA), blowfish, and the decryption time (DT) versus file size for RSA, AES, modified RSA (MRSA), and nth degree truncated polynomial ring units (NTRU) are also evaluated in this paper.

Post-quantum cryptography-driven security framework for cloud computing

Abstract Data security in the cloud has been a major issue since the inception and adoption of cloud computing. Various frameworks have been proposed, and yet data breach prevails. With encryption being the dominant method of cloud data security, the advent of quantum computing implies an urgent need to proffer a model that will provide adequate data security for both classical and quantum computing. Thus, most cryptosystems will be rendered susceptible and obsolete, though some cryptosystems will stand the test of quantum computing. The article proposes a model that comprises the application of a variant of McEliece cryptosystem, which has been tipped to replace Rivest–Shamir–Adleman (RSA) in the quantum computing era to secure access control data and the application of a variant of N-th degree truncated polynomial ring units (NTRU) cryptosystem to secure cloud user data. The simulation of the proposed McEliece algorithm showed that the algorithm has a better time complexity than the existing McEliece cryptosystem. Furthermore, the novel tweaking of parameters S and P further improves the security of the proposed algorithms. More so, the simulation of the proposed NTRU algorithm revealed that the existing NTRU cryptosystem had a superior time complexity when juxtaposed with the proposed NTRU cryptosystem.

Comparative Analysis of RSA and NTRU Algorithms and Implementation in the Cloud

The emergence of cloud computing platforms makes it easier to connect and collaborate globally without setting up additional infrastructures such as servers and data centers. This causes the emergence of threats to data security against digital information. This security threat can be overcome by cryptography. Examples of cryptographic algorithms are RSA and NTRU. The main concern that arises in this research is how to perform a comparative analysis between asymmetric cryptographic algorithms, RSA (Rivest-Shamir-Adleman) and NTRU (Nth-Degree Truncated Polynomial Ring) algorithms and their implementation in cloud storage. Comparison of performance between the RSA and NTRU algorithms at security levels 80, 112, 128, 160, 192, and 256 bits by running 5 – 1000 data, the results obtained that the running time of the key generation process and encryption of the NTRU algorithm is more efficient than the RSA algorithm. Wiener's Attack test on the RSA algorithm and LLL Lattice Basis Reduction on the NTRU algorithm. NTRU algorithm has a more secure level of resilience, so that it can be said that the NTRU algorithm is more recommended for cloud storage security.

Post-Quantum Blockchain-Based Secure Service Orchestration in Multi-Cloud Networks

Existing network service provisioning and lifecycle management (LCM) workflows rely on heterogeneous devices from multiple vendors and collaboration between multiple network actors, which can lead to numerous trust management and interoperability issues. Blockchain networks (BCNs) are a revolutionary way to establish trust in untrusted environments. In this complex environment, BCNs can help ensure transparency and security for network service LCM. However, BCNs are also vulnerable to quantum attacks. Advances in quantum computing will challenge the security of existing blockchain technology based on Public Key Infrastructure (PKI) technologies. In this paper, we explore how network services can be managed in a multiple administrative scenario. Our approach uses BCNs to track the operational steps of network service instantiation metrics while benefiting from the security features of post-quantum cryptography (PQC). Together with the use of N-th degree Truncated polynomial Ring Units (NTRU) as an example of a PQC algorithm that relies on the parallelization power of the Toom-Cook computation method with different security levels, we have shown that Quorum can provide a lower average time-to-write value compared to other BCNs considered (Ethereum and Hyperledger). At the end of the paper, we discuss the evaluation results and future directions regarding the coexistence of PQC algorithms and BCNs for network service orchestration and service federation between multiple administrative domains.

Quantum Secure Communication Between Service Provider and Sim

In resource-constrained devices such as Subscriber Identity Module (SIM), the possibility of using authentication and key exchange methods, which include the mutual verification of two parties to ensure the security of communication, provides a significant advantage. As the cost efficient computations may not be efficiently performed on resource-constrained devices, security is the main problem for SIMs. With the advancing technology and the use of quantum computers, it is predicted that this may lead to threats. It is a well-known fact that security on traditional public key cryptosystems will become vulnerable by using quantum computers due to Shor’s algorithm. In this paper, two different resistant to quantum attacks structures are proposed to ensure secure communication between SIM and service providers. In the proposed methods, Advanced Encryption Standard (AES-256) is used for communication with resource-constrained devices, and N-th degree Truncated polynomial Ring Units (NTRU) encryption system is used for communication with servers. Two methods are proposed, one with the private key and one without the private key in the production phase. This protocol provides authentication, data privacy and integrity for post-quantum SIM cards. The proposed method is inspired by the FLAT (Federated Lightweight Authentication) protocol. However, the main difference from the FLAT protocol is that it has more nodes and is resistant to quantum attacks.

Enhanced Health-Care Protection Using Advanced Encryption Standard and Diffie Hellman Key Exchange Algorithm

In the modern world as technology develops, it is very crucial to secure, share and store the data. Especially, when it deals with medical data it is very important to secure that sensitive information. Sensitive information might be the pulse, temperature, or any disease-related symptoms. Such factors must not be shared unless or until with the users’ permission. When any services or any user overcome the given act then the victim will be addressed in the court of law for the denial of service. Such activities must not be welcomed as in the world of modern technology; it is easy to secure the data as well as to share the data. Hence, it is significant to overcome privacy issues and security attacks. The co-factors associated with transmitting and securing data involves bandwidth and energy. Bandwidth and energy play a vital role during the transmission of data. Hence in this paper, we introduce a novel system of creating a portal for the patients who can enroll with their medical details and fix appointments with the doctor and get the prescription. First, the Nth Degree Truncated Polynomial Ring Unit (NTRU) method is used to encrypt the data collected where the private chats among the patient can be secured. Those data will be transmitted to the nearby cloud-let in an energy-efficient manner. Secondly, patients can communicate with other patients about their diseases where the disease based group can be created to share the information. Thirdly, patient’s information is divided into tables and stored in the cloud for proper protection. As group chats and personal chats are involved, security is a must during communication and as an additional feature, the data are collected in a buffer format in the mobile, which paves way for reducing the bandwidth and energy consumption.

Research perspectives on fully homomorphic encryption models for cloud sector1

As there is a continuous delivery of big data, the researchers are showing interest in the applications of cloud computing concerning privacy, and security. On the other hand, many researchers and experts of cybersecurity have commenced on a quest for improving the data encryption to the models of big data and applications of cloud computing. Since many users of the cloud become public cloud services, confidentiality turns out to be a more compound problem. To solve the confidentiality problem, cloud clients maintain the data on the public cloud. Under this circumstance, Homomorphic Encryption (HE) appears as a probable solution, in which the information of the client is encrypted on the cloud in such a process that it permits few manipulation operations without decryption. The main intent of this paper is to present the systematic review of research papers published in the field of Fully Homomorphic Encryption (FHE) over the past 10 years. The encryption scheme is considered full when it consists of plaintext, a ciphertext, a keyspace, an encryption algorithm, and a decryption algorithm. Hence, the review mostly concentrates on reviewing more powerful and recent FHE. The contributions using different algorithms in FHE like Lattice-based, integer-based, Learning With Errors (LWE), Ring Learning With Errors (RLWE), and Nth degree Truncated polynomial Ring Units (NTRU) are also discussed. Finally, it highlights the challenges and gaps to be addressed in modeling and learning about competent, effectual, and vigorous FHE for the cloud sector and pays attention to directions for better future research.

Zero-cycles with modulus and relative K-theory

Let D be an effective Cartier divisor on a regular quasiprojective scheme X of dimension d≥1 over a field. For an integer n≥0, we construct a cycle class map from the higher Chow groups with modulus {CHn+d(X|mD,n)}m≥1 to the relative K-groups {Kn(X,mD)}m≥1 in the category of pro-abelian groups. We show that this induces a proisomorphism between the additive higher Chow groups of relative 0-cycles and the reduced algebraic K-groups of truncated polynomial rings over a regular semilocal ring which is essentially of finite type over a characteristic zero field.

NTRU Over Galois Rings

Abstract As a cryptosystem, Nth Truncated Polynomial Ring (NTRU) is established on the fast and easy calculation. Improving the security is aimed by enlarging a ring where the processes execute and enhancing the number of a private key and a public key. In this study, NTRU takes over the Galois rings and is analysed by adding a new private key.