Number System Research Articles

Symmetric Encryption Algorithms in a Polynomial Residue Number System

In this paper, we develop the theoretical provisions of symmetric cryptographic algorithms based on the polynomial residue number system for the first time. The main feature of the proposed approach is that when reconstructing the polynomial based on the method of undetermined coefficients, multiplication is performed not on the found base numbers but on arbitrarily selected polynomials. The latter, together with pairwise coprime residues of the residue class system, serve as the keys of the cryptographic algorithm. Schemes and examples of the implementation of the developed polynomial symmetric encryption algorithm are presented. The analytical expressions of the cryptographic strength estimation are constructed, and their graphical dependence on the number of modules and polynomial powers is presented. Our studies show that the cryptanalysis of the proposed algorithm requires combinatorial complexity, which leads to an NP‐complete problem.

PaReNTT: Low-Latency Parallel Residue Number System and NTT-Based Long Polynomial Modular Multiplication for Homomorphic Encryption

PaReNTT: Low-Latency Parallel Residue Number System and NTT-Based Long Polynomial Modular Multiplication for Homomorphic Encryption

German nominal number interpretation in an impaired mental lexicon

Abstract There is an ongoing debate on how speakers and listeners process and interpret information in a morphological system that is very complex and not very transparent. A well-known test case is the German nominal number system. In this paper we employ discriminative learning (e.g., Ramscar & Yarlett, 2007; Baayen et al., 2011, 2019) to test whether discriminative learning networks can be used to better understand the processing of German number. We analyse behavioral data obtained from a patient with primary progressive aphasia (Domahs et al., 2017), and the unimpaired system. We test a model that implements the traditional cues borrowed from the schema approach (Köpcke, 1988, 1993; Köpcke et al., 2021), and compare it to a model that uses segmental and phonotactic information only. Our results for the unimpaired system demonstrate that a model based on only biphones as cues is better able to predict the number of a given word-form than a model using structural phonological cues. We also test whether a discriminative learning model can predict the number decisions by the aphasic patient. The results demonstrate that a biphone-based discriminative model trained on the patient’s responses is superior to a structure-based model in approximating the patient’s behavior.

A STUDY ON MATHEMATICIANS OF ANCIENT INDIA AND THEIR CONTRIBUTIONS

Indian mathematics has its roots in Vedic literature. It is imperative to know about the ancient, medieval and modern time Indian mathematicians and their contribution in the field of Mathematics. Indian mathematicians made significant early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed here. Ancient Indian mathematicians have contributed immensely to the field of mathematics. The invention of zero is attributed to Indians and this contribution outweighs all other made by any other nation since it is the basis of the decimal number system, without which no advancement in mathematics would have been possible. The number system used today was invented by Indians and it is still called Indo-Arabic numerals because Indians invented them and the Arab merchants took them to the western world. It is also noticed that there is a distinct and inequitable neglect off the contributions of the sub-continent. Many of the developments of Indian mathematics remain almost completely ignored, or worse or attributed to the scholars of other nationalities, often European. However a few historians (mainly European) are reluctant to acknowledge the contributions of Indian mathematicians. They believe Indians borrowed the knowledge of mathematics from Greeks. Even after all those criticisms it is a great pleasure for us to have been born in such a country where so many great mathematicians like Aryabhata, Brahmagupta, Srinivasa Ramanujan, P.C. Mahalanobis , Calyampudi Radhakrishna Rao, D.R. Kaprekar, Satyendra Bose ,Bhaskara I, Mahavira, Bhaskara II and many others born.

The Archimedean Origin of Modern Positional Number Systems

A symbolic analysis of Archimedes’s periodical number system is developed, from which a natural link emerges with the modern positional number systems with zero. After the publication of Fibonacci’s Liber Abaci, the decimal Indo-Arabic positional system was the basis of the algorithmic and algebraic trend of modern mathematics, but even if zero plays a crucial role in algebra and mathematical analysis, zeroless positional systems show the same capability of producing efficient arithmetical algorithms based on operation tables over digits. The crucial role of digits is assessed, by considering a representation of numbers based on strings in lexicographic order. A new algorithm for the determination of decimal periods is presented by remarking on the cruciality of this topic in number theory. Periods of ordinal numbers and enumerations of recursive enumerability are shortly recalled. Concluding remarks are formulated about the deep relationship between numbers and information, which shed new light on a red line passing through the whole history of mathematics.

A Hybridized Honey Encryption for Data Security using Residue Number System

Different encryption algorithms such as Advanced Encryption Standard (AES), Rivest, Shamir, Adleman (RSA) were proposed to protect the privacy of the data. However, most of these existing methods are vulnerable to a brute-force attack because the cipher text remains unintelligible until the original data is found. Consequently, this problem prompted researchers to introduce honey encryption (HE). HE helps to withstand the vulnerability of the encryption algorithms in brute force attacks making transmitted data more secure and efficient. Hence to reduce processing time we proposed a hybridized HE with Residue Number System. Traditional Moduli Set {2n-1, 2n, 2n+1} was used to generate the Key while Chinese Remainder Theorem (CRT), and HE technique were used for decrypting the data, based on the Distribution Transformation Decoder. Thus, anytime an attacker tries to access the data, so long the length of the guessed key is the same as the original key, the HE algorithm will generate meaningful but fake data, this helps to deter an attacker from further threat. The result showed that the proposed system was able to withstand brute force attacks with less processing time compared to other systems. N-values for the moduli set used were varied against encryption and decryption time. Comparing the obtained results with the existing system, the proposed system processing time was faster and more secure.

An inversion statistic on the generalized symmetric groups

In this paper, we construct a mixed-base number system over the generalized symmetric group G(m,1,n), which is a complex reflection group with a root system of type Bn(m). We also establish one-to-one correspondence between all positive integers in the set {1,⋯,mnn!} and the elements of G(m,1,n) by constructing the subexceedant function in relation to this group. In addition, we provide a new enumeration system for G(m,1,n) by defining the inversion statistic on G(m,1,n). Finally, we prove that the flag-major index is equi-distributed with this inversion statistic on G(m,1,n). Therefore, the flag-major index is a Mahonian statistic on G(m,1,n) with respect to the length function L.

Number system

We use numbers in daily life without examining a pattern in numbers [1]. We solve problems through equations everyday without generalizing our results [2]. This happens because we do not express numbers simply enough to get a sense of the numbers. In other words, numbers are not expressed in terms of their occurrences to understand the numbers and express them in the same form. This paper expresses numbers in the form of their occurrences.

Application of Residue Number Systems in enhancing the transmission of secured videos

Application of Residue Number Systems in enhancing the transmission of secured videos

Analytical model development for investigation on hydrogen management studies in the containment of a nuclear reactor

Containment is the ultimate physical barrier to prevent the spread of active air-borne fission products as well as radiation shielding in normal operation as well as accident condition and is designed to enclose whole reactor systems. The design of containment system for Fleet mode of 700 MWe Indian Pressurized Heavy Water Reactors (IPHWRs) incorporates a Primary Containment (PC) of pre-stressed concrete enveloped by a Secondary Containment (SC) of reinforced concrete. For the specification of the design pressure of the containment system, analysis is performed for various postulated accidents such as Loss of Coolant Accident (LOCA) and Main Steam Line Break Accident (MLSB). Further Studies on hydrogen generation and distribution for various accident scenarios in reactor containment of Indian Pressurized Heavy Water Reactors (IPHWR) indicate that during severe accident with core damage, the possibility of global deflagration in absence of mitigating features is high and may challenge the integrity of containment. Therefore, it is essential to incorporate mitigating provisions inside the containment to prevent the hydrogen reaching local detonation and global deflagration during such a scenario.In order to arrive at the design basis of containment system and simulation of large numbers of accident sequence a computer code is required, where mitigating features such as Passive Auto-Catalytic Recombiners, Containment Spray System & Containment Filtered Venting System can be modeled. A system thermal hydraulic computer code has been developed for containment response calculation during normal operation as well as accident conditions including, severe accident to include the new models for mitigating features. This article presents the details of various models of computer code PACSR-SI-2.0, its validation with data observed for hydrogen distribution & removal in Recombiner Test Facility,Battelle Spray Test for containment system & Spray test at IIT Mumbai.

Corrigendum: Effects of intelligence and approximate number system on the non-symbolic division ability in preschoolers

Corrigendum: Effects of intelligence and approximate number system on the non-symbolic division ability in preschoolers

Development of precision of non-symbolic arithmetic operations in 4-6-year-old children.

Children can represent the approximate quantity of sets of items using the Approximate Number System (ANS), and can perform arithmetic-like operations over ANS representations. Previous work has shown that the representational precision of the ANS develops substantially during childhood. However, less is known about the development of the operational precision of the ANS. We examined developmental change in the precision of the solutions to two non-symbolic arithmetic operations in 4-6-year-old U.S. children. We asked children to represent the quantity of an occluded set (Baseline condition), to compute the sum of two sequentially occluded arrays (Addition condition), or to infer the quantity of an addend after observing an initial array and then the array incremented by the unknown addend (Unknown-addend condition). We measured the precision of the solutions of these operations by asking children to compare their solutions to visible arrays, manipulating the ratio between the true quantity of the solution and the comparison array. We found that the precision of ANS representations that were not the result of operations (in the Baseline condition) was higher than the precision of solutions to ANS operations (in the Addition and Unknown-addend conditions). Further, we found that precision in the Baseline and Addition conditions improved significantly between 4 and 6 years, while precision in the Unknown-Addend condition did not. Our results suggest that ANS operations may inject "noise" into the representations they operate over, and that the development of the precision of different operations may follow different trajectories in childhood.

Using Descriptive Logic in Intelligent Educational Systems

The purpose of the article is to study and solve the problems of using the basic principles of the modern formalism of descriptive (terminological) logic in intelligent learning systems that use knowledge bases of educational content. The research methods are the main methodological approaches and technological tools for the development of intelligent learning systems. In particular, systemic and comparative analyses to identify the features of intelligent learning systems, the method of expert evaluation, which involves the analysis of literature and information resources, interviews and surveys of experts in the field of information technology and mathematical logic, as well as the process of developing and testing an intelligent system. The scientific novelty of the study is the development of a system for obtaining expert knowledge for an intelligent system that combines the formation of an expert knowledge base, and the formalism for representing which is descriptive logic. The proposed approach of formalized knowledge representation using descriptive logic can be used in the development of authors’ intelligent learning systems in the field of online education and cognitive skills development. Conclusions. The scope of application of descriptive logic in intelligent systems is constantly expanding, increasing the capabilities of these systems to formally derive new knowledge, including in conditions of incomplete certainty. Descriptive logic, used in intelligent learning systems, provides ample opportunities for building a conclusion (i.e., forming an answer), in particular in situations such as forming and answering various types of queries to the system’s knowledge base (e.g., fully defined and clearly formulated, incompletely defined and vaguely formulated, completely vague, incompletely defined and clearly formulated, completely vague, completely vague and vaguely formulated), helping to specify the query. To do this, it is necessary to formulate a rigorous number system (to build an axiomatic theory). Thus, as a means of solving semantic and logical problems, descriptive logic has become a unique and powerful mechanism for representing knowledge. The successful use of descriptive logic in the author’s intelligent educational system will ensure the improvement of the quality of the educational process; monitoring of the learning process and control of its results; will facilitate the integration of this system into many modern information educational systems; will provide various consultations, advice, interpretations, explanations regarding the educational content or educational process, as well as a wide range of topics of educational content and tests of different levels.

On Hybrid Hyper k-Pell, k-Pell–Lucas, and Modified k-Pell Numbers

Many different number systems have been the topic of research. One of the recently studied number systems is that of hybrid numbers, which are generalizations of other number systems. In this work, we introduce and study the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers. In order to study these new sequences, we established new properties, generating functions, and the Binet formula of the hyper k-Pell, hyper k-Pell–Lucas, and hyper Modified k-Pell sequences. Thus, we present some algebraic properties, recurrence relations, generating functions, the Binet formulas, and some identities for the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers.

Mathematical Model of the Process of Raising Integers to an Arbitrary Power of a Natural Number in the System of Residual Classes

It is known that the use of a non-positional number system in residual classes (SRC) in computer systems (CS) can significantly increase the speed of the implementation of integer arithmetic operations. The use of such properties of a non-positional number system in the SRC as independence, equality and low-bitness (low-digit capacity) of the residues that define the non-positional code data structure of the SRC provides high user performance for the implementation in the CS of computational algorithms consisting of a set of arithmetic (modular) operations. The greatest efficiency from the use of the SRC is achieved when the implemented algorithms consist of a set of arithmetic operations such as addition, multiplication and subtraction. There is a large class of algorithms and tasks (tasks of implementing cryptoalgorithms, optimization tasks, computational tasks of large dimension, etc.), where, in addition to performing integer arithmetic operations of addition, subtraction, multiplication, raising integers modulo and others in a positive numerical range, there is a need to implement the listed above arithmetic and other operations, in the negative numerical range. The need to perform these operations in a negative numerical range significantly reduces the overall efficiency of using the SRC as a number system of the CS. In this aspect, the lack of a mathematical model for the process of raising integers in the SRC in the negative numerical region makes it difficult to develop methods and procedures for raising integers to an arbitrary power of a natural number in the SRC, both in positive and negative numerical ranges. The purpose of the article is the synthesis of a mathematical model of the process of raising integers to an arbitrary power of a natural number in the SRC, both in positive and negative numerical ranges.

A Generalized Residue Number System Design Approach for Ultralow-Power Arithmetic Circuits Based on Deterministic Bit-Streams

The peak power consumption has become an important concern in the hardware design process of some of today’s applications, such as Energy Harvesting (EH) and Bio-Implantable (BI) electronic devices. The limited peak harvested power in EH devices and heating concerns in BI devices are the main reasons for power control’s importance in these devices. This paper proposes a generalized design approach for ultra-low-power arithmetic circuits. The proposed circuits are based on Residue Number System (RNS) combined with deterministic bit-streams. The resulting circuits can be used in systems with a restricted power budget. We suggest several approaches to design generic hardware-efficient adders, multipliers, multiply-accumulate units (MAC), forward converters (FC), and reverse converters (RC). Using the proposed approach, designing these components for any moduli of the RNS can be performed through simple bit-width adjustments in the circuits. The synthesis results show that the proposed adder achieves, on average, 69% and 2% lower area compared to the bit-serial and a state-of-the-art RNS adder, respectively. Furthermore, the proposed multiplier outperforms the bit-serial, interleaved, and a state-of-the-art design for multiplying RNS numbers by, on average, 57%, 60%, and 77% in terms of power consumption, respectively. The efficiency of our approach is shown via two essential applications, digital signal processing and machine learning. We implement an FFT engine using the proposed method. Compared to prior RNS implementations, our design achieves 47% lower power consumption. We also implement a CNN accelerator’s Processing Element (PE) with the proposed computation elements. Our design provides considerable speedup and lower power consumption compared to a state-of-the-art ultra-lower-power design.

Feature selection in intrusion detection systems: a new hybrid fusion of Bat algorithm and Residue Number System

ABSTRACT This research introduces innovative approaches to enhance intrusion detection systems (IDSs) by addressing critical challenges in existing methods. Various machine-learning techniques, including nature-inspired metaheuristics, Bayesian algorithms, and swarm intelligence, have been proposed in the past for attribute selection and IDS performance improvement. However, these methods have often fallen short in terms of detection accuracy, detection rate, precision, and F-score. To tackle these issues, the paper presents a novel hybrid feature selection approach combining the Bat metaheuristic algorithm with the Residue Number System (RNS). Initially, the Bat algorithm is utilized to partition training data and eliminate irrelevant attributes. Recognizing the Bat algorithm's slower training and testing times, RNS is incorporated to enhance processing speed. Additionally, principal component analysis (PCA) is employed for feature extraction. In a second phase, RNS is excluded for feature selection, allowing the Bat algorithm to perform this task while PCA handles feature extraction. Subsequently, classification is conducted using naive bayes, and k-Nearest Neighbors. Experimental results demonstrate the remarkable effectiveness of combining RNS with the Bat algorithm, achieving outstanding detection rates, accuracy, and F-scores. Notably, the fusion approach doubles processing speed. The findings are further validated through benchmarking against existing intrusion detection methods, establishing their competitiveness.

Compositionality and constituent structure in the analogue mind

AbstractI argue that analogue mental representations possess a canonical decomposition into privileged constituents from which they compose. I motivate this suggestion, and rebut arguments to the contrary, through reflection on the approximate number system, whose representations are widely expected to have an analogue format. I then argue that arguments for the compositionality and constituent structure of these analogue representations generalize to other analogue mental representations posited in the human mind, such as those in early vision and visual imagery.

The Conventionality of Real Valued Quantities

The original architects of the representational theory of measurement interpreted their formalism operationally and explicitly acknowledged that some aspects of their representations are conventional. In this paper we argue that the conventional elements of the representations afforded by the theory require careful scrutiny as one moves toward a more metaphysically robust interpretation by showing that there is a sense in which the very number system one uses to represent a physical quantity such as mass or length is conventional. This result undermines inferences which impute structure from the numerical representational structure to the quantity it is used to represent.

Two chromosome-level genome assemblies of Rhodiola shed new light on genome evolution in rapid radiation and evolution of the biosynthetic pathway of salidroside.

Rhodiola L. is a genus that has undergone rapid radiation in the mid-Miocene and may represent a typic case of adaptive radiation. Many species of Rhodiola have also been widely used as an important adaptogen in traditional medicines for centuries. However, a lack of high-quality chromosome-level genomes hinders in-depth study of its evolution and biosynthetic pathway of secondary metabolites. Here, we assembled two chromosome-level genomes for two Rhodiola species with different chromosome number and sexual system. The assembled genome size of R. chrysanthemifolia (2n = 14; hermaphrodite) and R. kirilowii (2n = 22; dioecious) were of 402.67 and 653.62 Mb, respectively, with approximately 57.60% and 69.22% of transposable elements (TEs). The size difference between the two genomes was mostly due to proliferation of long terminal repeat-retrotransposons (LTR-RTs) in the R. kirilowii genome. Comparative genomic analysis revealed possible gene families responsible for high-altitude adaptation of Rhodiola, including a homolog of plant cysteine oxidase 2 gene of Arabidopsis thaliana (AtPCO2), which is part of the core molecular reaction to hypoxia and contributes to the stability of Group VII ethylene response factors (ERF-VII). We found extensive chromosome fusion/fission events and structural variations between the two genomes, which might have facilitated the initial rapid radiation of Rhodiola. We also identified candidate genes in the biosynthetic pathway of salidroside. Overall, our results provide important insights into genome evolution in plant rapid radiations, and possible roles of chromosome fusion/fission and structure variation played in rapid speciation.