Сравнение чисел и анализ переполнения в модулярной арифметике

In the conditions of growing challenges in cyberspace, the information environment protection system is a preventive mechanism of protection against real and potential risks and threats to national interests. The study analyzed the position of Ukraine in the world rankings for cyber security and outlined promising directions for increasing its level, one of which is the improvement of information protection systems of critical infrastructure objects. A system of protection of the information environment against cyber attacks using of the modular number system based on non-positional code structures is proposed. Of the various options for the practice of intrusion detection and prevention systems, this chapter discusses the data comparison algorithm, which consists in continuously monitoring and scanning data by constantly comparing data with user and service information patterns, as well as threat patterns and indicators based on previous experience, not only own, within the local network or system, but also on a global scale. The chapter presents an improved method of fast comparison, which allows to carry out the comparison procedure in the modular number system, both in positive and negative numerical ranges. Improving the method of fast comparison of two integers is carried out by increasing the accuracy of comparison, by representing numbers in an artificial form, which expands the area of effective use of computer systems for processing integer economic data in the modular number system. The use of an improved method for fast comparison of data in the modular number system for one-byte, two-byte, three-byte, four-byte and eight-byte numerical bit grids of the computer systems, respectively, by 16 %, 37 %, 50 %. It is 58 % and 72 % more efficient in terms of number comparison time than using the fastest of the existing number comparison methods in the modular number system, which is based on the principle of nulling. The proposed method provides maximum comparison accuracy with a minimum amount of equipment for comparing devices. This circumstance makes it possible to obtain a reliable result of the data control operation. Its use makes it possible to identify potential cyber threats and take preventive measures, which will increase the level of protection of critical infrastructure objects.

In order to effectively plan and develop projects involving microcomputers, the industrial engineer requires background information on number systems, binary arithmetic, and binary logic. In this tutorial article, a basic introduction to these topics is provided. Specific topics include: positional number systems, conversion between number systems, addition and subtraction of binary numbers, the two's complement of negative numbers, the ASCII and Gray codes, the basic logic functions (AND, OR, NOT, NAND, NOR, XOR), and some applications of basic logic functions, including a decoder, an adder, and a binary latch. This is the first of an integrated series of articles dealing with concepts and techniques for interfacing microcomputers to industrial devices.

A SUCCESSFUL program in elementary mathej matics emphasizes the ability of pupils to do quan titative thinking and to understand arithmetic as a closely-knit system of ideas, principles and pro cesses. In order for elementary arithmetic pupils to think or communicate efficiently in the language of elementary mathematics, they must have an un derstanding of the nature and characteristics of the number system used. The Hindu-Arabic decimal number system is the foundation upon which the system of arithmetic is constructed. One of the basic objectives of the modern ele mentary arithmetic program is to have pupils de velop an understanding of the number system we use. The development of this understanding is often hampered by mental sets on the part of teachers and pupils and/or varying meanings of the terms used in explanation and discussion. Since, in many respects, children develop their understanding of our number system in much the same way that the system itself developed, one method of improving or increasing this under standing may be to allow elementary pupils to de velop a number system with base other than ten, using a minimum of old terms, and discovering relationships between the newly-developed system and the Hindu-Arabic system. The study of number systems with bases other than ten has generally been restricted to the en richment programs of pupils more advanced in grade level with the expressed purpose of increas ing their understanding of numbers and number system. A procedure which helps pupils to de velop a more complete understanding of our num ber system earlier in the elementary grades would have advantages in 1) classroom time and effort saved from reteaching and remedial work, 2) pu pils' successes in the arithmetical skills and con cepts to be achieved, and 3) possible increased motivation toward study of mathematics. Authors of texts dealing with methods of teach ing arithmetic discuss this problem very briefly, if at all, and generally infer that the properties | HAROLD H LERCH

Currently, the main directions of development of computer systems and components (CSC) in a positional binary number system (PNS) are improving the speed of implementation of arithmetic operations (including integer arithmetic operations) and reliability of their functioning. The complexity, scale, and volume of various computational tasks and tasks of managing large and complex information systems which are solved by computer-controlled systems (CS) necessitate the expansion of the CS functions and capabilities. It leads to an increase in the number and complexity of computing equipment and systems equipment, and the complexity of their software. This, in turn, necessitates the adoption of additional measures to ensure the high reliability of the CS and, first of all, the need to take additional measures to ensure the high fault tolerance of the CSC. The purpose of the article is to design of structures of a fault-tolerance computer system that functions in residue classes in the case of applying the active fault tolerance method.

This paper presents a new complex number representation called the Redundant Complex Number System (RCNS) as a useful basis for designing VLSI signal processors with complex arithmetic capability. RCNS described in this paper is defined as a positional number system that has a complex radix 2j and a digit set {-3,...,0,...,3} or {-2,...,0,...,2}. The use of complex radix 2j with the redundant digit set makes possible to construct high-speed complex number arithmetic circuits with highly regular structures. This paper discusses the elementary arithmetic algorithms of RCNS and their implementations.

Redundancy, the state of being in excess of what is necessary, as applied in the implementation of computer arithmetic is motivated by three design goals: to improve reliability, to increase speed of operation, and/or to provide structural flexibility. In achieving the first goal, improvement of reliability, hardware redundancy and/or redundant arithmetic codes are applied to the detection and correction of faults. Although this is an increasingly vital area it will not be discussed in this paper. Rather, the focus will be on the other two potential benefits: more specifically, on the judicious use of number systems employing redundancy in representation. A positional number system with fixed radix, r, is redundant if the allowable digit set includes more than r distinct elements, thereby affording alternate representations of a given numeric value. Uniqueness. of representation is sacrificed with hope of greater gains. A novel, rigorous treatment of redundant, radix polynomial representation is included in Reference 1.

The paper considers an approach to division modules of numbers by bases in number systems in residual classes (RNS). This may be required when solving specific tasks.The approach involves performing the following sequence of actions in the RNS:- reducing the modulus of numbers by the value of the deduction corresponding to the base by which the division is made;- dividing the reduced module by the value of the base;- formation the correction value and its addition to the quotient obtained.In total the division operation is performed through four accesses to the table memory two of which are overlapped in time.If one of the bases of the RNS is equal to pn then the approach under consideration allows to reduce the capacity of table memory by pn times in relation to the capacity of table memory containing the full set of division results, where pn is the base of the RNS by which the division is performed.

The information richness of the informatics course of the basic school makes you pay attention to the possibility for introducing the theme "Number systems" to students at the propaedeutic stage. The use in the fifth or sixth grades of the concept of a detailed form for recording a number to justify the studied algorithms, is impossible for obvious reasons. At this stage of training, you can arrange student actions using a digit grid, which allows you to include the concept of positionality in the orientation basis of actions with numbers in different number systems from the very beginning. The transition from work with a bit grid to the traditional design of algorithms is reasonable to be carried out in three stages. At the first stage, students are offered to operate with numbers in various number systems in a digit grid, replacing q units of the current digit with one unit of the next bit and vice versa. At the second stage, the correspondence between the actions performed at the first stage and the division (multiplication) operations is established, the digit grid is still used to mediate mental actions. At the third stage, the visual use of the bit scheme is refused, students move on to the traditional design of algorithms for actions with numbers in various number systems. For each of the stages, the article provides tasks and questions for organizing frontal solution.

Simple arithmetic processing: Surface form effects in a priming task

chapter 2 - VECTOR ALGEBRA

A versatile Montgomery multiplier architecture with characteristic three support

Community service is carried out by the Pamulang University lecturer team in Pamulang sub-district, South Tangerang city in an effort to increase understanding of mathematical concepts with the target of elementary school students. The activities carried out are divided into 3 stages, the first stage: first giving a pre-test to students to find out the students' initial abilities, the second stage providing mentoring and training in understanding mathematical counting concepts with the help of mathematical teaching aids, and the third stage conducting evaluations with how to give a test (post-test) to determine the results of the treatment in the second stage. The visual aids used in this activity are the nomographic aids, the visual aids weighing the car props. Each of these props provides an understanding of the mathematical counting concepts in the operations of addition, subtraction, multiplication and division. From the results of the post-test conducted an increase in students' understanding of the concepts of mathematical operations in the form of operations of addition, subtraction, multiplication and division after being given understanding with the help of mathematical teaching aids.

One of the main tasks of the modern school is the formation of a person with a sufficient level of digital competence. The formation of digital competence of schoolchildren will be facilitated by the inclusion of modern information and communication technologies in training as a subject of study, a means of interaction between participants in the educational process, a means of solving educational problems and planning educational activities. The use of a web quest at informatics lessons or for organizing independent work of students allows you to create conditions for the use of information and communication technologies in all the indicated aspects.The article describes an example of developing a web quest on the theme "Possibilities of the Python programming language in working with positional number systems" for students of 8th grade. This example made it possible to formulate general requirements for the structure of the web quest, for the information resource developed to accompany the web quest, for the forms of organizing the interaction of participants in the educational process, for ways of presenting the results of the work of students, for assessment procedures.

Signal processing and image processing applications require floating-point computation in digital circuit designs. The floating-point calculation is recommended in almost all digital design applications to improve the accuracy of results. Different mathematical operations use floating-point mathematical operations and can be used to perform calculations and different logical units. The proposed work sugestes a double-MAF unit that uses radix-8 for multiplication and uses a common addition block for addition and multiplication operations. Firstly, floating-point numbers are transformed into IEEE 754 format, next to both addition and multiplication calculations are achieved. Extracts the fractional part of a number and performs a calculation based on the exponential difference between the numbers. The proposed curriculum is designed with a parallel structure that first extracts the decimal and exponential values of the first unit and then performs multiplication and addition operations in the second block. At last, the output is made in the third block where normalization and zero detection are performed. The proposed approach is then compared to the baseline approach, which shows improvements in energy consumption and maximum common path delay. The results showed that the delay is reduced by about 57%.

Suppose that we have a computing machine that can only deal with the operations of addition, subtraction and multiplication, but not division, of integers. Can such a machine be used to find the decimal expansion of a given real number α to an arbitrary length? The answer is ‘yes’, at least in the sense that α is the limit of a sequence of rational numbers, and one can obtain the decimal expansion of a positive rational number a/b without division. For example, for any positive k , we try all non-negative c < b and d < 10 k a , and see if 10 k a = db + c . There will be success because all we are doing is reducing 10 k a modulo b in a particularly moronic way. This guarantees the existence and uniqueness of d , and d /10 k is then the desired expansion. Take, for example, = and k = 12; then, after a tedious search, we should find that, for c = 3, there is d = 230 769 230 769 because 3 × 10 12 = 230769230769 × 13 + 3, so that = 0.230769230769 + × 10 -12 . Such a procedure of searching for c, d is hopelessly inefficient, of course, and a more efficient method is given in the next section.