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Number Systems, Conversions and Codes
Published in Dale Patrick, Stephen Fardo, Vigyan ‘Vigs’ Chandra, Electronic Digital System Fundamentals, 2020
Dale Patrick, Stephen Fardo, Vigyan ‘Vigs’ Chandra
Hexadecimal numbers are used in digital systems to process large numbers. The base of this system is 16. The largest value displayed at a specific place is 15, which is displayed by digits 0 through 9 and the letters A, B, C, D, E, and F. The place values from the left of the hexadecimal point are powers of 16. Conversion of a hexadecimal number to a decimal number is similar to the other conversion processes. The hexadecimal number is recorded and the place values or powers of the base are then positioned under each digit. The values are then multiplied to indicate discrete place values. Conversion of decimal to hexadecimal is achieved by repetitive division. The quotient is transferred for the next division step and the remainder is recorded. The combined remainders form the hexadecimal number. Conversion of a binary number to hexadecimal is achieved by dividing the number into groups of four digits starting at the hexadecimal point. Each binary group is then changed into an equivalent hexadecimal value.
Very-Large-Scale Integration Implementations of Cryptographic Algorithms
Published in Tomasz Wojcicki, Krzysztof Iniewski, VLSI: Circuits for Emerging Applications, 2017
Modular multiplication is a slightly complicated arithmetic operation because of the inherent multiplication and division operations. Modular multiplication can be carried out by performing the modulo operation after multiplication or during the multiplication. The modulo operation is accomplished by integer division, in which only the remainder after division is taken for further computation. The first approach requires an n × n bit multiplier with a 2n-bit register followed by a 2n × n bit divider. In the second approach, the modulo operation occurs in each step of integer multiplication. Therefore the first approach requires more hardware while the second requires more addition/subtraction computations. Different number representations such as redundant number systems and higher radix carry-save form have been used for this purpose. A carry prediction mechanism has also been used for fast calculation of modular multiplication.
Basics of the central processing unit
Published in Joseph D. Dumas, Computer Architecture, 2016
The most basic algorithm for dividing binary numbers, known as restoring division, operates analogously to long division by hand; that is, it uses the “shift and subtract” approach. The bits of the dividend are considered from left to right (in other words, starting with the most significant bit) until the subset of dividend bits forms a number greater than the divisor. At this point, we know that the divisor will “go into” this partial dividend (that is, divide it with a quotient bit of 1). All previous quotient bits to the left of this first 1 are 0. The divisor is subtracted from the partial dividend, forming a partial remainder; additional bits are brought down from the rest of the dividend and appended to the partial remainder until the divisor can again divide the value (or until no more dividend bits remain, in which case the operation is complete and the partial remainder is the final remainder). Any comparison showing the divisor to be greater than the current partial dividend (or partial remainder) means the quotient bit in the current position is 0; any comparison showing the divisor to be smaller produces a quotient bit of 1. Figure 3.29 shows how we would perform the operation 29/5 (11101/101) using restoring division.
Invariant output feedback stabilisability: the scalar case
Published in International Journal of Control, 2022
Aristotelis Yannakoudakis, Michael Sfakiotakis
Consider the quotient and the remainder in the Euclidean division : Then and through the feedback transformation we obtain that and are -invariant. The first is known as Kalman’s Atom (see Vardulakis et al. 2021) and plays an important role in control design using dynamic output feedback. An -canonical form is: .
Divisibility tests for polynomials
Published in International Journal of Mathematical Education in Science and Technology, 2020
Let , with and . Setting , let and the quotient and the remainder of f modulo k, and and the quotient and the remainder of g modulo k. In the following, we denote with the symbol the determinant
Remainder and quotient without polynomial long division
Published in International Journal of Mathematical Education in Science and Technology, 2021
Let R be a commutative ring, let and let be a polynomial of degree . Then, the remainder of f modulo is the polynomial the quotient of f modulo is the polynomial .