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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.
Elvis Has Left the Building
Published in Ted G. Lewis, The Signal, 2019
For example, let the intended message be {1234}. Division by 33 yields a remainder of 13. If we encode and transmit both {1234} and 13, the receiver can verify that no error has occurred if it also divides by 33 and obtains r = 13 as a remainder. The long division table that follows illustrates this. The result of 1234/33 is 37 with a remainder of 13. Since we only care about the remainder, we discard 37. Mathematically, obtaining a remainder after integer division is called modulo arithmetic and its operation is the mod operation. Therefore, 1234 mod 33 is 37. The following tables show how grade-school long division yields the mod operation on 1234.
Computer Programming
Published in Quamrul H. Mazumder, Introduction to Engineering, 2018
The other operator that you might not be familiar with is %. The % sign is the symbol for the modulo operator. The modulus is simply the remainder after a division operation has been performed. For example, 13 divided by 4 is 3 with a remainder of 1. Thus, the modulus is 1. One practical application of the modulo operator is to determine if a particular number is an even or an odd number. In order to be even, the number has to be divisible by 2 with no remainder. A modulo operation with no remainder is 0, and thus, the program would declare a number, called numToTest, to be even if numToTest % 2 = 0.
Measuring 3- and 4-Moduli Sets Delay Per Bit in Residue Number System: A Survey
Published in IETE Journal of Research, 2023
Besides, to compare moduli sets fairly, the best realizations of them are reported. For each moduli set, the best RC is considered and specified by the reference number in the first column of Tables 2–4. As it is depicted in Table 1, modulo- addition/multiplication and modulo- addition are based on the work of Refs [15] and [14], respectively. Modulo-() addition is performed by fusing an end-around carry to an -bit carry-ripple adder. Modulo- multiplication consists of () levels of 4:2 compressor to reduce the partial product generations to two -bit numbers followed by a modular adder.
A Component-Position Model, Analysis and Design for Order-of-Addition Experiments
Published in Technometrics, 2021
Jian-Feng Yang, Fasheng Sun, Hongquan Xu
We need the concept of Galois fields in order to describe our next construction method for component orthogonal arrays. A Galois field (or finite field) is a field that contains a finite number of elements, on which the operations of multiplication, addition, subtraction and division are defined and satisfy the rules of arithmetic. The number of elements, called the order of a Galois field, must be a prime power. For any prime p and positive integer u, there is a unique Galois field of order pu up to isomorphism. Let be a Galois field. For a prime number p, is simply the ring of integers modulo p. For u > 1, . In other words, the elements of are polynomials with degree less than u and coefficients from GF(p). The addition of is ordinary polynomial addition with coefficients modulo p, and the multiplication is ordinary polynomial multiplication and then modulo a given irreducible polynomial of degree u. An irreducible polynomial (or primitive polynomial) is a polynomial that cannot be factored into the product of two non-constant polynomials. For a selected primitive polynomial over GF(p), see Table A.19 of Hedayat et al. (1999).
Observability and reconstructibility of bounded cellular automata
Published in International Journal of Systems Science, 2022
Théo Plénet, Samira El Yacoubi, Clément Raïevsky, Laurent Lefèvre
In Definition 2.1, one of the conditions for to be a field is that the state number k is prime. If it is not, the inverse of a number does not necessarily exist, however the inverse is required to reconstruct the initial state with Corollary 3.5. Moreover, in modular arithmetic, this inverse is not computed from the usual division but from multiplication and modulo: