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Introduction
Published in Vlad P. Shmerko, Svetlana N. Yanushkevich, Sergey Edward Lyshevski, Computer Arithmetics for Nanoelectronics, 2018
Vlad P. Shmerko, Svetlana N. Yanushkevich, Sergey Edward Lyshevski
The values a0,a1,…,aN-1 are called coefficients of the polynomial. Expression 9.1 means that there exist many polynomials that are distinguished by the properties of the fields, namely, by the types of operation being addition and multiplication. Example 9.5 The following fields are used for the representation of Boolean functions: (a) Galois field of order 2, GF(2). This field consists of two elements, 0 and 1. In GF(2), the sum and multiplication correspond to EXOR and AND operations, respectively. (b) A set of integer numbers that includes only the elements 0 and 1 . In this field, traditional sum and multiplication are used.
Finite Field Arithmetic Architecture
Published in Keshab K. Parhi, Takao Nishitani, Digital Signal Processing for Multimedia Systems, 2018
The finite field GF(2) has 2 elements, 0, 1. The addition and multiplication are performed modulo 2 as in one’s complement arithmetic. GF(2m) is the extension field of GF(2) and has 2m elements. Each of these elements is represented as a polynomial of degree less than or equal to m − 1 with coefficients coming from the ground field GF(2). For such representation, addition and subtraction are bit-independent and straightforward. However, multiplication and division involve polynomial multiplication and division modulo some primitive polynomial p(x), which are much more complicated. Hence the design of efficient architectures to perform these arithmetic operations is of great practical concern.
A Fast Image Encryption Algorithm Based on Convolution Operation
Published in IETE Journal of Research, 2019
The multiplication operation of Equation (1) is limited to the finite field GF(28), and the result of summation is processed by modulus 256, and then the convolution operation in GF(28) is obtained bywhere represents the convolution operation in GF(28), and represents the multiplication operation in GF(28) (the same meaning thereinafter). In GF(28), the irreducible polynomial is set to m(x) = x8 + x4 + x3 + x + 1.
Design of a digit-serial multiplier over GF(2m) using a karatsuba algorithm
Published in Journal of the Chinese Institute of Engineers, 2019
Trong-Yen Lee, Min-Jea Liu, Chia-Han Huang, Chia-Chen Fan, Chia-Chun Tsai, Haixia Wu
This study does not only use a traditional Karatsuba algorithm; it uses Karatsuba multiplier cutting, so when more sections are cut, more sub-multipliers are generated. However, the proposed method does not generate more sub – multipliers. The proposed multiplier cuts the two-part Karatsuba algorithm to derive a new digit-serial multiplication. It is assumed that a finite field GF(2m) is an irreducible polynomial F(x) and that polynomials A and B are a finite field GF(2m) of the two elements, which can be rewritten as:
Analysis for the reliability of computer network by using intelligent cloud computing method
Published in International Journal of Computers and Applications, 2019
Because analyzing the reliability of computer network in the finite field GF(28) is complicated, in the cryptography of intelligent cloud computing, we carry out the encryption transformation for the finite field GF(28). Four kinds of transformations used in encryption are: the byte substitutionSubBytes(), the line byte shift transformation ShiftRows(), the column byte mixed-transformation (Mix Columns), the addition transformation of round keyAddRoundKey().