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Algebraic Geometry
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
We will start with an overview of some concepts from abstract algebra, which will be used extensively in this chapter. A fieldK is a commutative ring in which every nonzero element has a multiplicative inverse. Some examples of fields are the real numbers ℝ, the field of rational numbers ℚ, and the field of complex numbers ℂ. The characteristic of K, denoted by char(K), is defined to be the smallest positive integer such that nx=0,forallx∈K.
Asymmetric Cryptography
Published in Khaleel Ahmad, M. N. Doja, Nur Izura Udzir, Manu Pratap Singh, Emerging Security Algorithms and Techniques, 2019
Rajiv Ranjan, Abir Mukherjee, Pankaj Rai, Khaleel Ahmad
Definition 5: A field F, denoted by (F, ⨁, ⨀), is an integral domain that has a multiplicative inverse for every element p ∈ F, except the 0 elements (i.e., additive identity). A multiplicative inverse of an element p ∈ F is an element p−1 ∈ F, such that p ⨀ p−1 = p−1 ⨀ p = 1. A field having a finite number of elements is called a finite field, or a Galois field. All finite fields must have an order that is a power of a prime number, i.e., the order must be pn where p is a prime and n is a positive integer. These finite fields are generally denoted by GF(pn), for some integer n.
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
In [2], RNS floating-point arithmetic (addition, subtraction, multiplication, division, and square root) for an interval number is discussed with the goal to achieve reliable computation when hardware representations of numbers have inadequate precision. In [8], the residue-to-binary number converters for the RNS 2n-1,2n,2n+1 were designed using 2n-bit or n-bit adders, that are twice as fast as generic ones, and achieve improvement in area and dynamic range as well. Particular modulo arithmetic involves Galois fields. Addition and multiplication in Galois fields, GF2n plays an important role in coding theory and is widely used in digital computers and data transmission or storage systems. The group theory is used to introduce algebraic system, called a field. A field is a set of elements in which we can do addition, subtraction, multiplication and division without leaving the set.
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).
Certificateless pairing-free authentication scheme for wireless body area network in healthcare management system
Published in Journal of Medical Engineering & Technology, 2020
Philemon Kasyoka, Michael Kimwele, Shem Mbandu Angolo
In studies by Kobiltz [26] and Miller [27], they independently proposed using group of points on an elliptic curve defined over a finite field in discrete log cryptosystems. A finite field is a field with a finite number of elements also known as Galois field [28]. The security of ECC is based on the mathematical difficulty of discrete logarithm in elliptic curve fields [29] rather than the discrete logarithm in prime fields.