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A Comparative Analysis of Classical Cryptography versus Quantum Cryptography for Web of Medical Things (WoMT)
Published in Aboul Ella Hassanien, Nilanjan Dey, Surekha Borra, Medical Big Data and Internet of Medical Things, 2018
K. Anitha Kumari, G. Sudha Sadasivam
Some of the concepts connected with the field are as follows: The number of elements in a finite field is called the order of ‘F’ and such order exists only if the order is a prime power pr where ‘p’ is a prime number and ‘r’ is a positive integer;The characteristic of any field can be either ‘0’ or a prime number. ‘p’ is called the characteristic of a field when adding an element ‘r’ times results in zero. A field is called a finite characteristic field or field of positive characteristic if it has a non-zero characteristic; andA multiplicative group is said to be a finite field consisting of non-zero elements. When all non-zero elements are expressed as powers of a single element, called a primitive element of the field, it is said to be cyclic. In field theory, a primitive element of a finite field Fp is a generator of the multiplicative group of the field. In other words, a ∈ Fp is called a primitive element, when all the non-zero elements of Fp can be written as ai for some positive integer.
Subfield codes and trace codes
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
12.2 Definition. Let f (X) ∈ Fq [X] be irreducible of degree r, and ∊ ∈ Fqrthe element corresponding to X. We say that f (X) is aprimitive polynomialif 1 = ∊0, ∊,..., ∊qr−2are different, equivalently if the powers of ∊ are all nonzero elements ofFqr. An element whose powers run through all nonzero field elements is called aprimitive element.
An Introduction to Error-Correcting Codes
Published in Erozan M. Kurtas, Bane Vasic, Advanced Error Control Techniques for Data Storage Systems, 2018
GF(8) can be obtained as a power of the element α. In this case, α is called a primitive element and the irreducible polynomial f(x) that defines the field is called a primitive polynomial. It can be proven that it is always the case that the multiplicative group of a finite field is cyclic, so there is always a primitive element. A convenient description of GF(8) is given in Table 1.1.
Series concatenation of 2D convolutional codes by means of input-state-output representations
Published in International Journal of Control, 2018
Joan-Josep Climent, Diego Napp, Raquel Pinto, Rita Simões
Let α be a primitive element of the Galois field with α3 + α + 1 = 0, and consider, for ℓ = 1, 2, the 2D convolutional code with ISO representation Σℓ = (A(ℓ)1, A(ℓ)2, B1(ℓ), B(ℓ)2, C(ℓ), D(ℓ)), where Note that and are systematic. In fact, the corresponding matrices defined by Equation (3)