|
Itoh–Tsujii Inversion Algorithm
The Itoh–Tsujii inversion algorithm is used to invert elements in a finite field. It was introduced in 1988, first over GF(2''m'') using the normal basis representation of elements, however, the algorithm is generic and can be used for other bases, such as the polynomial basis. It can also be used in any finite field GF(''p''''m''). The algorithm is as follows: : Input: ''A'' ∈ GF(''p''''m'') : Output: ''A''−1 :# ''r'' ← (''p''''m'' − 1)/(''p'' − 1) :# compute ''A''''r'' − 1 in GF(''p''''m'') :# compute ''A''''r'' = ''A''''r'' − 1 · ''A'' :# compute (''A''''r'')−1 in GF(''p'') :# compute ''A''−1 = (''A''''r'')−1 · ''A''''r'' −1 :# return ''A''−1 This algorithm is fast because steps 3 and 5 both involve operations in the subfield GF(''p''). Similarly, if a small value of ''p'' is used, a lookup table can be used for inversion in step 4. The majority of time spent in this algorithm is in step 2, the first expone ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Basis
In mathematics, specifically the algebraic theory of fields, a normal basis is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The normal basis theorem states that any finite Galois extension of fields has a normal basis. In algebraic number theory, the study of the more refined question of the existence of a normal integral basis is part of Galois module theory. Normal basis theorem Let F\subset K be a Galois extension with Galois group G. The classical normal basis theorem states that there is an element \beta\in K such that \ forms a basis of ''K'', considered as a vector space over ''F''. That is, any element \alpha \in K can be written uniquely as \alpha = \sum_ a_g\, g(\beta) for some elements a_g\in F. A normal basis contrasts with a primitive element basis of the form \, where \beta\in K is an element whose minimal polynomial has degree n= :F/math>. Group representation point of view A field e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Basis
In mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of all monomials. The monomials form a basis because every polynomial may be uniquely written as a finite linear combination of monomials (this is an immediate consequence of the definition of a polynomial). One indeterminate The polynomial ring of univariate polynomials over a field is a -vector space, which has 1, x, x^2, x^3, \ldots as an (infinite) basis. More generally, if is a ring then is a free module which has the same basis. The polynomials of degree at most form also a vector space (or a free module in the case of a ring of coefficients), which has 1, x, x^2, \ldots as a basis. The canonical form of a polynomial is its expression on this basis: a_0 + a_1 x + a_2 x^2 + \dots + a_d x^d, or, using the shorter sigma notation: \sum_^d a_ix^i. The monomial basis is naturally totally ordered, either by increasing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field Arithmetic
In mathematics, finite field arithmetic is arithmetic in a finite field (a field containing a finite number of elements) contrary to arithmetic in a field with an infinite number of elements, like the field of rational numbers. There are infinitely many different finite fields. Their number of elements is necessarily of the form ''pn'' where ''p'' is a prime number and ''n'' is a positive integer, and two finite fields of the same size are isomorphic. The prime ''p'' is called the characteristic of the field, and the positive integer ''n'' is called the dimension of the field over its prime field. Finite fields are used in a variety of applications, including in classical coding theory in linear block codes such as BCH codes and Reed–Solomon error correction, in cryptography algorithms such as the Rijndael ( AES) encryption algorithm, in tournament scheduling, and in the design of experiments. Effective polynomial representation The finite field with ''p''''n'' elements is de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Fields
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |