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GF(2)
(also denoted \mathbb F_2, or \mathbb Z/2\mathbb Z) is the finite field with two elements. is the Field (mathematics), field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity are denoted respectively and , as usual. The elements of may be identified with the two possible values of a bit and to the Boolean domain, Boolean values ''true'' and ''false''. It follows that is fundamental and ubiquitous in computer science and its mathematical logic, logical foundations. Definition GF(2) is the unique field with two elements with its additive identity, additive and multiplicative identity, multiplicative identities respectively denoted and . Its addition is defined as the usual addition of integers but modulo 2 and corresponds to the table below: If the elements of GF(2) are seen as Boolean values, then the addition is the same as that of the logical XOR operation. Since each element equals its opposite (m ... [...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 (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), 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 the integers mod n, integers mod p when p 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 p and every positive integer k there are fields of order p^k. All finite fields of a given order are isomorphism, 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 that is a fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integers Modulo A Prime
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 the integers mod p when p 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 p and every positive integer k there are fields of order p^k. All finite fields of a given order 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 that is a field; this means that multiplication, addition, subtraction and division (excluding divis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical XOR
Exclusive or, exclusive disjunction, exclusive alternation, logical non-equivalence, or logical inequality is a logical operator whose negation is the logical biconditional. With two inputs, XOR is true if and only if the inputs differ (one is true, one is false). With multiple inputs, XOR is true if and only if the number of true inputs is odd. It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true. XOR ''excludes'' that case. Some informal ways of describing XOR are "one or the other but not both", "either one or the other", and "A or B, but not A and B". It is symbolized by the prefix operator J Translated as and by the infix operators XOR (, , or ), EOR, EXOR, \dot, \overline, \underline, , \oplus, \nleftrightarrow, and \not\equiv. Definition The truth table of A\nleftrightarrow B shows that it outputs true whenever the inputs differ: Equivalences, elimination, and introduction Exclusive disjunction essentially me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Domain
In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include ''false'' and ''true''. In logic, mathematics and theoretical computer science, a Boolean domain is usually written as , or \mathbb. The algebraic structure that naturally builds on a Boolean domain is the Boolean algebra with two elements. The initial object in the category of bounded lattices is a Boolean domain. In computer science, a Boolean variable is a variable that takes values in some Boolean domain. Some programming languages feature reserved words or symbols for the elements of the Boolean domain, for example false and true. However, many programming languages do not have a Boolean data type in the strict sense. In C or BASIC, for example, falsity is represented by the number 0 and truth is represented by the number 1 or −1, and all variables that can take these values can also take any other numerical values. Generalizations Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even Number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Identity
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. Definitions Let be a set equipped with a binary operation ∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an . An identity with respect to addition is called an (often denoted as 0) and an identity with respect to multiplication is called a (often denoted as 1). The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after the Norwegian mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation ･ , that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication. The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'') (alternative notations: Mat''n''(''R'') and ). Some sets of infinite matrices form infinite matrix rings. A subring of a matrix ring is again a matrix ring. Over a rng, one can form matrix rngs. When ''R'' is a commutative ring, the matrix ring M''n''(''R'') is an associative algebra over ''R'', and may be called a matrix algebra. In this setting, if ''M'' is a matrix and ''r'' is in ''R'', then the matrix ''rM'' is the matrix ''M'' with each of its entries multiplied by ''r''. Examples * The set of all square matrices over ''R'', denoted M''n''(''R''). This is sometimes called the "full ring of ''n''-by-''n'' matrices". * The set of all upper triangular matrices over ''R''. * The set of all lower triangular matrices over ''R''. * The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Inversion
In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. An invertible matrix multiplied by its inverse yields the identity matrix. Invertible matrices are the same size as their inverse. Definition An -by- square matrix is called invertible if there exists an -by- square matrix such that\mathbf = \mathbf = \mathbf_n ,where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. Over a field, a square matrix that is ''not'' invertible is called singular or degenerat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
