Finite Field Arithmetic
Arithmetic in a finite field is different from standard integer arithmetic. There are a limited number of elements in the finite field; all operations performed in the finite field result in an element within that field.
While each finite field is itself not infinite, there are infinitely many different finite fields; their number of elements (which is also called cardinality) 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 and in cryptography algorithms such as the Rijndael encryption algorithm.
Read more about Finite Field Arithmetic: Effective Polynomial Representation, Addition and Subtraction, Multiplication, Arithmetic Example (where The Characteristic Is Not 2)
Famous quotes containing the words finite, field and/or arithmetic:
“Any language is necessarily a finite system applied with different degrees of creativity to an infinite variety of situations, and most of the words and phrases we use are “prefabricated” in the sense that we don’t coin new ones every time we speak.”
—David Lodge (b. 1935)
“The birds their quire apply; airs, vernal airs,
Breathing the smell of field and grove, attune
The trembling leaves, while universal Pan,
Knit with the Graces and the Hours in dance,
Led on th’ eternal Spring.”
—John Milton (1608–1674)
“I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction.”
—Gottlob Frege (1848–1925)