In mathematics, an all one polynomial (AOP) is a

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...

in which all

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...

s are one. Over the finite field of order two, conditions for the AOP to be

irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergen ...

are known, which allow this polynomial to be used to define efficient algorithms and circuits for

multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being ad ...

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, subt ...

s of characteristic two.. The AOP is a 1-

equally spaced polynomial An equally spaced polynomial (ESP) is a polynomial used in finite fields, specifically GF(2) (binary). An ''s''-ESP of degree ''sm'' can be written as: :ESP(x) = \sum_^ x^ for i = 0, 1, \ldots, m or :ESP(x) = x^ + x^ + \cdots + x^s + 1. Proper ...

Definition

An AOP of

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...

''m'' has all terms from ''x''^''m'' to ''x''⁰ with coefficients of 1, and can be written as :

AOP_m(x) = \sum_^ x^i

or :

AOP_m(x) = x^m + x^ + \cdots + x + 1

or :

AOP_m(x) = .

Thus the

roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...

of the all one polynomial of degree ''m'' are all (''m''+1)th

roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...

other than unity itself.

Properties

Over

GF(2) (also denoted \mathbb F_2, or \mathbb Z/2\mathbb Z) is the finite field of two elements (GF is the initialism of ''Galois field'', another name for finite fields). Notations and \mathbb Z_2 may be encountered although they can be confused with ...

the AOP has many interesting properties, including: *The

Hamming weight The Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string o ...

of the AOP is ''m'' + 1, the maximum possible for its degree *The AOP is

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...

''m'' + 1 is

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

and 2 is a primitive root modulo ''m'' + 1 (over GF(''p'') with prime ''p'', it is irreducible if and only if ''m'' + 1 is prime and ''p'' is a primitive root modulo ''m'' + 1) *The only AOP that is a primitive polynomial is ''x''² + x + 1. Despite the fact that the Hamming weight is large, because of the ease of representation and other improvements there are efficient implementations in areas such as

coding theory Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are stud ...

and

cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adve ...

. Over

\mathbb

, the AOP is irreducible whenever ''m'' + 1 is a prime ''p'', and therefore in these cases, the ''p''th

cyclotomic polynomial In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th primit ...

References

External links

*{{PlanetMath, urlname=AllOnePolynomial, title=all one polynomial Field (mathematics) Polynomials