Editing All one polynomial

{{Short description|Polynomial in which all coefficients are one}}
In [[mathematics]], an '''all one polynomial''' (AOP) is a [[polynomial]] in which all [[coefficient]]s are one. Over the [[GF(2)|finite field of order two]], conditions for the AOP to be [[irreducible polynomial|irreducible]] are known, which allow this polynomial to be used to define efficient algorithms and circuits for [[multiplication]] in [[finite field]]s of [[characteristic (algebra)|characteristic]] two.<ref name="hehcc">{{citation
 | last1 = Cohen | first1 = Henri
 | last2 = Frey | first2 = Gerhard
 | last3 = Avanzi | first3 = Roberto
 | last4 = Doche | first4 = Christophe
 | last5 = Lange | first5 = Tanja | author5-link = Tanja Lange
 | last6 = Nguyen | first6 = Kim
 | last7 = Vercauteren | first7 = Frederik
 | isbn = 9781420034981
 | page = 215
 | publisher = CRC Press
 | series = Discrete Mathematics and Its Applications
 | title = Handbook of Elliptic and Hyperelliptic Curve Cryptography
 | url = https://books.google.com/books?id=w6b0yhURTkQC&pg=PA215
 | year = 2005}}.</ref> The AOP is a 1-[[equally spaced polynomial]].<ref>{{citation
 | last1 = Itoh | first1 = Toshiya
 | last2 = Tsujii | first2 = Shigeo
 | doi = 10.1016/0890-5401(89)90045-X
 | issue = 1
 | journal = Information and Computation
 | pages = 21–40
 | title = Structure of parallel multipliers for a class of fields GF(2<sup>''m''</sup>)
 | volume = 83
 | year = 1989| doi-access = free
 }}.</ref>

==Definition==
An AOP of [[degree of a polynomial|degree]] ''m'' has all terms from ''x''<sup>''m''</sup> to ''x''<sup>0</sup> with coefficients of 1, and can be written as

:<math>AOP_m(x) = \sum_{i=0}^{m} x^i</math>

or

:<math>AOP_m(x) = x^m + x^{m-1} + \cdots + x + 1</math>

or

:<math>AOP_m(x) = {x^{m+1} - 1\over{x-1}}.</math>

Thus the [[root of a polynomial|roots]] of the '''all one polynomial''' of degree ''m'' are all (''m''+1)th [[roots of unity]] other than unity itself.

==Properties==
Over [[GF(2)]] the AOP has many interesting properties, including:

*The [[Hamming weight]] of the AOP is ''m'' + 1, the maximum possible for its degree<ref>{{citation
 | last1 = Reyhani-Masoleh | first1 = Arash
 | last2 = Hasan | first2 = M. Anwar
 | contribution = On low complexity bit parallel polynomial basis multipliers
 | doi = 10.1007/978-3-540-45238-6_16
 | pages = 189–202
 | publisher = Springer
 | series = Lecture Notes in Computer Science
 | title = Cryptographic Hardware and Embedded Systems - CHES 2003
 | volume = 2779
 | year = 2003| doi-access = free
 | isbn = 978-3-540-40833-8
 }}.</ref>
*The AOP is [[irreducible polynomial|irreducible]] [[if and only if]] ''m'' + 1 is [[prime number|prime]] and 2 is a [[primitive root modulo n|primitive root]] modulo ''m'' + 1<ref name="hehcc"/> (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 (field theory)|primitive polynomial]] is ''x''<sup>2</sup> + 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]] and [[cryptography]].<ref name="hehcc"/>

Over <math>\mathbb{Q}</math>, the AOP is irreducible whenever ''m'' + 1 is a prime ''p'', and therefore in these cases, the ''p''th [[cyclotomic polynomial]].<ref>{{citation
 | last1 = Sugimura | first1 = Tatsuo
 | last2 = Suetugu | first2 = Yasunori
 | doi = 10.1002/ecjc.4430740412
 | issue = 4
 | journal = Electronics and Communications in Japan
 | mr = 1136200
 | pages = 106–113
 | title = Considerations on irreducible cyclotomic polynomials
 | volume = 74
 | year = 1991}}.</ref>

==References==
{{reflist}}

==External links==
*{{PlanetMath|urlname=AllOnePolynomial|title=all one polynomial}}<!-- This was originally taken from the WP article so it can't be used as a ref. -->

[[Category:Field (mathematics)]]
[[Category:Polynomials]]