All one polynomial

In mathematics, an all one polynomial is a polynomial in which all coefficients are one. Over the finite field of order two, conditions for the AOP to be irreducible are known, which allow this polynomial to be used to define efficient algorithms and circuits for multiplication in finite fields of characteristic two. The AOP is a 1-equally spaced polynomial.

Definition

An AOP of degree m has all terms from x^m to x⁰ with coefficients of 1, and can be written as
or
or
Thus the roots of the all one polynomial of degree m are all 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
The AOP is irreducible if and only if m + 1 is prime and 2 is a primitive root [modulo n|primitive root] modulo m + 1
The only AOP that is a primitive polynomial is x² + 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.
Over, the AOP is irreducible whenever m + 1 is a prime p, and therefore in these cases, the pth cyclotomic polynomial.