All one polynomial

An all one polynomial (AOP) is a polynomial used in finite fields, specifically GF(2) (binary). The AOP is a 1-equally spaced polynomial.

An AOP of degree m has all terms from x^m to x⁰ with coefficients of 1, and can be written as

<math>AOP(x) = \sum_{i=0}^{m} x^i<math>

or

<math>AOP(x) = x^m + x^{m-1} + \cdots + x + 1.<math>

Properties

Over GF(2) the AOP has many interest properties, including:

• The Hamming weight of the AOP is m + 1
• The AOP is irreducible if and only if m + 1 is prime and 2 is a 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