In mathematics, an all one polynomial (AOP) 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<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 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
  • The AOP is irreducible if and only if m + 1 is prime and 2 is a primitive root modulo m + 1

References

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