All one polynomial

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.^[1] The AOP is a 1-equally spaced polynomial.^[2]

Definition

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

$AOP_m(x) = \sum_{i=0}^{m} x^i$

or

$AOP_m(x) = x^m + x^{m-1} + \cdots + x + 1$

or

$AOP_m(x) = {x^{m+1} - 1\over{x-1}}$

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^[3]
• The AOP is irreducible if and only if m + 1 is prime and 2 is a primitive root modulo m + 1^[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 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.^[1]

Over $\mathbb{Q}$, the AOP is irreducible whenever m + 1 is prime p, and therefore in these cases, the pth cyclotomic polynomial.^[4]

References

1. Cohen, Henri; Frey, Gerhard; Avanzi, Roberto; Doche, Christophe; Lange, Tanja; Nguyen, Kim; Vercauteren, Frederik (2005), Handbook of Elliptic and Hyperelliptic Curve Cryptography, Discrete Mathematics and Its Applications, CRC Press, p. 215, ISBN 9781420034981.
2. Itoh, Toshiya; Tsujii, Shigeo (1989), "Structure of parallel multipliers for a class of fields GF(2^m)", Information and Computation 83 (1): 21–40, doi:10.1016/0890-5401(89)90045-X.
3. Reyhani-Masoleh, Arash; Hasan, M. Anwar (2003), "On low complexity bit parallel polynomial basis multipliers", Cryptographic Hardware and Embedded Systems - CHES 2003, Lecture Notes in Computer Science 2779, Springer, pp. 189–202, doi:10.1007/978-3-540-45238-6_16.
4. Sugimura, Tatsuo; Suetugu, Yasunori (1991), "Considerations on irreducible cyclotomic polynomials", Electronics and Communications in Japan 74 (4): 106–113, doi:10.1002/ecjc.4430740412, MR 1136200.

External links

• all one polynomial at PlanetMath.org.