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. The AOP is a 1-equally spaced polynomial.
An AOP of degree m has all terms from xm to x0 with coefficients of 1, and can be written as
thus the roots of the all one polynomial of degree m are all (m+1)th roots of unity other than unity itself.
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 (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.
- 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.
- Itoh, Toshiya; Tsujii, Shigeo (1989), "Structure of parallel multipliers for a class of fields GF(2m)", Information and Computation, 83 (1): 21–40, doi:10.1016/0890-5401(89)90045-X.
- 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.
- 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.