Primitive Polynomial

A primitive polynomial is a polynomial that generates all elements of an extension field from a base field. Primitive polynomials are also irreducible polynomials. For any prime or prime power q and any positive integer n, there exists a primitive polynomial of degree n over GF(q). There are


primitive polynomials over GF(q), where phi(n) is the totient function.

A polynomial of degree n over the finite field GF(2) (i.e., with coefficients either 0 or 1) is primitive if it has polynomial order 2^n-1. For example, x^2+x+1 has order 3 since

(x+1)/(x^2+x+1)=(x+1)/(x^2+x+1) (mod 2)
(x^2+1)/(x^2+x+1)=1+x/(x^2+x+1) (mod 2)
(x^3+1)/(x^2+x+1)=x+1 (mod 2).

Plugging in q=2 to equation (◇), the numbers of primitive polynomials over GF(2) are


giving 1, 1, 2, 2, 6, 6, 18, 16, 48, ... (OEIS A011260) for n=1, 2, .... The following table lists the primitive polynomials (mod 2) of orders 1 through 5.

nprimitive polynomials
31+x+x^3, 1+x^2+x^3
41+x+x^4, 1+x^3+x^4
51+x^2+x^5, 1+x+x^2+x^3+x^5, 1+x^3+x^5, 1+x+x^3+x^4+x^5, 1+x^2+x^3+x^4+x^5, 1+x+x^2+x^4+x^5

Amazingly, primitive polynomials over GF(2) define a recurrence relation which can be used to obtain a new pseudorandom bit from the n preceding ones.

See also

Finite Field, Irreducible Polynomial, Polynomial, Polynomial Order, Primitive Element, Primitive Root

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Primitive Polynomial

Cite this as:

Weisstein, Eric W. "Primitive Polynomial." From MathWorld--A Wolfram Web Resource.

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