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  and any positive integer , there exists a primitive polynomial of degree  over GF(). There are

(1)

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

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

			(2)
			(3)
			(4)

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

(5)

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

	primitive polynomials
1
2
3	,
4	,
5	, , , , ,

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

REFERENCES:

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Church, R. "Tables of Irreducible Polynomials for the First Four Prime Moduli." Ann. Math. 36, 198-209, 1935.

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O'Connor, S. E. "Computing Primitive Polynomials." http://seanerikoconnor.freeservers.com/Mathematics/AbstractAlgebra/PrimitivePolynomials/overview.html.

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Sloane, N. J. A. Sequence A011260/M0107 in "The On-Line Encyclopedia of Integer Sequences."

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