Primitive polynomial (field theory)

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In finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field GF(p^m). This means that a polynomial F(X) of degree m with coefficients in GF(p) = Z/pZ is a primitive polynomial if it is monic and has a root α in GF(p^m) such that ${\displaystyle \{0,1,\alpha ,\alpha ^{2},\alpha ^{3},\ldots \alpha ^{p^{m}-2}\}}$ is the entire field GF(p^m). This implies that α is a primitive (p^m − 1)-root of unity in GF(p^m).

• Because all minimal polynomials are irreducible, all primitive polynomials are also irreducible.
• A primitive polynomial must have a non-zero constant term, for otherwise it will be divisible by x. Over GF(2), x + 1 is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by x + 1 (it has 1 as a root).
• An irreducible polynomial F(x) of degree m over GF(p), where p is prime, is a primitive polynomial if the smallest positive integer n such that F(x) divides xⁿ − 1 is n = p^m − 1.
• A primitive polynomial of degree m has m different roots in GF(p^m), which all have order p^m − 1, meaning that any of them generates the multiplicative group of the field.
• Over GF(p) there are exactly φ(p^m − 1) primitive elements and φ(p^m − 1) / m primitive polynomials, each of degree m, where φ is Euler's totient function.^[1]
• The algebraic conjugates of a primitive element α in GF(p^m) are α, α^p, α^p2, …, α^pm−1 and so the primitive polynomial F(x) has explicit form F(x) = (x − α) (x − α^p) (x − α^p2) … (x − α^pm−1). That the coefficients of a polynomial of this form, for any α in GF(pⁿ), not necessarily primitive, lie in GF(p) follows from the property that the polynomial is invariant under application of the Frobenius automorphism to its coefficients (using α^pn = α) and from the fact that the fixed field of the Frobenius automorphism is GF(p).

Over GF(3) the polynomial x² + 1 is irreducible but not primitive because it divides x⁴ − 1: its roots generate a cyclic group of order 4, while the multiplicative group of GF(3²) is a cyclic group of order 8. The polynomial x² + 2x + 2, on the other hand, is primitive. Denote one of its roots by α. Then, because the natural numbers less than and relatively prime to 3² − 1 = 8 are 1, 3, 5, and 7, the four primitive roots in GF(3²) are α, α³ = 2α + 1, α⁵ = 2α, and α⁷ = α + 2. The primitive roots α and α³ are algebraically conjugate. Indeed x² + 2x + 2 = (x − α) (x − (2α + 1)). The remaining primitive roots α⁵ and α⁷ = (α⁵)³ are also algebraically conjugate and produce the second primitive polynomial: x² + x + 2 = (x − 2α) (x − (α + 2)).

For degree 3, GF(3³) has φ(3³ − 1) = φ(26) = 12 primitive elements. As each primitive polynomial of degree 3 has three roots, all necessarily primitive, there are 12 / 3 = 4 primitive polynomials of degree 3. One primitive polynomial is x³ + 2x + 1. Denoting one of its roots by γ, the algebraically conjugate elements are γ³ and γ⁹. The other primitive polynomials are associated with algebraically conjugate sets built on other primitive elements γ^r with r relatively prime to 26:

${\displaystyle {\begin{aligned}x^{3}+2x+1&=(x-\gamma )(x-\gamma ^{3})(x-\gamma ^{9})\\x^{3}+2x^{2}+x+1&=(x-\gamma ^{5})(x-\gamma ^{5\cdot 3})(x-\gamma ^{5\cdot 9})=(x-\gamma ^{5})(x-\gamma ^{15})(x-\gamma ^{19})\\x^{3}+x^{2}+2x+1&=(x-\gamma ^{7})(x-\gamma ^{7\cdot 3})(x-\gamma ^{7\cdot 9})=(x-\gamma ^{7})(x-\gamma ^{21})(x-\gamma ^{11})\\x^{3}+2x^{2}+1&=(x-\gamma ^{17})(x-\gamma ^{17\cdot 3})(x-\gamma ^{17\cdot 9})=(x-\gamma ^{17})(x-\gamma ^{25})(x-\gamma ^{23}).\end{aligned}}}$

Primitive polynomials can be used to represent the elements of a finite field. If α in GF(p^m) is a root of a primitive polynomial F(x), then the nonzero elements of GF(p^m) are represented as successive powers of α:

${\displaystyle \mathrm {GF} (p^{m})=\{0,1=\alpha ^{0},\alpha ,\alpha ^{2},\ldots ,\alpha ^{p^{m}-2}\}.}$

This allows an economical representation in a computer of the nonzero elements of the finite field, by representing an element by the corresponding exponent of ${\displaystyle \alpha .}$ This representation makes multiplication easy, as it corresponds to addition of exponents modulo ${\displaystyle p^{m}-1.}$

Primitive polynomials over GF(2), the field with two elements, can be used for pseudorandom bit generation. In fact, every linear-feedback shift register with maximum cycle length (which is 2ⁿ − 1, where n is the length of the linear-feedback shift register) may be built from a primitive polynomial.^[2]

In general, for a primitive polynomial of degree m over GF(2), this process will generate 2^m − 1 pseudo-random bits before repeating the same sequence.

The cyclic redundancy check (CRC) is an error-detection code that operates by interpreting the message bitstring as the coefficients of a polynomial over GF(2) and dividing it by a fixed generator polynomial also over GF(2); see Mathematics of CRC. Primitive polynomials, or multiples of them, are sometimes a good choice for generator polynomials because they can reliably detect two bit errors that occur far apart in the message bitstring, up to a distance of 2ⁿ − 1 for a degree n primitive polynomial.

A useful class of primitive polynomials is the primitive trinomials, those having only three nonzero terms: x^r + x^k + 1. Their simplicity makes for particularly small and fast linear-feedback shift registers.^[3] A number of results give techniques for locating and testing primitiveness of trinomials.^[4]

For polynomials over GF(2), where 2^r − 1 is a Mersenne prime, a polynomial of degree r is primitive if and only if it is irreducible. (Given an irreducible polynomial, it is not primitive only if the period of x is a non-trivial factor of 2^r − 1. Primes have no non-trivial factors.) Although the Mersenne Twister pseudo-random number generator does not use a trinomial, it does take advantage of this.

Richard Brent has been tabulating primitive trinomials of this form, such as x^74207281 + x^30684570 + 1.^[5][6] This can be used to create a pseudo-random number generator of the huge period 2^74207281 − 1 ≈ 3×10^22338617.

• Weisstein, Eric W. "Primitive Polynomial". MathWorld.