Primitive polynomial (field theory)

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 $GF(p) = Z / p Z$ is a primitive polynomial if it is monic and has a root $α$ in $GF(p m)$ such that $\{0,1,\alpha ,\alpha ^{2},\alpha ^{3},\ldots \alpha ^{p^{m}-1}\}$ is the entire field $GF(p m)$ . This implies that $α$ is a primitive ( $p m - 1$ )-root of unity in $GF(p m)$ .

Properties

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.

Over GF(p) there are exactly φ(p^m − 1)/m primitive polynomials of degree m, where φ is Euler's totient function.

A primitive polynomial of degree m has m different roots in GF(p^m), which all have order p^m − 1. This means that, if α is such a root, then α^{p^m−1} = 1 and αⁱ ≠ 1 for 0 < i < p^m − 1.

The primitive polynomial F(x) of degree m of a primitive element α in GF(p^m) has explicit form F(x) = (x − α)(x − α^p)(x − α^p²)⋅⋅⋅(x − α^{p^m−1}).

Usage

Field element representation

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:

\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 $\alpha .$ This representation makes multiplication easy, as it corresponds to addition of exponents modulo $p^{m}-1.$

Pseudo-random bit generation

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.[1]

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.

CRC codes

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.

Primitive trinomials

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.[2] A number of results give techniques for locating and testing primitiveness of trinomials.[3]

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.[4][5] This can be used to create a pseudo-random number generator of the huge period 2^74207281 − 1 ≈ 3×10^22338617.

References

C. Paar, J. Pelzl - Understanding Cryptography: A Textbook for Students and Practitioners
Gentle, James E. (2003). Random number generation and Monte Carlo methods (2 ed.). New York: Springer. p. 39. ISBN 0-387-00178-6. OCLC 51534945.
Zierler, Neal; Brillhart, John (December 1968). "On primitive trinomials (Mod 2)". Information and Control. 13 (6): 541, 548, 553. doi:10.1016/S0019-9958(68)90973-X.
Brent, Richard P. (4 April 2016). "Search for Primitive Trinomials (mod 2)". Retrieved 4 June 2020.
Brent, Richard P.; Zimmermann, Paul (24 May 2016). "Twelve new primitive binary trinomials". arXiv:1605.09213 [math.NT].

External links

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

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[1] C. Paar, J. Pelzl - Understanding Cryptography: A Textbook for Students and Practitioners

[2] Gentle, James E. (2003). Random number generation and Monte Carlo methods (2 ed.). New York: Springer. p. 39. ISBN 0-387-00178-6. OCLC 51534945.

[3] Zierler, Neal; Brillhart, John (December 1968). "On primitive trinomials (Mod 2)". Information and Control. 13 (6): 541, 548, 553. doi:10.1016/S0019-9958(68)90973-X.

[4] Brent, Richard P. (4 April 2016). "Search for Primitive Trinomials (mod 2)". Retrieved 4 June 2020.

[5] Brent, Richard P.; Zimmermann, Paul (24 May 2016). "Twelve new primitive binary trinomials". arXiv:1605.09213 [math.NT].