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 for a degree n primitive polynomial. Primitive trinomials A useful class of primitive polynomials is the primitive trinomials, those having only three nonzero terms: . Their simplicity makes for particularly small and fast linear-feedback shift registers. A number of results give techniques for locating and testing primitiveness of trinomials. For polynomials over GF(2), where 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 . 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 . This can be used to create a pseudo-random number generator of the huge period ≈ . References External links Field (mathematics) Polynomials