Gauss's lemma for polynomials states that the product of primitive polynomials ( with coefficients in the same unique factorization domain ) also is primitive.
12.
Customarily, the LFSRs use primitive polynomials of distinct but close degree, preset to non-zero state, so that each LFSR generates a maximum length sequence.
13.
A 16-bit Galois LFSR . The register numbers above correspond to the same primitive polynomial as the Fibonacci example but are counted in reverse to the shifting direction.
14.
This shows that every polynomial over the rationals is associated with a unique primitive polynomial over the integers, and that the Euclidean algorithm allows the computation of this primitive polynomial.
15.
This shows that every polynomial over the rationals is associated with a unique primitive polynomial over the integers, and that the Euclidean algorithm allows the computation of this primitive polynomial.
16.
The latter part follows from the former since is certainly a common divisor of the coefficients of the product, so one can divide by and to reduce and to primitive polynomials.
17.
In general, for a primitive polynomial of degree " m " over GF ( 2 ), this process will generate pseudo-random bits before repeating the same sequence.
18.
A useful class of primitive polynomials is the primitive trinomials, those having only three nonzero terms, because they are the simplest and result in the most efficient pseudo-random number generators.
19.
If " R " is a B�zout domain ( so in particular if it's a principal ideal domain ) then also every primitive polynomial in " R " is co-maximal.
20.
We check that A ^ 4 + A ^ 3 + 1 = 0 mod f, and so our primitive polynomial is x ^ 4 + x ^ 3 + 1, agreeing with the table in LFSR.
