Period of a pseudo-random sequence generated using an LFSR

Question

I was trying to generate maximal length pseudo random sequence using an linear feedback shift register (LFSR). I have read from many sources that the length of the pseudo random sequence generated from the LFSR would be maximum if and only if the corresponding feedback polynomial is primitive.

I have tried to generate the pseudo random sequence for the primitive polynomial (4,1,0) (corresponding to polynomial $x^4 + x + 1$) by tapping at bits 4 and 1, and it has resulted in a sequence with a period of $2^4$ - 1 = 15. Similarly I have tried primitive polynomial given by (3, 1, 0) (corresponding to polynomial $x^3 + x + 1$) by tapping at bits 3 and 1 and it has resulted in a sequence of period $2^3$ - 1 = 7.

When I tried to apply this same logic to the primitive polynomial (5, 2, 0) (corresponding to polynomial $x^5 + x^2 + 1$) it yields a sequence with period 15. While it should have yielded a sequence with period 2^5 -1 = 31. Following is the circuit diagram for (5, 2, 0)

1) What is the exact procedure to convert the given polynomial into a maximal length LFSR?

2) How do I determine the correct tap bits for a given polynomial?

Answer 1

The LFSR that you give is really of length $4$ rather than $5$, since $b_1$ never gets used. The correct way to implement a "Fibonacci" LFSR is explained in the Wikipedia article: $b_1$ should be the leftmost bit rather than the rightmost bit.

There is also an alternative kind of LFSR knows as a "Galois" LFSR, also described in Wikipedia, which is dual to a Fibonacci LFSR: at each point, the contents of the register is shifted, and if the bit which is shifted out is 1, then a mask is XORed to the contents. In both types of LFSR, the resulting sequence has maximum period iff the feedback polynomial is primitive.

---

The polynomials that you give as examples are known as trinomials, polynomials with three non-zero monomials. Not all trinomials are primitive. There are algorithms for deciding whether a trinomial, or any other polynomial, is primitive. As far as I know, there is no explicit construction of primitive polynomials.

How do we test whether a given polynomial $P$ of degree $n$ is primitive? The first step is verifying that it is irreducible. The second step verifies that $x^{(2^n-1)/p} \not\equiv 1 \pmod{P}$ for every prime divisor $p$ of $2^n-1$ – this step requires factoring $2^n-1$. A list of primitive polynomials can be found on Wikipedia.