# Building an LFSR logical bitmap using characteristic polynomials

How would one build an LFSR logical map, with a characteristic polynomial such as

$$P(x) = x^4 + x^3 + 1$$?

Also, how would you interpret it, and what would you do to do the reverse it in order to create a characteristic polynomial from an LFSR map?

• Hint: calculate the reciprocal of $P(x)$ then use feedback polynomial as in here Apr 8 '19 at 19:06
• Thank you for the link, I am able to understand this much better, thanks kelalaka! Apr 8 '19 at 19:32
• Use sageMath . It has great functions to calculate characteristic polynomial. Apr 9 '19 at 9:56

## 1 Answer

The recipocal $$P^*(x)$$ of polynomial $$P(x)= a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$$ of degree $$n = deg(P(x))$$ is given by;

$$P^*(x) = a_n + a_{n-1}x + \cdots + a_0x^n = x^n P(x^{-1})$$

The feedback polynomial defines the tap points of LFSR. The characteristic polynomial and is the reciprocal of feedback. Therefore;

Let $$P^*(x)= x^4 \cdot P(1/x) = 1 + x + x^4$$ be the feedback polynomial LFSR calculated by the recipocal of characteristic polynomial $$P(x)$$. Then the taps of the LFSR as follows;

For further reading in this subject, Golomb's classic book is advised.

The image is produced with the LFSR drawing library.