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I want to prove the following theorem in LFSR:

If $f(x)$ is a polynomial in $\text{GF}(2)$ with exponent $e$, which means $f(x) \mid x^e + 1$.

I want to prove that $e \le 2^n -1$.

I had some attempts in special case that $f(x)$ is irreducible polynomial: if we get the period as $p$ we can easily prove $e \le p$ and from $p \le 2^n - 1$. We can then conclude the desired result but only for the special case.

Let me explain the proof I know:

For the case $f(x)$ is irreducible: $S_1(x) = \sum_{i=0}^{p-1} s_i x^i$ and we know: $G(x) = \frac{S_1(x)}{1 + x^p}$, $G(x) = \frac{\emptyset(x)}{f^\ast(x)}$ from this we know that $\deg(\emptyset) < \deg(f)$, $\frac{\emptyset(x)}{f^\ast(x)} = \frac{S_1(x)}{1 + x^p}$ so we conclude $\emptyset(x)(1 + x^p) = f^\ast(x)S_1(x)$, $(\emptyset, f) = 1$ so we have $f(x) \mid (1 + x^p) \rightarrow e \le p$

and we thereachieve the desired result.

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    $\begingroup$ Suggestion: if $p(x)$ is not irreducible, then $p(x) = q_1(x) q_2(x) ... q_k(x)$ for some multiset of irreducible polynomials $q_1, q_2, ..., q_k$. Would that help? $\endgroup$
    – poncho
    Commented Apr 24 at 17:19
  • $\begingroup$ you've omitted the definition of $n$, is it the degree of the polynomial? $\endgroup$
    – kodlu
    Commented Apr 24 at 17:57
  • $\begingroup$ Yes, sorry I didn't notice. n is the degree of polynomial $\endgroup$ Commented Apr 24 at 18:04
  • $\begingroup$ poncho I guess your suggestion doesn't help. If you have any opinion I will acclaim $\endgroup$ Commented Apr 24 at 18:05

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