Let $T_1$, $T_2$ be two $m$-sequences over $\mathbb{F}_q$ of length $q^n-1$, say $T_1 = (\text{Tr}_{q^n | q}(\alpha^i))_{i \geq 0}$, $T_2 = (\text{Tr}_{q^n | q}(\beta^i))_{i \geq 0}$, for some primitive elements $\alpha, \beta$ of $\mathbb{F}_{q^n}$. Let $f_1(x_0,\ldots, x_{n-1})$, $f_2(x_0,\ldots, x_{n-1})$ be the FSR functions of $T_1, T_2$, respectively.
Question: What are the FSR functions of the sequences $T_1 \cdot T_2$, $T_1 + T_2$, where the $\cdot, +$, mean component-wise multiplication, addition, of the sequences. Can we write these FSRs in terms of $f_1, f_2$? Are these NLFSRs? My apologies if these questions are trivial; they are not mentioned in my book. Thanks!
Here's what's known to the best of my knowledge. If one multiplies two nonzero sequences of periods $T_i$, $i=1,2$ over the same field, one obtains a sequence with period in the interval $$\left[\frac{T_1 T_2}{gcd(T_1,T_2)},T_1 T_2\right].$$
If two $m-$sequences over $GF(q)$ whose minimal polynomials have relatively prime degrees $d_i$ are multiplied then we get the lower bound on the period given by $$(q^{d_1}-1)(q^{d_2}-1)/(q-1)^2.$$
In general, we need to go to the splitting field, where we need to consider all products of the form $\alpha_i \beta_j$ as possible roots of the new minimal polynomial for the product sequence, if the $\alpha_i$ are roots for the first $m-$sequence and $\beta_j$ are for the second. Sometimes, we may end up getting the all zero sequence, and such cases must be avoided.
However, more is known, and not just for m-sequences. Obviously, repeated roots and non irreducible polynomials cause some difficulties, and Hasse derivatives are utilized in some of the work. A few references:
Products of linear recurring sequences, Neal Zierler, W.H Mills Journal of Algebra 10/1973; 27(1-27):147-157.
A general lower bound for the linear complexity of the product of shift-register sequences, Rainer Göttfert, Harald Niederreiter, Advances in Cryptology — EUROCRYPT'94, Lecture Notes in Computer Science Volume 950, 1995, pp 223-229.
Sequences of Period 2^N –2, Rainer Göttfert, Sequences and Their Applications – SETA 2006, Lecture Notes in Computer Science Volume 4086, 2006, pp 223-236