How to calculate period of stream cipher with multiple LFSR and NLFSR I am interested in knowing how to calculate period of a stream cipher with more than one FSR (linear as well as non linear). I know for a single FSR, period can be calculated with farmula q^L-1 where q is elements in finite field and L is length of feed back polynomial. However what I am confused in is, is this period is considered period of whole cipher if cipher has only one LFR?

Also if cipher has multiple LFR (both linear and non-linear), how can I calculate overall period? will it be equal to peroid of LFR with minimum period or there is some thing else?

Thank you.
While I do believe I understand what you are asking for, I will rephrase my own interpretation of the question, and apologize if I have misunderstood the question.

Question: How do you calculate the period of a stream cipher whose output is determined by multiple feedback shift registers (linear or nonlinear)?

"Determined by multiple feedback shift registers" is somewhat vague, so I will give some concrete examples.

Nonlinear combiners

First off, let's talk about nonlinear combiners. That is, the output of the feedback shift registers (linear or nonlinear) are combined in some way. The scheme would look something like this, where $R_1$, ... , $R_n$ are the shift registers and F is some boolean function.



Let $a_{1}$, ... , $a_{n}$ be the periodic sequences produced by $R_1$, .. , $R_n$ respectively and $b$ be the output sequence from F. Now let $P(a_i)$ denote the period of sequence $a_i$ $\forall i \in \{1, .. , n\}$ and $P(b)$ be the period of sequence $b$. Then the following holds:

$$
P(b) | lcm[P(a_1), ... P(a_n)]
$$
Where lcm denotes the least common multiple. Now, if the FSRs $R_1$, ... , $R_n$ were chosen such that $P(a_i)$ are pairwise coprime, we get a stronger bound:

$$
P(b) = \prod_{i = 1}^{n} P(a_i)
$$

In fact this condition ONLY holds if $P(a_i)$ are pairwise coprime. This result is due to Lidl and Niederreiter.

Decimation

Another way to construct a stream cipher from multiple LFSRs is by letting one LFSR clock another LFSR. In this section, I will not discuss NLFSRs.

We will consider two well-known examples, however in order to keep the answer somewhat brief I will not give full descriptions of them. Links will be given to resources below. Only two LFSRs are used in these examples, $R_1$ and $R_2$.

The Binary Rate Multiplier

A theorem due to Chambers and Jennings (1) show that under the following assumptions:


R1 has a primitive feedback polynomial of degree m, and hence a period of $P(a_1) = 2^m - 1$
R2 has a primitive feedback polynomial of degree n, and hence a period of $P(a_2) = 2^n - 1$
All prime factors of $P(a_1)$ divide $P(a_2)$
$gcd[\sum_{i=0}^{2^m-2}x_i, 2^n - 1] = 1$
Then we have that $P(b) = (2^m - 1)(2^n - 1)$


The Shrinking Generator

Shrinking generators are another way of combining LFSRs by decimation. In the original paper by Coppersmith, Krawczyk and Mansour (2), the following bound is given on the period:


$R_1$, $R_2$ have primitive feedback polynomials of order $r_1$, $r_2$ respectively.
$P(a_1)$, $P(a_2)$ are coprime.
Then $P(b) = (2^{r_1} - 1)2^{r_2 - 1}$.


Note that this is only to give you an idea, and only holds for the cases described above. Below I link to a couple resources for further reading on binary-rate multipliers (BRMs) and shrinking generators, if they are of interest.

(1): W. G. Chambers and S. M. Jennings, "Linear equivalence of certain BRM shift-register sequences," in Electronics Letters, vol. 20, no. 24, pp. 1018-1019, 22 November 1984. 

(2): Coppersmith D., Krawczyk H., Mansour Y. (1994) The Shrinking Generator. In: Stinson D.R. (eds) Advances in Cryptology — CRYPTO’ 93. CRYPTO 1993. Lecture Notes in Computer Science, vol 773. Springer, Berlin, Heidelberg