Let’s consider 128-bit pseudorandom generator. From Harris 1960, if we model uniform random permutation, we know that every period length $l$ has equal probability $1/n$ (Eq.Â 5.2) for any particular starting point, where $n = 2^{128}$ is the size of the domain, so the expected cycle length is $sum_{i=1}^n i/n = (n + 1)/2 approx 2^{127}$.

We know that in typical LCG’s modulo $2^{128}$ it is not always the case:

https://en.wikipedia.org/wiki/Linear_congruential_generator

We can use Hullâ€“Dobell Theorem to achevie period equal modulus. But what period we could expect in general in LCG, when we will choose multiplier and increment randomly? Let’s consider only LCG’s modulo $2^{n}$. And the main question. Let’s consider some simple modification of LCG – compute output in little endian, in other words our output is reversed block of ordinary LCG. What period or expected cycle length we can expect with such modification?

My preliminary experiments indicate that these cycles will be larger than in regular LCG’s. But I checked not much LCG’s that way. Also I’m looking for some mathematical aproach or proof.