It also dawned on me your method is basically the same as vonNeumann's 1946 middle square method, which turned out to be terrible from short cycles. His was bas

It also dawned on me your method is basically the same as vonNeumann's 1946 middle square method, which turned out to be terrible from short cycles. His was base 10; yours is base 2. His dropped the bottom and top "bits". Yours drops the top (during the mult). His permuted each time via a different number; yours uses a fixed number. Both use a mult to scramble, and then retain a subset of the bits. Adding more rotates and mults does not change the underlying issues. This type of PRNG has been pretty thoroughly mined over the past 70+ years. You should dig through this work and the following literature on why PRNGs of this structure fell out of favor. I suspect exactly yours appears somewhere following this work. https://en.wikipedia.org/wiki/Middle-square_method https://en.wikipedia.org/wiki/Middle-square_method