Here's my layman's take - basically it's similar to how computers convert binary integer representations to integers. For instance, a binary number of 0b1111 c

Here's my layman's take - basically it's similar to how computers convert binary integer representations to integers. For instance, a binary number of 0b1111 could be represented as 2^3 + 2^2 + 2^1 + 2^0 = 15, or 8+4+2+1=15. You now have a typical polynomial which can also be factored algebraically and quickly using something like Mathmatica. Not all of these binary conversion type polynomials can be factored however as they're irreducible. There's a lot of shifting around in big numbers when you compare the normal binary conversion and what is algebraically viable to be factored. In my conjecture - it's not clear to me the exact mathematical connection between the normal binary conversion and the perfect polynomial factors we're searching for. I believe it still may be possible to find these unknown polynomial factors by migrating some of the polynomial terms through redistribution of higher value terms and/or performing some sort of hard search. So instead of 2^4+2^1, you can rewrite as 2^3+2^3+2^1, for example. The factors are exponentially smaller and the search space is reasonable considering that it's basically possible to reduce the polynomial to a large matrix. I gave up on this path a year ago, so I may be a bit off in my recollection. I was using SymPy (a computer algebra system) but the speed was an issue.