Can you un-rest it for a second and tell me what you mean by that? For our purposes, a lattice is just a specialization of a vector space, and LLL is (1) not a whole lot harder to grok than Graham-Schmidt and (2) available in every serious library and in Sage, which is how people generally do this. If you have zero linear algebra, this sounds forbidding, but the fundamentals you need before tackling lattices and LLL are like, 1st semester linear algebra, and you can self-study your way to it.

Sean Devlin has talked a bunch of people through actually writing these attacks in cryptopals set 8. We talked English professors through the "number-theoretic" attacks on RSA in cryptopals set 6. It's fine if you don't want to dip into this stuff, but I'm not OK with the pretense that this intuition is somehow unattainable.

We need more people playing with these attacks, and fewer people trying to assemble new cryptosystems out of libraries they understand only from the documentation on the web page.