it's really fucking annoying how cryptographic keypairs were literally supposed to be just an arbitrary flat string of bits which can be directly and unambiguously interpreted into a bignum as a sequence of machine-size integers. and then in 1982 IBM had lodged sufficient propaganda around "average-case runtime" that nobody was investigating how to leverage the hardness of modular exponentiation into a direct proof that P < NP anymore.
and they decided finite fields were too simple to understand and manipulate without extensive training. this risked anyone having strong crypto they wrote by hand! we can't have that! time to add some fucking polynomial bullshit
the one thing i haven't seen from ECC or normal finite field arithmetic is a way to identify and group all the generating sets starting from a given seed. because that's what i think is necessary to use modular exponentiation as a """true""" RNG