Suddenly realized why Turing, in On Permutations, was so focussed originally on wanting his subgroup to have terms that sum to zero. One sees, from the bit-focused example built into the fano plane, how te test for point membership in a line is reduced to a two-term xor, each term of which it itself two terms (xored). This is Tunny delating, of course.
As the cryptanalytical process reduces noise (or better said, detects a given noise signal) one gets a “test” for the detection process for the needle (in the haystack): the delta’ing would now show properties of the symmetry for the needle case, only. Or, rather, for the needle and its automorphic shadows.
Fun to see group theory concering subgroup and homomorphism relations become goemetry, become rotor rotating,become permutations groups, become bit strings, and become masked bitstrings that impose a parity constraint.
Mix all that then with the quantum harmonic oscillator and its algebra-generating power, where moving a rotor left and right is the basic raising and lowering operators, one sees how it all leads to fractal math and bifucation points that just change the complexity game.