perhaps, one can look at du/dXi as the xor differential, in DES, whereas d2u/dXidXj is the differential between the outputs of the 2 neighboring boxes. Do we now have a “theory” for the diffusing properties of DES, when putting together (i) the likelihood framework for cryptanalysis, (ii) the idea that the 2 4bit sboxes are a manifestation of the 8-element quaternion group, and (iii) one now has a means to “control” the evolution of the conditional probabilities, so as to drive the first-order differences towards the uniform density?
its almost as as if having decided that one determined that one must defeat the colossus-era probabilistic attack, so then on turns on what makes that attach difficult and exploits it in cipher design, now. This means concocting an SP network that acts as a generator of fields, that act as differencing operators/actors. IN the case of DES, and its neighboring sbox properties, this means looking at output line dependencies as the field, where one continues to evolve the field and so one continuously evolves the conditional probabilities to fall within SNR + 1/n. It’s the goals of the quaternion algebra to ensure that there is no inherited algebraic structures from one round to another, despite the characteristics of the SPN and its differential trail resulting from the dependency on subkey keybits.