We know that the joint cipher machine of wwii required each of typex and sigaba to swap out components…to effect the same process. Given to all allies for batallion tactical communication security, one assumes that the uk/us kept things weak mostly so as to spy on their Allies. Thus, a particular stepping scheme was used, rotating the typex or sigaba enigma wheels …that were obviously wired in common and used in wheel orders that formed part of the daily key. In the sigaba case, this meant swapping out the cage with the us-eyes-only wheel sets that generated sigaba’s own native stepping sequence.
If we look at the sigaba native stepper, we note the timeline. Its first version comes about in the mid 1920s, a couple of decades into the new math of quantum mechanics, measurable functions, functions on measuring operators, group theory for modelling a qm system, and the general notion of density evolution. By mid 1930s, the keytape used on an operational machine is replaced by the stepper cage (and its theory of operation), generating key as needed. Which leads us to consume consider the theory, assuming that what was first used to generate the keytapes was the same as that used later in the design of the stepper action.
A constant function is applied to the first wheel of the set of index rotors, that we assume are identically wired for now. These wheels are set to wheel positions, using keysheet “indicator” instructions. The net offsets sum to zero, reflecting the intent that the resulting action upon the constant function acts as does turings normal subgroup, inducing a 1930s theory style qm measuing operator (or a sequebce of such, rather, since there are several wheels). In dual space, this induces a stochastic operator on the wave function. As with chaos theory of the day, measures are rescaled and distances are expanded; with the result that the containing space of density functions is itself contracted. In the special case that the contraction is a one way function over lipschitz spaces (think non unitary qm) this has the effect of decaying the norms, which is that driving the generation of distances/measures that differentiate/remeasure the density and drive the correlations between denstities towards the normal pdf.
Now assune that the main function of the second set of wheels is to qualify the generated density, with certain wheel not only transposing wires thing confuse each sensitivity class independenty but also to use further “concentrate” the variance, with “better” reserved for higher sensitivities of data.
We can now turn to 1930s american spying on imperial Japan ciphers, having created an art of crytpanalysis (of correlations in stepper actions, over time) founded in the knowledge of the how the design principles behind sigaba had bounds (for the asymptotic convergence). Knowing that lack of ideal concentration of variance is the weak point (motivating the second wheel sets indicator to be divided up so the better settings, with more ideal concentrations, are reserved for the more secret data flow) of course machinary of exploiting this cryptanakytical secret is leveraged, when breaking japanese steppers driving their polyalphabetic rotors. Of course thus meant translating abstract theory into something practical, which meant counting things – first by hand and then using hollywood film projectors moving high speed tapes (film on whose frames countable spots are exposed, photograhically) over each other to effect different counting algorithms.
By the time ah folk are analyzng how collosus is being applied (in the attack on tunny only, note) one notes how they recognize what they already know (in what they still believe are restricted from the uk american-eyes-only “core” ciphering and cryptanalytical secrets) they are revealing their theories (of correlations, uncertainty due to commutator actions, and log likelihood methods).