So how did cryptologists in 1938 type period design the wheels for the UK Typex machine? What were the design principles? What were the methods?

Well even today, the answers to whose question are still “secrets” – even in the core knowhow is now well known. After all, the answer has to be called “a secret” BECAUSE its on the topic of crypto. Whether or not the entire planet knows the answer is irrelevant. Its Crypto, and is designated for “information control” for the purpose of keeping secrets from the “public” because of that fact, alone. If 1 + 1 = 2 was a crypto matter, it’s would be classified as a (ridiculous) UK “secret” – despite one or two million five year olds knowing the identity.

Well lets take some guesses at some of the answers.

First, “good” wheel wirings required machines (or rooms of people, playing machine) to “search out” those wirings that met certain conditions. While the math could eliminate entire sets of non-starters to get to a starting set (upon which a search-based elimination process could further refine the candidates based on “testing” the candidate, in practice).

So let’s assume that the drivel about the first Turing bombe “suddenly” coming into being in 1939 is just that (propaganda drivel). Lets assume that classes of search machine (for wiring design) already existed, as did the method of “setting up” the search. Yes! I’d expect the history of that to be suppressed (if true..) today; still. It’s not only that its been a convenient lie for so long (that would motivate perpetuating the lie yet still behind a modern secrecy wall, mostly to control the “I’m a UK liar” embarrassment factor and the consequences of official lies being unmasked). Its also important to realize that WHEN things were known (even if 50 years ago) is sensitive information. Knowing how one lie was timed and what areas of knowhow of technology it was masking gives good hints to the next…in the line. Now one can start to see WHERE micro-computer technology, for example, was being applied. Or today, where banks of asics built on a lithography level far beyond where folks think it could even “could” realistically/economically be. Once the dam breaks, folks start to look out of the box and suddenly one can fore-see todays “secrets technology” – ”levels”. It just required undermining the social conditioning set up “not to look” (or think).

So, in the late 1930, lets assume that folks could choose “starter wheel” wirings (almost at random). And then, counting machines would go looking for aligned with the even members of the the alternating group (i.e. count the inversions). Then folks would consider each members of that group as a commutator and attempt to use it (along with all the others, later), as a generator (of another group). One would consider what happens mathematically when, as in an enigma/typex machine, the first wheel is stepped along one. One has to recall that the stepping is relative to the wiring plan coming IN to the wheel (whose pads are also labeled…). Each wiring plan or pad labeling is static – until you rotate/step the wheel. Without explaining it (but see any general enigma book on Friedman squares), the net result is that the group elements of the pre-stepped wheel are “conjugated” (to another rod, in the rod square). In programming terms, stepping elicits the “action” that generates the next wheel permutation, almost as if one just “ran the program” implied within the wheel. So one SHOULD (these days) look at a wheel’s wiring as the encoding of a program (that has within the elicited action).

Having conceived of a enigma wheel as an encoded program, we want self-describing programs – or programs that, upon eliciting the action, producing… more programs…whose own “rod” actions continue the chain (of actions, derived actions, and actions that imply actions from which one can derive yet more actions…).

What we want thus a starter action than can generate an entire “family” of groups (with the action-generating property). Or, more properly, graphs (as in Cayley graphs) that generate further graphs in the family (each having nodes/group-elements and edges/group-generators that retain the desired property of “perpetuating the family”).

Now, to go further here, we have to more firmly distinguish between commutators, conjugators and why the distinction matters IN the world of wheel writing. If one wants the math, just read any text!

For now, its worth simply noting the notion explained well in http://en.wikipedia.org/wiki/Cayley_table in which an identity “skeleton” (a term found in WWII cryptanalysis docs note) can be explored, building upon certain matters (such as inverse) to leverage ASSUMPTIONS (that may yet yield a contraction, and thus a backtracking in the search process). We see to have, as would have been apparent even in 1920, the elements of a search process (looking for contradictions) and a reason to be doing some searching (wheel wirings).

http://en.wikipedia.org/wiki/Dihedral_group_of_order_6 gives clear background to the whole issue of why Turing was so focussed in 2-cycles and 3-cycles. WE are really dealing with lagrangians (which relate to path walking in adjacency matrices). We are also dealing with the “search conditions” that assurace the that matrix gives a fully connected graph, which gets to expanders (and thus to families of expanders derived by the action).