Lets see if we can relate alpha, below
to the earlier work of the paper expressed in permutation group theory
We look at a “small wheel’s” 2-element rotation (α0 α1) as – under the conjugating effect of R powers) – a means of inducing an arbitrary motion of the “bigger wheel” of s elements”, in which element s moves to s+1 (for each s, obviously). Obviously, this should feel a little like the world of semi-direct products of groups. Or, better still, the world of quantum computing.
We see, in Dirac notation, Turing’s concept. S is Turing action, too ; taking a compound state to some thing that satisfies the “possibilities specified by” the equation S, given. That is, view it as a specification of random walking space, for a token wandering around turing’s (α2 α-2 ) ring of points (see his diagram, above) under the action induced by a random coin being thrown and coming up 0/1 (i.e. his “core” cycle (α0 α1) )
The typesetting of S is awful – helping making quantum theory appear hard and awfully mathematical (when its not!) S is only saying is what Turing was saying (50 years earlier!) Or more fully, a coin is thrown, coming up heads. If so, a sequence of motions happens of the nominal tokens on i moving to i+1 as the large wheel rotates anti-clockwise, which can alternatively be expressed as making the point labeled –2 become -1 (as it goes around the possible labels, adding one). If the coin comes up tails, motion happens from i to i-1 (in which –1 would become –2, and –2 would become…. +2). Of course, being a wheel, the “collective motion” of the space happens as a “sum of” several “parallel” motions (a collective set of re-labellings).
S says some more, which is VERY turing like. The very pure math “branching conditional” between the impulse and the motion is represented by the blue cross. That the impulse induces a set of “parallel-processed” relabelings (point movements) is represented by the Sigma over i. (Yes this is NOT a numerical counting-sum, but a “specification of parallelism”). i.e. . Dirac notation helps, since i (bra) is the input state, and i+1 is the output state (ket) as one performs the walking step. One doesn’t need diagonalized matrices, now!
Of course, what is an automorphism… but one of the set of exactly these relabeling functions (of the points on the ring)?
The red plus in S means that the labels of the wheel can move over time per the rule on the left of the plus –a (n unmentioned) number of) positions in which point labeled i becomes i+1 (i.e. the +1 generator was applied, giving a clockwise relabeling of points). “Or” the rule on the right occurs, in which the contrary wheel rotation occurs, relabeling backwards (from i to i-1) – based on the –1 generator.. In some sense we could have an entire sequence of some forward and some backward movements, leading to the current “state” of the labels, which is the notion of (red) plus.
Hey, alternatively we can think of a token on the points walking the points, similarly, so a “random walk”. Its better to think in terms of point re-labelings, however, since then conjugations and automorphisms “explain” and “calculate” the random walk of the particle/token in more “modern” quantum computing theory (circa 1954!). A particle moves forward a step… when – alternatively – the descriptor-labels move backwards – RELATIVELY. (REmember his 1920-30 math training.. when relativity is hot!) And of course several such re-labeling can be composed, using the red plus, which is of course the automorphism GROUP.
Ok this is lovely – since lots of really badly taught pure math is actually something entirely intuitive. if you have the intuitive model (a wheel motion), the formal modeling language is trivial!
Now, later we see Turing talk about the coin function being the subgroup H1 – where H1 can have different definitions – according to the types of biases one wants of a (multi-dimensional) coin, doing n-dimensional flipping. With just a classical coin (with 2 states), we get 1 and k (and –1 and –k) – when H1 is the quaternion group. The 1 term gives us +1 and -1 generator, which induces the actions as desired. The k term nominally gives +k and –k “generators” – which MAY BE what happens when you have the –1 generator act on –2 (when k=2 or -k=-2) which does not give –3 but induces –2 to d-flip over to 2 (since 2 follows –2, around the ring). Which aligns PERFECTLY with the sign flipping nature of the quaternion group as you go around a particular line.
Ok I get it (a year later).
Boy you are crap at explaining things, Turing. Or, rather, you assume context – probably per the “oral paper reading” culture infamous in the Cambridge tutorial system – in which you can assume a “reading context” – missing in the paper itself. I had a boss once trained in that, and it was infuriating until you got used to it (at which point it became superb way of advanced folks to interact).
so his 1-d projection of the “walk specifier” Phi()*Phi()*… is really nothing more than an allusion to such as
now K makes sense, being a random walk… inducing what he probably wanted – a stream generator for use in an additive cipher. So, is the delilah key stream generator? IS this the guts of the wheel wirings “design theory” of ECM (when attempting to produce a random stepping sequence for its polyalphabetic wheels).
Essentially, he is “taking the history” of the plaintext and using each to pick an automorphism that rotates his labels, inducing conjugations (in a more “refined” manner than enigma does it). Once projected onto a frequency distribution, the group theory ensures that the choice of re-labellings eventually converges to uniform selection of leaves. Or for a quaternion basis, H1 cycles creating a factor group of s=2. I think of the sign change of 1 to –k (say) as a clockpulse from low to high …which maps onto the S, nicely. If the clock is in a hadamard mixed state, then the action of the H1 “memory/history” terms induces the pulse (before returning to being balanced), which conditionally induces the dynamics, which take into consideration the memory of the plaintext – rather like a Tunny goback2onV, etc – which tells use a little of what the designers of Tunney may have been aiming for! It also makes sense now why a Turing, particularly if the work was post Tunny, would be modeling such a thing.
So he has function generator for automorphism expressions, cued off the plaintext, which uniformly eventually chooses the mapping of the current plaintext to its output. Thus, there is perhaps no “additive” of this “generated stream” to a plaintext stream – which would have been REALLY hard with the limited electronics of Delilah. He has a block cipher of 1 char!, rather like Tunny and Hagelin. its just that his “tunny motor” is entirely the plaintext – which aligns with the theories of the day for auto-inameiveforgotten crypto designs. It also aligns with step-and-go and other similar means to conditionally step (or sample) a stream to produce an additive.