GCHQ apparently posted a Turing paper on repeat rates (which doesn’t appear that interesting, to be honest, from the public view we have). Without having seen what was billed as a theoretical case for the method of repeats we can yet see the clear lineage between elements of Turingismus and Banburismus. From the Prof’s book (ellsbury edition)
Prof’s Book, transcribed at http://www.ellsbury.com/profsbk/profsbk-134.htm
Having guessed that the operators producing several enigma ciphered messages have chosen the same “position” (of rotors) or that the position chosen on one happens to overlap later with the other’s start, once things have stepped a bit, Turing advocates building a matrix — with n rows and columns whose number would have been 100-200, given the length of typical messages. For a given column, count the number of identical pairs and the number of all pairs. Averaging many such counts, form the ratio of the two to act as a discriminant of enigma traffic (of a certain era, contingent on the value).
James Coughlan does a great job of leading us back in time to understand how folks thought about cryptanalysis, in 1939 – while actually discussing application of techniques to stereoscopic vision problems. Presumably the term BP was originally a joke…(Bletchley Park, Belief Propagation)
citations for all quotes, below
You have to start out in outer-space. No, not the NASA canned type with pretty pictures but Newton’s outer space, where the nutty professor spent his mental time. You are in a gassy “fluid”, perhaps similar to steamy steam room at a spa. While you the machine may move, your impact on the fluid is pretty minimal, but at least you define an inner space. At most, you move some molecules around outer space, and the thermal energy re-distributes in accord with your use of it as you change the boundaries of your inner space. Using a thermal camera, one might imagine being able to see multi-colored “flux lines” connecting regions of the fluidic gas to your motion, reflecting the change in energies in the density. If you a nutty math type, you can think that fluxion particles are released and undergo motion, to account for the energy change (in the “vortex”). The imaginary fluxion is the infamous x.dot (of calculus fame). It strikes you with a certain momentum that reflects the very energy redistribution induced by you and the steam cloud interacting – thus describing what just happened. It also has math properties that “keep the invariants” of fluxions in a particular energy band, per the master plan of Newton’s god, now proven to have a fluidic hand in (or on) everything.
It’s not clear to me whether Turing invented BP in the early days of his BP tenure, brought it with him (from Cambridge or Princeton), or it was already at BP in genus form. The latter is quite plausible, note. But what is it?
Well to Turing trained in real analysis, he is well used to an idea: the idea of the L2 space (of finite energy functions) and integration using horizontal bands across the area-under-the curve (rather than the vertical bands know to any 14 year old math student). The change of perspective means you are used to counting up multiple components sliced out by the band to form the composite area, formed as the curve interacts with the band at multiple points. He is also used to using math for engineering-modeling – including considering the molecules in the super-misty steam room to each be a meandering functions, or a tiny histogram representing one tiny probability function of said molecule as it wanders “in a density”. The very idea of the totting up of the various contributions “area under a curve for a given band” can be generalized – to adding up the contributing functions that are located in a given multi-dimensional band, cross-crossing the multi-dimensional fluidic space.
Depending whether we want to model from the standpoint of the inner space (probability domain) or outer space (energy domain) we get:
f g lower case
F G upper case
Think of f, g and h as places in the integration systems “horizontal band” to be accumulated. Think of F and G as Newton’s fluxions – that “describe the changes” (as does the spec of any “differential” gearbox).
Now compress the two worlds into one, removing the distinction between information and meta-information. Its all the same now, so that (formally imaginary) fluxions actually interact (imparting “causal momentum” that induces changes in probabilities on molecules and other fluxions, alike).
Now my intuition tells me that Turing though in terms of tree-like “boolean networks” , rather than factor graphs associated with the edges in a network. But its also clear that he thought of roup theory as a connected graph, whose edges (collectively) expressed the group properties. Those groups and quotient groups that naturally express linear coding calculations must not have strayed too far from his mind, as he focused on “expression trees”. For the group-graph and the expression tree allow us to see, specifically for cryptanalysis two sides of the same coin, and see them at the same time.
Anyways, let’s reduce fluxions and gassy fluids to the steam room model:
all 4 particles are modeled as a combinatoric-product (in some inner space)
red particles have mutual group properties with each other (and their implications too)
blue particles express their innate causality relations with one or more red particles, being thus “conditional probabilities”
Now at this point we can go back to the 1938/1939 crypto world, a time when folks are able to use the methode de batons to cryptanalyse certain enigma machine wheels using rod squares; but for the rest featuring additional countermeasure BP staff are focused on inventing such as depth metrics (i.e. pairwise notions) and stecker theories (i.e. more pairwise notions). The permutation group theory and its cycle-notation conventions are very much in mind, given their role in forming a catalogue of “box shapes” (conjugate classes) used when cryptanalysing the indicator function of a given enigma graph-puzzle.
So when we see
one should look at (w, x, y) as 1 cycle (i.e. a cycle expression) or “compartment” in the “boxing” of a given (13*2 bigram) alphabet for the wheel. Think of the f not as a function of (w,x,y) as 3 arguments producing a value, but as the 1 fluxion which constrains the 3 molecules in the denoted cycle. It’s the generator matrix or its parity matrix constraint, if you will. We have take the side of fluxions, as it were, looking at the stream room from the perspective of the power shifts occurring along flux lines.
Our goal in cryptanalysis is model the flux lines, denoting each by a weighting or probability. To this end we need to introduce the first calculation model which wants to sum up the individual flux lines and express them as a final probability of the overall state space conformed of an entire system of fluxions driving causality. Of course the fluxion is the bit on the Tunny wheel (or wire in the Enigma wheel). And the cycles are the n ciphertexts in depth, whose first 6 characters are bound up in a “parity expression” relationship – that is of course the infamous Polish discovery not about parity constraints (infact) but congjuate-class constraints in a world of algebras defined by rod squares bound up with the symmetric (and reversible) permutation group.
We can now return to a picture we omitted.
The first element gave us the model of conditionality. The second version of the same thing is more group-like. It says there is archetypical structure involved, with a top row (wheel bits recall) that is a sequence (left to right say), a bottom row (a similar structure, modeling the sequences in cipher/text), and the fluxions’ natures (in the connectives). Or, we might say there is history (in the wheel), present (in the flux), and future (in the ciphertext sequencing). Since the purple wheel “setting” is hidden, we can think of the history (as regards the particular green line) as being hidden and that which we seek to discover.
In this form, the diagram gives us one more thing. Looking at the edges, we can given them labels. Pairwise labels. Though our architypes have imposed structure on the stereotypes of the ciphertext, we can always reduce the model to one of simple connectivity between nodes, all playing some stream room role as peons in a thermodynamic dance. And, in a local sense, we can always look at any one node and consider its particular neighbourhood of connectivity. Call it a clique, or call it a cliché (from 1915 world of older cipher cryptanalysis based on pattern matching and edge detection), it doesn’t matter. We have, per Turing’s later mental model about computers as giant brains, the brain neuron, its connectively to other neurons, and the role of DNA that pre-forms certain connectivity (a starter network whose design “primes the pump” beyond the threshold point in differential signaling that induces all further fluxes to converge).
we saw I<j constraint in Banburismus and Turingismus…focusing on “pairwise” metrics
Having seen how the whole is a sum of its parts (where cause-inducing fluxions and causal-relating molecules have equal bit parts on the stage), enter parallelism (and tensor products)
which derives into a debate about evaluation order and its impacts on convergence:
Once all is said and done and the system has settled down, you can use your “magnetic resonance device” to slice the inner space, so either looking out from a subspace or onto the particular subspace – giving you the cryptanalyst a video game “controller” with the space that can give either the first-person interaction or third-person interaction with the objects in the space. Of course, in crypto, we go beyond 3d and our visualizations are somewhat different.
ok, so without ado, the overall inference world of banburimus and turingismus gets reduced to some plain words (and steam room metaphors, perhaps befitting Turing’s inner life).
But on this stage we never saw the weight of evidence, the ban, etc. SO how does this play in?
Its really quite easy. First, its 1940 and time is of the essence. So lets reduce the calculation to only what we MOST need (when sieving and sifting “wheel breaking stories” derived from the field of received data)
Since we have finally got to scoring, we get to go one final step:
Its fun to see how little this particular explanation focuses on conditionality, hypothesis, and odds. Of course, this is a vital element of the story. For that, the Judea Pearl material seems to do a better job of showing how to calculate with those conditional expressions that, even in binary fields, are really focused on conditional probabilities (not merely the if-then sequential logic of American computing). And, one sees how to calculate in correlation space (a cousin of convolution space), which like a modern turbocode decoder accomodates the idea that you need to recall that may be actually wrong, when you think you are right (or vice-versa). So plan for a pairwise world i.e. 4-state group table (a bit like coset addition tables for quotient groups!)