## From crypto wheel wiring to bilinear transforms and log-likelihood basis

When I studied conditional probability, methods and notation , from the department of statistics (of which our computer science “unit” was originally a part), I struggled. No,  let’s be truthful: I hated it. Studying the same ideas, notation and methods in the context of tunny cryptanalysis (and coding theory) is just fun!

IN particular, the disclosures about the cryptanalysis of Tunny all point to conditional probability applications in which one is interested in something QUITE intuitive: bigrams. Does E follow A, etc? Or, does wheel bit dot follow the previous bit on the Tunny wheel?

Studying LDPC theory really helps, for we see the modern theory of bulges and probability.

From circuits and the paths that form up from particular wiring of wheels, we can get to bulges and distance – as theory of wheel wirings. Not all wheels are wired one pin, on left side, to one pad, on right side! Certain pads on the right may be wired together on the right (aswell as connect to a pin or two  the left…

Building upon the notion that certain circuits designs (aka algorithms) can increase certainty about bit values in such as wheels being broken  –  a process now known generally as belief propagation or message passing – we see that two complementary design components are in fact linked, by a transform.

First the likelihood aspects of a wiring can be thought of how pins on the left (lambda) relate to pads on the other wheel side . Or, how the next bit (possibly 1 or 0) relates to the state of the current wheel bit. Second, the flip side of the same coin considers the probability difference (of these scenarios). IN each case, one can look at a value like lambda or mu; or cast such in terms of a set of basis states. One should be immediately thinking quantum mechanics math, here. I can be looking at the bulge for the “generating” case that controls evolution/analysis; or I can be looking at evidence of two particular cases (once generated).

Given two objects that are in a sense dual to each other, its fun to the role of the bilinear transform that processes one into the other. This reminds one, as in quantum mechanics, that one can be calculating in the adoint space of bras or the world of kets – not that it makes much difference.

Its also fun then to see how one can also change basis from abstract to concrete – and the same relationship holds when working in log-likelihood differences or ratios.