## projecting roots of unity onto 1 dimension

yesterday, I thought of  Turing’s Q function in terms of providing borel-type subsets (or open intervals) to a group engine – all in all modeling the wandering through a combinatoric space via left-recursion.

But, its also possible to look at things in terms of roots of unity.

We know that we can always swap our thinking module out, when doing this kind of work, and start “thinking” in polar and complex planes. If we see turing drawing a wheel with –1, 0, 1, 2 terms…and then drawing a (–1,1) wheel, we might be tempted to see him thinking in terms of roots of unity – and group algebras thereof.

Take the ith roots (at 2pi.i. over n) as point on the unit circle. Assume an even number, and draw a line between points that cuts the axis at 90 degrees, at point x=t. Look at this t distance from the origin as a “1-dimensional” measure of those ith roots.

If you rotate the circle (under the action of smaller wheel, wreath product style), not only do the ith roots get new names (viewing each ith root as a generator, that may be positive or negative in a symmtric generator set) it also “shift their spectrum” on that 1 d line – which is of course germane to Tunny skills, reading the output of the approximate counting done by Colossus to see if the identified setting for the newly-guessed impulse stream is supported by the right spectral weights, in the right places, looking at the combined spectrum.

One now starts to think a little of bent functions in a sbox, that must work with the OTHER bent functions conforming the sbox, to ensure that they are mutually supportive when doing their “one-way function” thing.

we also have to think “more like Tunny” realizing that the distribution at t = (pi/4)/n is the uniform distribution, on the hypercube walk – remembering that in Colossus process folks were doing spectral scanning for each eigenvector with weight 1. This is pertinent to modern quantum computing of cryptanalysis and its associated speedup, leveraging the “continuity” of the infinite impulse response to drive a quantum-search – rather than punt and wave hands.