three observations about math and crypto.
1. Quaternion “Algebras”
Its fun to look at quaternions as a special kind of polynomial sum, with terms weighted as is usual. Then, its interesting to see how to abstract H, the quaternions, in quaternion algebras – for different basis sets.
In light of how the theory of albelianization of streams, one can feel, quite intuitively, just how this is so crucial to even modern cryptanalysis.
The main point is that F can be anything (including huge number systems) good for cryptanalytical discriminant finding.
2. Wave “functions”
Also fun to review even the basics of quantum mechanics, so well reviewed by Susskind.
He was able to put succinctly how a wave function is just a function of x (just as the are the functions we all learn about, aged 12). Its just that the calculation formulation for that function is an inner product, where x varies (just as it does in the functions we all…).
What he doesn’t do well is just say what Turing said: a wave function is just an array of proportions!
3. hyperbolic geometries and generating algebras
its from hyperbolic geometry that we can a glimpse of how Turing saw quantum modeling and simulation AT AN INTUITIVE level. IN particular, in his on permutations paper, we say how he leveraged just 2 and 3 (2 steps between 3 points) to create normed spaces (of 6 elements); then argued how energy functions (i.e. Hamiltonians evolving the energy component of quantum systems over time ) can be represented simply in terms of permutation groups, leveraging the projection of those functions in a hyperbolic geometry onto a projective plane that supports calculation in terms of long expressions of (ordered) swaps, using just the Newman’s core topological knowledge in foundational groups and homotopic equivalence.
We got to see how conjugation in a hyperbolic geometry is the reflection operation (when the geometry is the interior, or inner space, of the unit circle). Similarly, we got to see how external point relate to the circle too, and how this quickly gets us not only to the notion of duality but to external product space where constraints between two intertwined systems create a hilbert space where quadratures and spreads are preserved; and in which one can do quantum calculations using only proportions.