We have seen how to use the Hadamard transform to make a coin flip into an operator that induces a shift operation. And we saw how Turing, back in 1954 or earlier had a shift operator for a unitary swap (called an upright in GCCS terminology) – comparing the inter-point distances before and after conjugation, noting the “invariant distance” before related to that after, by a shift of the digits. Lets put the 2 notions together, in the context of understand DES.
Fortunately, someone seems to help us. For we recall that the P() function of the DES basically encodes 3 graphs, the first acting (by diffusion) on the sbox to the left (of current), the last acting (by diffusion) on the sbox (to the right of current) and the middle one (we assume) acting as a a way of putting the two previous results into superposition – ready for the next round.
The following explanation does better!
We see H transform applied not to put a coin flip state into superposition but to the n-1 phase (sbox) and the n+1 phase (sbox) and the current phase (sbox) at the current time. Lets assume that it acts similarly to the coin flip = putting the three phases (before, current and after) into superposition. Lets note how this allows one to transform the wave equation itself such that we now have a statement of what the current position will be (having diffused out, to a couple of degree in various directions) at the next point in time. WE think of this as a special-coin that, post diffusing and loss of energy, takes the now-lesser energy state and rejiggers it back …into the superposition form suited to the next round of calculation.
We see how the “operator form of hadard” that acted a single classical random bit has become a more-evolved hadamard (for want of a missing term): with now the left and the right forms of Phi acting on state 0 (and state 1). That is we have the before-after-sbox-coin, as a class of randomness that will act as a coumpound-generator, inducing a compound-action on the compound-hamiltonian of the boolean/bent functions stored within the sboxes.