## circulants, key schedules eigenvalues, and quantum walks

Lets assume that the main purposes of the DES-like key schedule to to parameterize a general purpose computing graph (i.e. the rest of the DES method). So what form do the parameters take?

Lets assume that the “rest of the DES method” is a (parameterizable) means to evolve quantum walks. That is quantum walks, not random walks; where the quantum world requires reversibility. Despite that constraint, we still want to perform a quantum walk on a ring, so our equivalent-world of a stationary distribution is uniform.

Now lets assume that the subkeys really are rows of a circulant matrix, that gives us a sequence of eigenvalues. Assume that the distances between the values is what then parameterizes the “quantum walker”- affecting the (nearest-neighboring style) averaging process. Should one step over a row when generating subkeys, this is a akin to ignoring the eigenvalue associated with that row in the ordered sequence of eigenvalues, increasing the distance between its predecessor and successor values. This gap along with the others between ALL the values, rather like the 1 gap used in mere-random walk theory, controls the Quantum-mixing.

we can think of the two halves of the DES Mixer as the 24-term-bra and 24-term ket; and each get a distinct set of mixing parameters. At the end of the day, we have to conserve probability, but we want to have the quantum mixer do the work. Since the bra, acted on to the left by the H.dagger gets a different set of eigenvalue parameters to that of the ket, after on to the right by H, evidently one can conserve probability despite the difference between the two separate key schedules – that don’t obviously have any dual relationship to each other.

ok. so subkey schedule is having two impacts. In this theory, the first impact is that of parameterizing quantum walking, in two dual worlds that act of conserve probability and impose unitary rules. Second, it works WITH the plaintext to control conditional operators, where the calculated conditions are retained as one expands from 48 to 64 bit flows. This suggests that there are only 8 conditional operators…for each duality.

If there is a backdoor, its in the particular sequencing of the conditional operators and their dependency on original key. The backdooring has, as always, to facilitate a world in which any one key bit break affords ever greater advantage to breaking the next, as it further prunes the conditional operator search space. With backdooring, one has an advantage in knowing which branches to prune.