https://yorkporc.wordpress.com/2011/03/10/understanding-des-by-improving-it/

We can look at the above in two ways:

First, the inputs (in the middle) are dispersed towards the codebook (left) and the dual codebook (to the right).

Second, the codebook is conformed from the 7 columns (those left-most). Similarly, the dual code is conformed of the rightmost columns.

Now one can think simply. To generate a coin-tossed (half) set of heads, with equal improbability, use 8 cayley groups representing the cycles of a 8-term symmetric group, for which each ith line (of 7 __leftmost__ terms) is the i’th cayley group’s generator term. Similarly, generate a coin-tossed (half) set of tails, with equal improbability, use 8 cayley groups …for which each i’th line (of 7 __rightmost __terms) is that cayley group’s generator term.

You can think of this as a physical process in which electrons and hole flows are being create in a semi-conductor material, suitably doped to create the tension. At the end of the day things need to balance out. in the crypto world, the pool of heads and the pool of tails than square off, being “EQUIVALENT” in improbability metrics.

Of course, Turing modeled taking 7 of 8 terms, too, getting to cosets that become a group algebra in their own right.

We might look at the distances between 1 7 4 2 5 6 8, and 3 8 6 5 2 4 7.

i.e. 632312, and 521323, then remember turing trick (rotate to a lowest numeric value):

126323, and 132352.

interesting to see just how close those two are!, if you now reverse the first.

126323 reversed is 323621, which rotated into lowest numeric value is 132362, which is only a single digit different to 132352 (which is not totally suprising!!) and its in the 5th column.

I wonder if the rule happens to hold for the next line!

so 2 6 8 3 7 5 1 and 4 1 5 7 3 8 6

had distance 4 2 5 4 2 4 and 3 4 2 4 5 2 then remember turing trick (rotate to a lowest numeric value):

2 4 4 2 5 4 and 2 3 4 2 4 5

2 4 4 2 5 4 reversed is 4 5 2 4 4 2 which rotated into lowest numeric value is 2 4 4 2 4 5, which is different in the second column, off by one.

so 3 5 1 4 6 7 2 and 8 2 7 6 4 1 5

had distance 2 4 3 2 1 5 and 6 5 1 2 3 4 then remember turing trick (rotate to a lowest numeric value):

1 5 2 4 3 2 and 1 2 3 4 6 5

1 5 2 4 3 2 reversed is 2 3 4 2 5 1 which rotated into lowest numeric value is 1 2 3 4 2 5, which is different in the fifth digit, off by 4.

Well, obviously, there is a pattern here!

its almost as if the common space is inducing two sets of row swaps, changing the basis orders as the code book or the dual book is generated.