Magic square 2 of 13 shared secret scheme

the centrosymmetric magic square has 12 lines along which the magic number (in our case 1033), 5 vertical, 5 horizontal and 2 diagonals. Now because of the centrosymmetric property, the matrix has 13 elements. Any such structure then can be represented by one number and the position of the number. This is because if the magic sum and one specific element is known the above properties describe a 12 variable linear equation system with a unique real solution. For arbitrary numbers integer solutions won't exist, but for any x the magic square of x*1033 the elements will be x times the repective element of our original.

Now taken our particular magic 1033 matrix, our integers are relative primes so if you have any 2 elements their ratios will determine their position. Therefore our matrix can be used as a 2 of 13 cipher the following way. Take your secret, turn it into a number S, multiply it by 1033 then calculate all 13 elements of the magic matrix of base S*1033, then release them. Upon catching 2 of those, one can simply take the ratio, map it to the known ratios of our favourite magic square and voila, you identified the position. From one element and its position you can calculate the secret number S.


Rosecrucian compass cipher - segment permutation

Instead of squaring the circle lets circle the square. Our magic square can be represented as an onion like structure with 3 concentric layers. This is basically a compass layout, I suggest you imagine the drawing here :
https://infotomb.com/25z8v

This structure can be construed with the help of a compass and ruler very easily. Now because of the rotational symmetries of the matrix, the compass can be turned 180 degrees, both layers independently. Let's look at how we fill in our magic square. Any missing elements can be supplied by rotations if one side had the number and the other side didnt. For instance if we got 341 at north in the outer layer but south is covered, the rotation will uncover the south value. The other way to uncover a number is if we know it is in a line with 4 (or 2 in case of rotational symmetry) uncoveredd elements since we know they need to add up to 1033.

Now we can ask the question how many elements and in which position are needed to fill (or uncover) the full magic square if we allow only rotations (R) and line completion (L). The answer is interesting. For instance almost all cases when the outer layer has 6 out of 8 uncovered positions, and the middle layer or center is the 7th. It turns out altogether 6 positions can be enough too (positions 1,2,4,5,6,10, using continuous left-right downwards reading style indexing).
But for the canonical case we can for instance take the arrangement with positions 1,2,4,5,10,11,14 revealed. Now this is a minimal representation in that no shown positions can be covered and still allow full completion of the magic squar using rotations and line completion. After rotating both layers 180 degrees we obtain the drawing on the original page 4 jpeg, where the runes on the matrix represent the covered positions. If you draw this in the circular compass format, it is obvious that it is a rosecrucian cross. It even looks like a sophisticated cipher device with a lock push from the middle layer outwards (cover moves from 245 to 226).

lets see now if we can say anything about the particular order of operations.

An observation is that 22 on the left hand side (position 11) is open while 226 on the right at position 15 it is covered. However all the south section of the outer layer is covered. If we assume this configuration results from a minimal representation we need to assume that instead of 180 rotation of the compass layer we individually flip opposite sides of each layer . This corresponds to rotation of an imaginary perpendicular wheel in 3d by 180 degrees (just for cool visualisation).

Another impportant realisation is that 341 is not eliminated before we reach clear the south bit, this suggests that line completion comes after all rotations.

Note further that 245 is not flipped so middle layer rotations come after outer layer ones.

These pieces of information together mean that for a minimal cover scheme there is a unique canonical way to fill in the matrix recording every position that is flipped or uncovered. Obviously using only the first 13 indexes.
Take the example of the above north oriented crescent, the steps are the following, 1,2,4,5,10,11,12,3,13,7,8,9
           R R R R R  R  R  C C  C C C
It s easy to see that because of the rotational symmetry the geometric path is isomorphic if you got a cross pointing say NorthEast. The permutation itself is nontrivial since the matrix positions do not reflect the circular symmetry of the compass. Though nontrivial, it is still deterministic once we fix the pattern. Fixing the pattern here means fixing the first flip and the orientation, so each permutation (3bit information) can be equally characterized by a starting point and a direction. I take it that the hint for all primes are sacred (plural) and the totient function (singular) is sacred means 131, 151, 18, which is an unambiguous reference to position 4 and clockwise direction (key=4R).

If the hex strings are encrypted this could be segment transposition corresponding to the permutation described by the magic square cipher with key 4R. It is not necessary that the ciphertext has all the necessary 13 parts. In fact if we got 2 of them (one with key the other without), then 12 segments is both necessary and suffiecient to fully decode 13 segments.
If the hex strings are what we are looking for as ciphertext, then one segment is 256/12 = 63 byte long. Alternatively the ciphertext is the p4 garbage outguess