“The Hill variants Bauer left open: blocks chained to their neighbours, and two matrices taking turns”
Four variants of the Hill cipher instead of the plain and affine forms: each block also adds the previous carved block, or the previous message block, or is mixed together with it, or alternates between two matrices. The exact solver of experiment 15 handles every one.
Experiment 15 ruled outdead Said of a period, grid or method when the known letters contradict it: no secret word of that kind can produce the carved letters. the plain Hilla Hill cipher A cipher that turns letters into numbers, takes them two, three or four at a time, and mixes each group with a fixed table of numbers. cipher, where every group of letters is mixed by the same table of numbers on its own. A cryptologist who studied the sculpture listed the variants nobody had tried: mixing each group together with the group before it, or using two tables that take turns. Because the carved lettersthe carved letters The 97 letters cut into the copper at the CIA, shown at the top of every experiment page. are all known, and the known message letters come in two long runs, each variant can be solved directly from the known groups rather than guessed.
Words in the margin are defined on the glossary page.
1.1
The method at work
The carved letters go in at the top; the reading comes out at the bottom.
the variants
Experiment 15 tried plain Hill ciphers. Here each block is mixed with the previous block's output or input, or two tables alternate, for blocks of 2 to 4 under four ways of numbering the letters.
the solve
Whole blocks inside the known letters give exact equations for the table; the known letters in partial blocks are the test.
the bound
144 settings: 2 solve on their whole blocks, and both then miss 8 of the 24 known letters.
2
Best candidate against the known letters
This experiment produced no candidate: the search ruled everything out before a full text was built. Its product is the bounded negative in section 7.
3
Walkthrough, by hand
Redo the method on a few letters with pencil and paper.
Number the letters A=0 to Z=25, blocks of two from position 1.
Ciphertext feedback: the carved pair at 25-26 equals M times the message pair at 25-26 plus the carved pair at 23-24.
Subtract the carved 23-24 from the carved 25-26: that difference is M times TN.
Do the same for the next whole pairs inside EASTNORTHEAST and BERLINCLOCK: six equations in four unknowns.
Solve; if no M fits all of them, this setting is dead.
For two matrices taking turns, split the pairs into even and odd and solve each separately.
the worksheet: the first steps by hand, on squared paper
4
Worked example, generated
The same steps run by the code on the real letters.
ciphertext feedback, blocks of two from position 1, A=0: the equation for the pair at 25-26
carved 25-26 VQ = [21, 16]; carved 23-24 LR = [11, 17]; difference [10, 25]
message 25-26 TN = [19, 13]
so M x (19, 13) = (10, 25) mod 26; the pairs 27-28, 29-30, 31-32 and 33-34 give four more such equations.
5
Hypothesis, for a cryptanalyst
Family, key space, the constraint under test, and the clues it reads.
Bauer (knowledge/sources/bauer-2017-unsolved-chapter-notes.md, C21) lists chaining and paired 2x2 matrices as untested Hill variants. Chained forms: c_i = M p_i + c_(i-1) (ciphertext feedback), c_i = M p_i + p_(i-1) (plaintext feedback), c_i = M (p_i + p_(i-1)) (summed input), with the first block's predecessor taken as zeros. Paired form: M_A on even blocks, M_B on odd. Each is linear in the matrix given the known blocks, since the previous carved block is always known and the previous message block is known inside a crib run, so the whole crib blocks give rows of equations and the exact mod-13 and mod-2 solver of experiment 15 enumerates every matrix. Block sizes 2-4, four numberings, every alignment, matrix encrypting or decrypting. A matrix consistent with all its blocks decrypts the text and the partial-block cribs test it. Bounds: complete for these four variants under the stated settings.
6
Prediction
Written before the run; the result decides it.
No variant, size, numbering, alignment or direction gives a matrix consistent with all its whole crib blocks that also reproduces the known letters in partial blocks. Zero candidates pass.
7
Method
For each numbering, block size, alignment and variant, list the whole crib blocks and their partners (previous carved block or previous message block).
Build the linear rows for the matrix, in each direction.
Enumerate all solutions mod 26 by elimination mod 13 and mod 2.
Decrypt the text with each invertible solution and count known letters right.
Submit any decryption with all 24.
8
Result and bounds
The harness's record: candidates, gates, loss, and the negative result this experiment leaves behind.