Experiment 193

“F8.1: Hill 2x2 with an unknown one-to-one letter numbering (any of 26!), every invertible matrix, both alignments: exact; 3x3 counted and blocked”

F8.1. Earlier Hill tests fixed the numbering (4 standard, 126 vocabulary, 200 K1-K3 words; 0015, 0044, 0055). Here the numbering is free. Counting first: at either alignment 11 digraphs of K4 lie wholly inside the known runs, 22 linear equations mod 26 (about 103 bits) in the numbers of 20-21 letters (79-81 bits) and the 4 matrix entries (17 bits): a surplus of 5-7 bits, so an exact test has power. For 3x3, 6-7 trigraphs give 18-21 equations against 76-81 + 41 bits: 21-32 bits short, no checks, blocked. For each of the 157,248 invertible 2x2 matrices and each alignment, a depth-first search with propagation looks for a one-to-one numbering satisfying all 22 equations (compiled). K4 against 5 shuffled copies (each takes minutes); plants with random numberings.

errorstatus
a new rootparents
0 stime, all phases
0candidates
0/24known letters
unclocked
1

The experiment method

What the method does with the 24 known letters.

The tableauthe table The right-hand half of the sculpture, 26 rows of the alphabet each shifted one place further, starting from the word KRYPTOS.'s last column spells 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., the name of a cipher that mixes letters in pairs with a little matrix of numbers. It was tried with the usual ways of numbering the letters. Here the numbering itself is a secret: any order of the alphabet. The known words still give enough pairs to decide it exactly, for every possible matrix.

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.

OBKRUOXOGHULBSOLIFBBWEASTNORTHEASTOTWTQSJQSSEKZZWATJKLUDIAWINFBBERLINCLOCKWGDKZXTJCDIGKUHUAUEKCARIN: THE COPPER, ALL 97 LETTERS, THE 24 KNOWN PLACES IN ACCENTHill 2x2, any letter numbering, pairs from place 1: does any of the 157,248 matrices admit a numbering? K4, then 5 copiesK45not run: the code stoppedHill 2x2, any letter numbering, pairs from place 2: does any of the 157,248 matrices admit a numbering? K4, then 5 copiesK45the run stopped with an error before its summary; experiment 0194 reran itOUT: NO READING; THE RUN DID NOT FINISHEASTNORTHEASTBERLINCLOCK
the idea

The tableau's last column spells HILL. Here the numbering of the letters is a secret too: any order of the alphabet. For every invertible 2x2 matrix, the known pairs decide exactly whether a numbering exists, on K4 and 5 scrambled copies.

what happened

The run stopped at once, in 0.2 seconds: the job could not be handed to the worker processes. Nothing was decided.

the rerun

0194 is the same test with the job fixed.

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.

  1. Pair the message from place 1: places 21-22 are not both known, 23-24 (A S) are, over the carved L R.
  2. With matrix (a b; c d): an(A) + bn(S) = n(L) and cn(A) + dn(S) = n(R), where n is the secret numbering.
  3. Eleven such pairs give 22 equations; choosing numbers for two or three letters forces the rest.
  4. If no one-to-one numbering survives, that matrix is out.
  5. All 157,248 invertible matrices, both pairings, K4 and scrambled copies.
4

Worked example, generated

The same steps run by the code on the real letters.
5

Hypothesis, for a cryptanalyst

Family, key space, the constraint under test, and the clues it reads.

C21(a) (the HILL column of the tableau), F8. Model: x a one-to-one map letters -> Z26; for each aligned block, (x[c1], x[c2]) = M (x[p1], x[p2]) mod 26 with M invertible. Encryption and decryption matrices, row and column conventions are all covered by enumerating every invertible M. Statistic: matrices admitting a numbering, K4 against 5 copies; expected on a random text about 2^-5 to 2^-7 per alignment in total.

6

Prediction

Written before the run; the result decides it.

No matrix admits a numbering on K4 at either alignment; copies mostly none; plants found. Zero candidates.

7

Method

  1. For each alignment: the digraphs wholly inside the known runs (11 each).
  2. For each invertible 2x2 matrix: two equations per digraph; search for a one-to-one numbering with propagation (node limit 200,000, undecided counted).
  3. K4, 5 shuffled copies, 3 planted Hill texts with random numberings and matrices.
8

Result and bounds

The harness's record: candidates, gates, loss, and the negative result this experiment leaves behind.

0candidates returned
0passed every gate
0/24known letters, best
–lowest loss
nonechampion before

The run stopped with an error: when serializing tuple item 3

How this experiment sits among the others is on the lineage page; its family's open hypotheses are on the families page.

Rendered from the record on 2026-10-18 12:26 UTC; updated after every experiment.
view on GitHub