Experiment 194

“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.

revertedstatus
experiment 193parents
8 mintime, 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 ACCENTletters paired from place 1: 11 pairs lie wholly inside the known words, 22 equationsEASTNORTHEASTBERLINCLOCKletters paired from place 2: 11 pairs lie wholly inside the known words, 22 equationsEASTNORTHEASTBERLINCLOCKof 157,248 invertible matrices, those that some numbering of the letters fitsK45PPPno matrix fitsone matrix fits, its ownK4 0, five scrambled copies 0; each planted Hill text (P) fits its own matrix and no otherHill 3 x 3 with a free numbering: 21-32 bits short of a single check, so blockedOUT: NO READING; HILL 2 X 2 FITS K4 UNDER NO NUMBERING OF THE LETTERSEASTNORTHEASTBERLINCLOCK
the idea

The tableau's last column spells HILL, a cipher that mixes letters in pairs with a small matrix of numbers. It had been tried with the usual numberings of the letters; the numbering could itself be secret.

the check

For every invertible 2 x 2 matrix and both ways of pairing the letters, a search for any one-to-one numbering of the alphabet that makes all 22 equations from the known pairs hold.

the bound

No matrix fits K4 or a scrambled copy under any numbering; planted texts are recovered exactly. F8.1 closed for 2 x 2; 3 x 3 has too few equations.

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.
the worksheet: the first steps by hand, on squared paper
PAIRS INSIDE THE KNOWN WORDSPairing from place 1, the pairs whollyinside the known words are AS->LR,TN->VQ, OR->QP, TH->RN, EA->GK, ST->SS,ER->YP, LI->VT, NC->TM, LO->ZF, CK->PK.With matrix (a b; c d) and numbering n:a n(A) + b n(S) = n(L), c n(A) + d n(S)= n(R), and so on. ST->SS ties S and Tto S itself, which already fixes a lot.
4

Worked example, generated

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

digraphs wholly inside the known runs, pairing from place 1

AS -> LR
TN -> VQ
OR -> QP
TH -> RN
EA -> GK
ST -> SS
ER -> YP
LI -> VT
NC -> TM
LO -> ZF
CK -> PK
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

matrices admitting a numbering: K4 0 (copies 0-0); undecided K4 0; plants: [(True, 1), (True, 1), (True, 1)]

What was searched, as the harness recorded it.

family
Hill 2x2, any invertible matrix, any one-to-one numbering, alignments 0 and 1, exact
matrices
157,248
blocks per alignment
0
11
1
11
fitting matrices
k4
0
copies
0, 0, 0, 0, 0
undecided
k4
0
copies
0, 0, 0, 0, 0
max nodes
40,301
k4 fits
none
plants
  1. matrix
    9, 21, 18, 19
    alignment
    0
    true matrix fits
    yes
    fitting matrices
    1
  2. matrix
    5, 20, 3, 15
    alignment
    1
    true matrix fits
    yes
    fitting matrices
    1
  3. matrix
    17, 13, 5, 16
    alignment
    0
    true matrix fits
    yes
    fitting matrices
    1
count 3x3
6-7 trigraphs, 18-21 equations (85-99 bits) against 76-81 bits of numbering and 41 of matrix: 21-32 bits short; blocked
copies
5 shuffled copies (seed 1930)
seconds
496
exhaustive
yes

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.
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