Experiment 116

“A Hill matrix and a repeating key on the table, in either order: 'matrix codes', then 'back to this table'”

0105 solved Hill and affine Hill alone. Sanborn's words, 'matrix codes Ed gave me' and 'double matrixes... then went back to this table to do the ultimate', and the HILL column read literally give a Hill matrix followed by the Vigenere table, or the table followed by the matrix. On one numbering both orders give p = B c + u, where the shift vector u depends on the block's place modulo the key length: still linear, so each row of B is solved exactly from every known letter for key lengths 1-26.

revertedstatus
experiment 105parents
2 mintime, all phases
0candidates
0/24known letters
unclocked
1

The experiment method

What the method does with the 24 known letters.

Sanborn said he used the matrix codes Ed Scheidt gave him, and then went back to the tablethe table The right-hand half of the sculpture, 26 rows of the alphabet each shifted one place further, starting from the word KRYPTOS.. A Hill ciphera 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. is literally a matrixa grid Writing the message into rows of some width and reading it out in another order, column by column for instance, which shuffles the letters without changing them. code: blocks of letters are multiplied by a small square of numbers. Here a Hill matrix of size 2, 3 or 4 is combined with a short repeating secret word on the table, in either order. Every known letter gives one equation, and the equations are solved exactly.

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 ACCENTa 2x2 Hill matrix and a key of 4, either order: message = matrix x carved block + a shift that depends on the block's place in the keyOBKRUOXOGHULBSOLIFBBWEASTNORTHEASTOTWTQSJQSSEKZZWATJKLUDIAWINFBBERLINCLOCKWGDKZXTJCDIGKUHUAUEKCARblocks of 2 from place 1; boxed in ochre: blocks starting at 1, 5, 9 ... (class 0), the rest class 2. Ochre letters: the 11 first letters, equations for row 1block 23-24, LR = [11, 17], class 2: row 1 x (11, 17) + shift of class 2 = A = 0every size and key length, all alignments and four numberings: dead where the equations contradict, open where too few to decide12345678910111213141516171819202122232425262x24/88/88/84/83x36/1212/1212/126/1212/1212/1212/1212/1212/126/1212/1212/124x42/1612/162/1616/1610/1612/168/162/1610/1616/1616/1612/1616/1610/1616/1612/1616/168/1616/1612/1616/1610/16key lengthall deadpartly open (open/total)all openK4: 536 settings dead, 400 open, 0 alive; ten scrambled coppers: 0 alive; a planted Hill-and-key text comes out alive (recovered)OUT: NO READING; NO HILL MATRIX WITH A SHORT KEY FITSEASTNORTHEASTBERLINCLOCK
the idea

Sanborn spoke of matrix codes and then going back to the table. A Hill cipher is a matrix code; here a Hill matrix of size 2 to 4 is joined to a repeating key of 1 to 26 letters, in either order, which comes to one model: message = matrix x carved block + a shift.

the check

Every known letter gives one equation for one row of the matrix and one shift. Solved exactly mod 2 and mod 13, a setting is dead if the equations contradict or the matrix cannot be undone.

the bound

Nothing is alive on K4 or on ten scrambled coppers. 536 settings are dead; 400 stay open where the key is long and the equations too few. A planted text of this kind is recovered.

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. Cut the copper into blocks of 2 from place 1 and use a key of length 4: blocks starting at places 1, 5, 9, ... share one key pair, those at 3, 7, 11, ... the other.
  2. For each known letter, write: its number = row of the matrix times the block's two carved numbers + the shift for its row and block class.
  3. Row 1 then has 2 matrix entries and 2 shifts as unknowns, against about 12 equations.
  4. Solve mod 2 and mod 13; if no solution, or no invertible matrix, the setting dies.
  5. Repeat for sizes 2-4, key lengths 1-26, alignments and numberings, and on scrambled coppers.
the worksheet: the first steps by hand, on squared paper
2X2 HILL WITH A KEY OF LENGTH 4, BLOCKS FROM PLACE 1, LETTERS A = 0: EQUATIONS FOR ROW 1block placescarvednumbersblock classknown letter23-24LR11 172A = 025-26VQ21 160T = 1927-28QP16 152O = 1429-30RN17 130T = 1931-32GK6 102E = 433-34SS18 180S = 1865-66YP24 150E = 467-68VT21 192L = 1169-70TM19 120N = 1371-72ZF25 52L = 1173-74PK15 100C = 2Blocks starting at 1, 5, 9 ... shareone pair of shifts (class 0), blocksat 3, 7, 11 ... the other (class 2).Row 1 has two matrix entries and oneshift per class: four unknownsagainst eleven equations, so sevenchecks.Each line: a x first number + b xsecond number + shift of its class =the known letter, mod 26.
4

Worked example, generated

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

2x2 Hill with a key of length 4, blocks from place 1, letters A = 0: equations for row 1

block places  carved  numbers   block class  known first letter
  23-24          LR  [11, 17]            2  A = 0
  25-26          VQ  [21, 16]            0  T = 19
  27-28          QP  [16, 15]            2  O = 14
  29-30          RN  [17, 13]            0  T = 19
  31-32          GK  [6, 10]             2  E = 4
  33-34          SS  [18, 18]            0  S = 18
  65-66          YP  [24, 15]            0  E = 4
  67-68          VT  [21, 19]            2  L = 11
  69-70          TM  [19, 12]            0  N = 13
  71-72          ZF  [25, 5]             2  L = 11
  73-74          PK  [15, 10]            0  C = 2

row 1: two matrix entries and one shift per block class (2 classes) against the equations above
5

Hypothesis, for a cryptanalyst

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

C1 ('double matrixes and triple matrixes... then went back to this table'), C21 (the HILL column; Bauer), C3 ('matrix codes'). Model, one numbering for both stages (standard or KRYPTOS order, 0- or 1-based): blocks of n = 2-4 carved letters; message block p = B c + u_kappa mod 26, B invertible, kappa = (block start) mod L classes, L = 1-26 the key length, u_kappa free. This covers Hill then a periodic Vigenere key (u = -B k) and a periodic key then Hill (u = w - k) alike, and an affine Hill. Row r of (B, u) has n + #classes unknowns; each known letter in row position r gives one equation. Solved exactly mod 2 and mod 13 with the Chinese remainder theorem; a setting dies if a row has no solution or no choice makes B invertible; rows with more unknowns than equations leave it open. Block alignments 0 to n-1. Ten shuffled coppers.

6

Prediction

Written before the run; the result decides it.

No setting with spare equations survives for any size, key length, alignment or numbering; settings with no spare equations are reported as open. Counts match the coppers. Zero candidates.

7

Method

  1. For each size, key length, alignment and numbering, group the known letters by block, row position and block class.
  2. Solve each row mod 2 and mod 13; join by the Chinese remainder theorem.
  3. Require an invertible B.
  4. Count alive (with spare equations), dead and open settings.
  5. Repeat on ten shuffled coppers.
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

78 (size, key length) cells x alignments x 4 numberings. K4: alive 0, dead 536, open 400; controls alive [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]. Largest key length with any decided setting per size: 2x2: 26, 3x3: 24, 4x4: 26.

What was searched, as the harness recorded it.

family
Hill matrix with a periodic tableau key, either order (p = B c + u_kappa), sizes 2-4, key lengths 1-26
by size and key length
2x2 L=1
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=2
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=3
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=4
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=5
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=6
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=7
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=8
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=9
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=10
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=11
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=12
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=13
k4
alive
0
dead
4
open
4
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=14
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=15
k4
alive
0
dead
0
open
8
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=16
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=17
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=18
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=19
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=20
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=21
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=22
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=23
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=24
k4
alive
0
dead
8
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=25
k4
alive
0
dead
0
open
8
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2x2 L=26
k4
alive
0
dead
4
open
4
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=1
k4
alive
0
dead
12
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=2
k4
alive
0
dead
12
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=3
k4
alive
0
dead
12
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=4
k4
alive
0
dead
12
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=5
k4
alive
0
dead
12
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=6
k4
alive
0
dead
12
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=7
k4
alive
0
dead
12
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=8
k4
alive
0
dead
6
open
6
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=9
k4
alive
0
dead
12
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=10
k4
alive
0
dead
0
open
12
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=11
k4
alive
0
dead
0
open
12
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=12
k4
alive
0
dead
12
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=13
k4
alive
0
dead
12
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=14
k4
alive
0
dead
12
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=15
k4
alive
0
dead
12
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=16
k4
alive
0
dead
6
open
6
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=17
k4
alive
0
dead
0
open
12
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=18
k4
alive
0
dead
12
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=19
k4
alive
0
dead
0
open
12
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=20
k4
alive
0
dead
0
open
12
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=21
k4
alive
0
dead
12
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=22
k4
alive
0
dead
0
open
12
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=23
k4
alive
0
dead
0
open
12
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=24
k4
alive
0
dead
6
open
6
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=25
k4
alive
0
dead
0
open
12
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3x3 L=26
k4
alive
0
dead
0
open
12
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=1
k4
alive
0
dead
16
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=2
k4
alive
0
dead
16
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=3
k4
alive
0
dead
14
open
2
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=4
k4
alive
0
dead
16
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=5
k4
alive
0
dead
4
open
12
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=6
k4
alive
0
dead
14
open
2
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=7
k4
alive
0
dead
0
open
16
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=8
k4
alive
0
dead
16
open
0
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=9
k4
alive
0
dead
6
open
10
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=10
k4
alive
0
dead
4
open
12
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=11
k4
alive
0
dead
8
open
8
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=12
k4
alive
0
dead
14
open
2
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=13
k4
alive
0
dead
6
open
10
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=14
k4
alive
0
dead
0
open
16
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=15
k4
alive
0
dead
0
open
16
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=16
k4
alive
0
dead
4
open
12
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=17
k4
alive
0
dead
0
open
16
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=18
k4
alive
0
dead
6
open
10
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=19
k4
alive
0
dead
0
open
16
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=20
k4
alive
0
dead
4
open
12
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=21
k4
alive
0
dead
0
open
16
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=22
k4
alive
0
dead
8
open
8
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=23
k4
alive
0
dead
0
open
16
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=24
k4
alive
0
dead
4
open
12
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=25
k4
alive
0
dead
0
open
16
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4x4 L=26
k4
alive
0
dead
6
open
10
controls alive
0, 0, 0, 0, 0, 0, 0, 0, 0, 0
k4 alive
none
n k4 alive
0
controls
10 shuffled coppers (seed 116), identical test

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