“Gromark with three-term rules and the bases 16 to 26 that experiment 28 left alive”
Rules with three terms, k[i] = k[i-a] + k[i-b] + k[i-c], alongside the two-term rules, lags up to 13; bases 16 to 26 only, since experiment 28 showed every alphabet triple forces a shift of 15 or more, which kills every smaller base before any rule is checked.
GromarkGromark A cipher whose secret is a short number that grows into a long stream of digits by adding pairs of earlier digits; each message letter is pushed along the alphabet by the next digit. grows a long number stream from a short seed by addingVigenere The three rules of arithmetic a table can use: add the key letter to the message letter, subtract the message letter from the key letter, or subtract the key letter from the message letter. earlier digits. Earlier we tried adding two earlier digits; now we also try adding three, with any spacing up to thirteen back, and we count in every base from 16 to 26, the only ones the known lettersthe known letters Sanborn has confirmed that positions 22 to 34 of the message read EASTNORTHEAST and positions 64 to 74 read BERLINCLOCK. still allow. The 13 pushes in a row that EASTNORTHEAST pins down must follow the rule, or the rule is out.
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 shifts
A Gromark key is a stream of digits, each the sum of earlier digits. Each known letter under its carved letter tells the shift used there; a shift of 10 or more cannot be a digit.
the rule
Drawn for the rule 'five back plus four back' in base 26 on the standard and KRYPTOS alphabets: the stream must reproduce each later shift, and it does not.
the bound
184,512 alphabet triples, 364 two- and three-term rules with lags up to 13, bases 16 to 26: no rule survives six checks under any triple.
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.
Take the standard alphabet for the plaintext and the KRYPTOS alphabet for the ciphertext, Vigenere.
Position 22: plaintext E is index 4 in the standard alphabet; ciphertext F is index 11 in KRYPTOS; the shift is 11 - 4 = 7.
Position 23: A is 0, L is 18; shift 18.
Continue to position 34 for 13 shifts in a row.
Try the Gromark rule, a = 5, b = 4: the sixth shift must equal the first plus the second, mod 10, and all shifts must be digits. The shift 18 already fails base 10.
In base 26, check the sixth against the first plus the second mod 26, the seventh against the second plus the third, and so on: eight checks.
One failure kills the rule under this alphabet pair; try the next pair.
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.
plain alphabet standard, cipher alphabet KRYPTOS, Vigenere: forced shifts at 22-34
pos plain cipher shift
22 E F 8
23 A L 17
24 S R 9
25 T V 3
26 N Q 7
27 O Q 6
28 R P 12
29 T R 8
30 H N 12
31 E G 9
32 A K 0
33 S S 14
34 T S 13
shifts: 8 17 9 3 7 6 12 8 12 9 0 14 13
three-term rule lags 5, 3, 1 in base 26: shift 6 should equal shifts 1 + 3 + 5 = 24, it is 6
5
Hypothesis, for a cryptanalyst
Family, key space, the constraint under test, and the clues it reads.
Experiments 13 and 28 killed two-term lagged-Fibonacci rules up to lag 12 in bases 10 and 26, and every base up to 15 outright, under the 248-alphabet pool. Bean (knowledge/05-attacks.md) names other expansion rules as untested. Here: two-term rules with lags up to 13 and three-term rules k[i] = k[i-a] + k[i-b] + k[i-c] with c < b < a <= 13, bases 16 to 26, under the same pool and three modes. For each triple the largest forced shift fixes the admissible bases; each rule is checked on the 13 and 11 consecutive forced shifts, at least six checks required, chance below 26 to the minus six per check set. Survivors extend to a full stream, decrypt and submit. Bounds: complete for these rules and bases over the pool; free alphabets remain Bean's.
6
Prediction
Written before the run; the result decides it.
No rule with six or more checks survives under any alphabet triple and base; zero candidates pass.
7
Method
Build the alphabet pool: standard, KRYPTOS and the four mixings of each vocabulary word.
For each plain alphabet, cipher alphabet and mode, compute the forced shift at each of the 24 crib positions.
For base 10, drop any triple with a shift above 9.
For each lag pair (a, b) check k[i] = k[i-a] + k[i-b] mod B on both crib runs; keep triples with every check passing and at least six checks.
For each survivor, extend the stream over all 97 positions from the 22-34 window, confirm it reproduces 64-74, decrypt and submit.
8
Result and bounds
The harness's record: candidates, gates, loss, and the negative result this experiment leaves behind.