Moshe Rubin
(mosher@mountainvistasoft.com)

Last updated: 25 February 2009

Last updated: 25 February 2009

A pt/ct identity will not occur within a distance of N if a key does not occur again for the next N letters.

I'm now looking at a contrived hypothetical system which would explain the distance < 9 AND the second ct letters of doubled plaintext letters. It does this by guaranteeing that a key is not encountered for 8 more letters after being used. This system is decidedly not the Chaocipher. The point is to concoct a system that will prevent pt/ct identities in distances less than 9 and will allow more latitude than the system in report #1. This will give us a model somewhat closer to the Chaocipher, allowing us to hopefully hypothesize in the right direction.

The system assumes:

- Two alphabetic slides, pt and ct. The ordering of the alphabets is not relevant for this discussion.
- A list of the possible key designations (e.g., 1-26, 0-25, or A-Z) in any order.
- We have a reproduceable method for generating a number greater than 8.

- Use the key designator at the head of the list as the key.
- Encipher the plaintext letter using the alphabetic slides and the key letter.
- Generate a number 9 <= N <= 26
- Remove the head of the key designator list, leaving 25 letters in the list.
- Insert the just-discarded key designator as the Nth element of the list
- Repeat from step (1)

pt: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

ct: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

pt: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

ct: R S T U V W X Y Z A B C D E F G H I J K L M N O P Q

` C A K W U B M X L D N Y P F O Z E G Q H S I T R J V`

[(numerical value of pt) + (numerical value of ct) mod 18] + 9

pt: ABCDEFGHIJKLMNOPQRSTUVWXYZOur number generator gives ((pt+ct) mod 18)+9 = ((23+25) mod 18)+9 = 12+9 = 21. Removing the head of the key designator list (the 'C') and inserting it as the 21st element gives:

ct: CDEFGHIJKLMNOPQRSTUVWXYZAB

key designator list: A K W U B M X L D N Y P F O Z E G Q H S C I T R J V

pt: ABCDEFGHIJKLMNOPQRSTUVWXYZOur number generator gives ((pt+ct) mod 18)+9 = ((23+23) mod 18)+9 = 10+9 = 19. Removing the head of the key designator list (the 'A') and inserting it as the 19th element gives:

ct: ABCDEFGHIJKLMNOPQRSTUVWXYZ

key designator list: K W U B M X L D N Y P F O Z E G Q H A S C I T R J VThe following table shows the full encipherment:

Step | Key Designator List | Key | Enciphering Alphabets | Pt | Ct | Decimation
factor | New Key Designator List |

1 | CAKWUBMXLDNYPFOZEGQHSITRJV | C | pt: ABCDEFGHIJKLMNOPQRSTUVWXYZ ct: CDEFGHIJKLMNOPQRSTUVWXYZAB | X | Z | ((23+25) mod 18)+9 = 21 | `(C)AKWUBMXLDNYPFOZEGQHSCITRJV` |

2 | AKWUBMXLDNYPFOZEGQHSCITRJV | A | pt: ABCDEFGHIJKLMNOPQRSTUVWXYZ ct: ABCDEFGHIJKLMNOPQRSTUVWXYZ | X | X | ((23+23) mod 18)+9 = 19 | (A)KWUBMXLDNYPFOZEGQHASCITRJV |

3 | KWUBMXLDNYPFOZEGQHASCITRJV | K | pt: ABCDEFGHIJKLMNOPQRSTUVWXYZ ct: KLMNOPQRSTUVWXYZABCDEFGHIJ | X | H | ((23+7) mod 18)+9 = 21 | (K)WUBMXLDNYPFOZEGQHASCKITRJV |

4 | WUBMXLDNYPFOZEGQHASCKITRJV | W | pt: ABCDEFGHIJKLMNOPQRSTUVWXYZ ct: WXYZABCDEFGHIJKLMNOPQRSTUV | X | T | ((23+19) mod 18)+9 = 15 | (W)UBMXLDNYPFOZEGWQHASCKITRJV |

5 | UBMXLDNYPFOZEGWQHASCKITRJV | U | pt: ABCDEFGHIJKLMNOPQRSTUVWXYZ ct: UVWXYZABCDEFGHIJKLMNOPQRST | X | R | ((23+17) mod 18)+9 = 13 | (U)BMXLDNYPFOZEUGWQHASCKITRJV |

6 | BMXLDNYPFOZEUGWQHASCKITRJV | B | pt: ABCDEFGHIJKLMNOPQRSTUVWXYZ ct: BCDEFGHIJKLMNOPQRSTUVWXYZA | X | Y | ((23+24) mod 18)+9 = 20 | (B)MXLDNYPFOZEUGWQHASCBKITRJV |

7 | MXLDNYPFOZEUGWQHASCBKITRJV | M | pt: ABCDEFGHIJKLMNOPQRSTUVWXYZ ct: MNOPQRSTUVWXYZABCDEFGHIJKL | X | J | ((23+9) mod 18)+9 = 23 | (M)XLDNYPFOZEUGWQHASCBKITMRJV |

8 | XLDNYPFOZEUGWQHASCBKITMRJV | X | pt: ABCDEFGHIJKLMNOPQRSTUVWXYZ ct: XYZABCDEFGHIJKLMNOPQRSTUVW | X | U | ((23+20) mod 18)+9 = 16 | (X)LDNYPFOZEUGWQHAXSCBKITMRJV |

9 | LDNYPFOZEUGWQHAXSCBKITMRJV | L | pt: ABCDEFGHIJKLMNOPQRSTUVWXYZ ct: LMNOPQRSTUVWXYZABCDEFGHIJK | X | I | ((23+8) mod 18)+9 = 22 | (L)DNYPFOZEUGWQHAXSCBKITLMRJV |

10 | DNYPFOZEUGWQHAXSCBKITLMRJV | D | pt: ABCDEFGHIJKLMNOPQRSTUVWXYZ ct: DEFGHIJKLMNOPQRSTUVWXYZABC | X | A | ((23+0) mod 18)+9 = 14 | (D)NYPFOZEUGWQHADXSCBKITLMRJV |

11 | NYPFOZEUGWQHADXSCBKITLMRJV | N | pt: ABCDEFGHIJKLMNOPQRSTUVWXYZ ct: NOPQRSTUVWXYZABCDEFGHIJKLM | X | K | ((23+10) mod 18)+9 = 24 | (N)YPFOZEUGWQHADXSCBKITLMNRJV |

12 | YPFOZEUGWQHADXSCBKITLMNRJV | Y | pt: ABCDEFGHIJKLMNOPQRSTUVWXYZ ct: YZABCDEFGHIJKLMNOPQRSTUVWX | X | V | ((23+21) mod 18)+9 = 17 | (Y)PFOZEUGWQHADXSCBYKITLMNRJV |

13 | PFOZEUGWQHADXSCBYKITLMNRJV | P | pt: ABCDEFGHIJKLMNOPQRSTUVWXYZ ct: PQRSTUVWXYZABCDEFGHIJKLMNO | X | M | ((23+12) mod 18)+9 = 26 | (P)FOZEUGWQHADXSCBYKITLMNRJVP |

14 | FOZEUGWQHADXSCBYKITLMNRJVP | F | pt: ABCDEFGHIJKLMNOPQRSTUVWXYZ ct: FGHIJKLMNOPQRSTUVWXYZABCDE | X | C | ((23+2) mod 18)+9 = 16 | (F)OZEUGWQHADXSCBYFKITLMNRJVP |

15 | OZEUGWQHADXSCBYFKITLMNRJVP | O | pt: ABCDEFGHIJKLMNOPQRSTUVWXYZ ct: OPQRSTUVWXYZABCDEFGHIJKLMN | X | L | ((23+11) mod 18)+9 = 25 | (O)ZEUGWQHADXSCBYFKITLMNRJVOP |

16 | ZEUGWQHADXSCBYFKITLMNRJVOP | Z | pt: ABCDEFGHIJKLMNOPQRSTUVWXYZ ct: ZABCDEFGHIJKLMNOPQRSTUVWXY | X | W | (23+22) mod 18)+9 = 18 | (Z)EUGWQHADXSCBYFKITZLMNRJVOP |

- The new system prevents pt/ct identities within distances less than 9. It does this by "throwing" a key used for enciphering at least 9 positions into the future.
- When plaintext doubles (e.g., "XX" in the above example) are enciphered, the second ciphertext letter can be any letter other than the first ciphertext letter). This was a problem with the old system in progress report #1, but is no longer a problem here.
- A mistake made while enciphering will garble the remaining message.

- Letters at the end of the key designator list tend to stay at the end for quite some time. For example, the last letter in the key designator list will remain at the end until a key letter is positioned in slot 26.
- An enciphering mistake will not necessarily immediately garble everything from this point onwards. It could take quite a few encipherings until an incorrectly decimated key letter makes itself felt. This may be counter to the comment in Deavours and Kruh: "And, an important shortcoming of the cipher -- make one error during encipherment and the message is garbled beyond repair ..." [1].

Copyright (c) 2009 Moshe Rubin