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04R9'"9IAn appraisal of the present enciphering models for Chaocipher.
Introduction.
This brief discussion paper puts forward results from trials that suggest none of the present model enciphering systems adequately represent Byrne's machine and system. It appears that we need to keep searching.
Argument.
Chaocipher has two key features that are proving very difficult to replicate together in model machines.
1st feature. When a letter repeats within an interval of 9 places in the plaintext of Exhibit 1, the cipher letters in these positions never repeat. This is true throughout the lengthy 13,598 letters of Exhibit 1. At intervals of 9 and over, the incidence of such pt/ct hits varies in a set pattern, described by Jeff Hill in his paper "Chaocipher: analysis and models".
Jeff Hill has taught us that this feature can largely be achieved by rotating a cipher disk with moves of [1,1,2,4] in a random sequence. I say "largely" because, while this sequence nicely produces the required pattern, it also produces a small but regular number of unwanted hits at intervals less than 9 in the lengthy text of Exhibit 1 (see Appendix 1 for explanation)
2nd feature. There is a large variety of different cipher bigrams produced for a given plain bigram. For example, the 339 instances of the plain bigram 'th' in Exhibit 1 are enciphered to 268 different cipher bigrams in Chaocipher. This variety is impossible to achieve from a single disk system, whether moved directly as in C941 or indirectly (ie through the mediation of a key disk) as in C98A, using moves of [1,1,2,4] at random.
When a 2-disk enciphering system is used, the required cipher bigram variety can be achieved, using moves of [1,1,2,4] for disk 1 and moving disk 2 by 1 step periodically. The more frequently that disk 2 is moved, the more cipher bigram variety results, until a limit is reached, as is shown below:
Number of different ct bigrams for plain 'th' in Exh. 1
Turnover C98U, with random My 2-disk machine
period, 1,1,2,4 moves with textual key
no turnover 75 75
5000 letters 150 155
1000 226 230
500 240 240
250 241 241
150 242 243
26 244 243
In order to achieve the necessary variety, the period cannot be longer than some 500 letters, implying at least 26 turnovers when enciphering Exhibit 1. This now adds to the problem of unwanted hits at intervals<9. At every turnover there is a further risk of making such a hit because the turnover disrupts the regular change in the effective enciphering alphabets (explanation in Appendix 2).
It is almost certain that existing models will have 1 or more hits at intervals<9 due to the two factors of repeated state and turnover. This is illustrated by the following trial data:
Results of 1000 runs enciphering Exhibit 1 plaintext.
C98U, fed by random [1,1,2,4]:
-no turnover 89.8% of runs have hits<9
-turnover at 500 98.3
My 2-disk system fed indirectly with [1,1,2,4] moves from textual key:
-no turnover 31.9%
-turnover at 500 92.9
Conclusion.
It seems to me that these high probabilities of hits at intervals<9 from existing models makes it most unlikely that they are good models for Byrne's machine and enciphering system.
Varying the mixed alphabets and changing the size of the turnover stepping make only marginal changes to the results and thus no change to the conclusion.
While the sequence of [1,1,2,4] moves seems just too relevant to be discarded, I can only conclude that the disk enciphering system is at fault and that we need to search for an alternative way of achieving the high variety of cipher bigrams while reducing the likelihood of the offending hits to much lower levels.
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footnote 1: C94, C98A, C98U are all model machines described by Jeff Hill in his paper ""Chaocipher: analysis and models", available at The Chaocipher Clearing House.
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Appendix 1 regaining the same enciphering state within an interval of 9 letters.
I think this mechanism is well understood, but for the record let me say that moves of [1,1,2,4] can rotate a cipher disk back to an original position in one of two ways:
-for a sequence of 7 letters, there are 7 ways that moves of 4,4,4,4,4,4,2 can be made to total a displacement of 26, out of a total of 16,384 ways of arranging a sequence of 7 moves of [1,1,2,4]. Thus one expects that 26/16384 = 0.04% of the 493 repeated plain letters at interval 7 in Exhibit 1 to be a hit which would give an average of 0.2 hits per encipherment of Exhibit 1.
-for a sequence of 8 letters, there are similarly 168 ways of making a total displacement of 26, out of 65,536 ways of arranging [1,1,2,4] moves. This implies that 168/65536 = 0.26% of the 796 plain repeats at interval 8 in Exhibit 1 are expected to result in hits which would lead to an average of 2 hits.
Together, 2.2 hits at interval<9 are expected in Exhibit 1 from this cause as an average over many encipherments.
Appendix 2 regaining the same individual plain-cipher relationship after a turnover.
When enciphering with a two-disk system, an initial sequence of 26 enciphering alphabets is created before a turnover, in which any plain letter is always enciphered to different cipher letters. If the turnover is set to 500 letters, then during the encipherment of the first 500 letters a hit can only occur if after 7 or 8 letters have been enciphered the disk returns to the same position which we have seen in Appendix 1 will occur only infrequently, on average 3 times in 1000 consecutive letters.
However once a turnover is made the situation changes because the pattern of the enciphering alphabet is disrupted, as can be seen below. Now any plain letter enciphered before the turnover can be enciphered to the same cipher letter after the turnover. As a result more hits at interval<9 will be made. It is difficult to calculate the expected number of such hits, as was possible in Appendix 1, but from trial results the number is 2.5 hits at interval<9 for the text in Exhibit 1 (viz about 0.1 hit per turnover).
ring abcdefghijklmnopqrstuvwxyz
disk 1 XSHDLOIYECPKNGMJWRAZUTBVFQ
disk 2 UNSRDYFOPKGAELBMCXHTJZWIVQ
Effective enciphering alphabets
abcdefghijklmnopqrstuvwxyz
0 IHORABPVDSMGLFEKWXUQJTNZYC
1 HORABPVDSMGLFEKWXUQJTNZYCI
2 ORABPVDSMGLFEKWXUQJTNZYCIH
3 RABPVDSMGLFEKWXUQJTNZYCIHO
4 ABPVDSMGLFEKWXUQJTNZYCIHOR
5 BPVDSMGLFEKWXUQJTNZYCIHORA
6 PVDSMGLFEKWXUQJTNZYCIHORAB
7 VDSMGLFEKWXUQJTNZYCIHORABP
8 DSMGLFEKWXUQJTNZYCIHORABPV
9 SMGLFEKWXUQJTNZYCIHORABPVD
10 MGLFEKWXUQJTNZYCIHORABPVDS
11 GLFEKWXUQJTNZYCIHORABPVDSM
12 LFEKWXUQJTNZYCIHORABPVDSMG
13 FEKWXUQJTNZYCIHORABPVDSMGL
14 EKWXUQJTNZYCIHORABPVDSMGLF
15 KWXUQJTNZYCIHORABPVDSMGLFE
16 WXUQJTNZYCIHORABPVDSMGLFEK
17 XUQJTNZYCIHORABPVDSMGLFEKW
18 UQJTNZYCIHORABPVDSMGLFEKWX
19 QJTNZYCIHORABPVDSMGLFEKWXU
20 JTNZYCIHORABPVDSMGLFEKWXUQ
21 TNZYCIHORABPVDSMGLFEKWXUQJ
22 NZYCIHORABPVDSMGLFEKWXUQJT
23 ZYCIHORABPVDSMGLFEKWXUQJTN
24 YCIHORABPVDSMGLFEKWXUQJTNZ
25 CIHORABPVDSMGLFEKWXUQJTNZY disk 2 turns over for next alphabet
26 VTPDEMKQYRCABOLGIHNUZJSWFX
27 TPDEMKQYRCABOLGIHNUZJSWFXV
28 PDEMKQYRCABOLGIHNUZJSWFXVT
29 DEMKQYRCABOLGIHNUZJSWFXVTP
30 EMKQYRCABOLGIHNUZJSWFXVTPD
31 MKQYRCABOLGIHNUZJSWFXVTPDE
32 KQYRCABOLGIHNUZJSWFXVTPDEM
33 QYRCABOLGIHNUZJSWFXVTPDEMK
34 YRCABOLGIHNUZJSWFXVTPDEMKQ
35 RCABOLGIHNUZJSWFXVTPDEMKQY
36 CABOLGIHNUZJSWFXVTPDEMKQYR
37 ABOLGIHNUZJSWFXVTPDEMKQYRC
38 BOLGIHNUZJSWFXVTPDEMKQYRCA
39 OLGIHNUZJSWFXVTPDEMKQYRCAB
40 LGIHNUZJSWFXVTPDEMKQYRCABO
41 GIHNUZJSWFXVTPDEMKQYRCABOL
42 IHNUZJSWFXVTPDEMKQYRCABOLG
43 HNUZJSWFXVTPDEMKQYRCABOLGI
44 NUZJSWFXVTPDEMKQYRCABOLGIH
45 UZJSWFXVTPDEMKQYRCABOLGIHN
46 ZJSWFXVTPDEMKQYRCABOLGIHNU
47 JSWFXVTPDEMKQYRCABOLGIHNUZ
48 SWFXVTPDEMKQYRCABOLGIHNUZJ
49 WFXVTPDEMKQYRCABOLGIHNUZJS
50 FXVTPDEMKQYRCABOLGIHNUZJSW
51 XVTPDEMKQYRCABOLGIHNUZJSWF
52 QJKYLCGUFDXEMPBAVTSNWZRIOH
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