Moshe Rubin
(mosher@mountainvistasoft.com)

Received 11 March 2009 |

Hi
Moshe, Great analysis and written summarization of your work on Chaocipher. I, too, have enjoyed(!) the challenge of Byrne's ingenious scheme. After reading Kruh's Cryptologia Article, I noted that in Exhibit 5 - Message #3 the ciphertext string BBNKF appeared twice. Once as a complete, independent block, the other as as split block (TBBNK FUCBP). Interestingly, this Message has an anomaly as presented in the article - namely, while all other full blocks are 5 characters, this one contains a 4 letter block (UHWA) between the 2 instances of BBNKF. I can't determine if this was an error during the writing of the article (i.e., a character got inadvertently dropped) or an intended ciphering decision. If intended, then there are 29 characters between the two occurrences (From the 2nd B in BBNKF to the first B of the next instance). If a 5th character was left out of the noted block, then you'd have 30 characters between the BBNKF occurrences. It's unlikely that such a 5-character string would occur entirely randomly. ... Keep up the good work - I wonder if Byrne Jr., Kruh and Deavours will ever divulge what they know (Non-disclosure agreement notwithstanding)? Jeff Calof |

Jeff was referring to this web site. Whoever scanned in the article inadvertently dropped a letter:

Sent 13 March 2009 |

Hi
Jeff, First off, thank you very much for your kind words. The best compliment you could give me is that the site is of value to you. I had a feeling that much more Chaocipher research was being done but there was no focal point. From the responses I'm getting I believe this is true. I hope you will take part in this global effort. I look forward to any other comments, research, etc. you may have. Truth be told, I had not noticed the 5-letter repetition -- thanks for pointing it out. I'd like to compute the probability of such an occurrence. The other four exhibits do not show any causal repetitions (Mellen points out a 5-letter repetition, XXACN, in Exhibit 1 but notes that the plaintexts are different, so the repetition is non-causal). Could Deavours and Kruh have knowingly selected a plaintext so that an underlying plaintext repetition *was* duplicated in the ciphertext? I hope to pursue this repetition in the future and look forward to any research you may do on it. . . . You mention that there is a 4-letter block, UHWA, in message #3. You may have lost a letter while transcribing or scanning the message in. The article clearly shows: ... TTXJX UHWQA PTSNB ... Therefore the distance between the repetitions is 31. The repetition is definitely not expected and should be examined carefully. . . . I agree with you that Byrne may have used gear-like sequences or cipher alphabets whose periods are prime to each other. This may be what he meant when he wrote that the ancient Egyptians and Babylonians could have been completely familiar with the principle. Regarding Byrne, Deavours and Kruh divulging the mechanism, I asked Deavours and Kruh this same question in a recent e-mail to them (they haven't answered yet). In my opinion there is no commercial value in Byrne's system given the power, security, and ease of public-key systems today. In this light Byrne would do best to publicize the system. The cryptologic community will verify whether it has any commercial value. Thank you once again for you kind words and for taking the time to write them. I look forward to hearing from you again. Best regards, Moshe Rubin |

After uploading Progress Report #6 I received this e-mail from Jeff (20 March 2009):

Here's my response to him on 21 March 2009:

Hi
Jeff, Many thanks (again!) for your very kind words (I wasn't sure if you received my reply on 13 March to your first e-mail, so I'm attaching it just in case). It's great to hear from one's peers, especially on a subject we're both highly interested in. I know my progress reports are circling around the target, but I get a strong feeling that they're homing in on it. My thoughts are becoming clearer and more focused re Chaocipher the more I commit thoughts to paper. I hope I can help others like you to see your ways clear to solving this challenge. << The "fractionate" reference supported something I've always suspected was a missed clue from Byrne - but seeing this supports my suspicion >> I'm glad to hear you've thought along the same lines. I'm wondering how two concentric disks can be used to fractionate a letter and recombine it again. Any ideas? There is a difference between Byrne and Callimahos: Byrne refers to splitting a word while Callimahos refers to fractionating a letter. They might be referring to the same thing ... << I believe this reference is to Hero, or Heron's, treatise "Metrica". The highlight of this is the Babylonian method for finding Square Roots (also known as the Heron Method) >> Your idea is a valid one, as there is an infinite number of numbers whose square roots are irrational. I believe there was also a "letter of influence" element in the cipher, so two people using the same seed number N would diverge almost immediately based on the plaintext, fulfilling Byrne's description of an infinite number of keys. Vis-a-vis Chaocipher, such a system would have to address the following: * As you say, it would have to qualify as "simple". * We know the system fit in a cigar box. Could Byrne have created a square root calculator in some form? * It would have to explain the "pt/ct identities > 9" phenomenon in Exhibit 1. Can you think of such a scheme? In my reply to you last week I thanked you for pointing out the highly significant repetition in Exhibit 5, Message 3. It was your observation that launched into investigating the repetition in the exhibits. (I pointed out that you may have dropped a letter accidentally when transcribing the message.) I'd like to thank you again . . . Regarding the Exhibit 5 repetition, as I wrote, it certainly looks highly significant. I'm wondering what one can do with it. It has a distance of 31, so you have your prime number theory back again <g>. Once again, Jeff, thank you so much for following my progress reports and for taking the time to write. It's a great feeling knowing others are enjoying them. Best regards and looking forward to hearing from you, Moshe P.S. You might find NSA's Declassification Initiative page of general cryptologic interest: http://www.nsa.gov/public_info/declass/index.shtml |

So, for example, we encounter this portion of juxtapositioning with a shift of 182:

1: ALLGOODQQUICKBROWNFOXESJUMPOVERLAZYDOGTOSAVETHEIRPARTYW ...

CLYTZPNZKLDDQGFBOOTYSNEPUAGKIUNKNCRINRCVKJNHTOAFQPDPNCV ...

2: NFOXESJUMPOVERLAZYDOGTOSAVETHEIRPARTYWALLGOODQQUICKBROW ...

OWHKECRMLYWIQIFIKSCYJGCVXNSKYHRYVYEDSZRIFFZAQNHSOMJPORW ...

Here's the resulting graph:

The first time I saw this graph I was amazed by the tall lines in the left-hand side of the graph. They were precisely 55 positions apart -- was I on to something?! The mystery was solved: the "All good, quick" phrase is exactly 55 letters long. Having 100 such phrases greatly raises the probability of a non-causal pt+ct coincidence. If we ignore these spikes, the graph looks non-causal, with the number of coincidences slowly decreasing as the number of overlapping letters decreases.

Just to drive the point home, I normalized the preceding graph by dividing the number of coincidences by the number of overlapping letters:

The rise at the end is to be expected: with relatively few overlapping letters, the graph becomes much more sensitive to coincidences.

Conclusion: I cannot identify a point where the machine returns to its original settings by examining coincidences.

- Print out the page
- Cut out the two largest disks
- Write the alphabet components on the disk rims
- Get a piece of corrugated cardboard of, say, 6 inches by 6 inches.
- Stick a pin through the centers of the disks, connecting them to the cardboard.

Having a physical cipher disk has helped me work on the question: how can one create a sophisticated cipher from a cipher disk. I've learned a lot by just fiddling around with the cipher disk.

In an upcoming progress report I'd like to share with you some of the schemes I've come up with to create sophisticated ciphers from a standard cipher disk. I highly advise reading F. L. Bauer's [6] chapter 3 ("Encryption Steps: Simple Substitution") and chapter 7 ("Polyalphabetic Encryption: Families of Alphabets") to get a basic mathematical basis for dealing with alphabets: shifted, rotated, power alphabets, cyclic notation, iterated substitutions, mixed alphabets, and more.

Here are quotes I believe are relevant to the question:

(A) "The
ancient Egyptians and Babylonians could have been completely familiar
with the principle, a fact that is readily deducible from a treatise on
mathematics written by Hero of Alexandria in the second century B.C"
[1, page 265]

(B) "The first device, or machine, which I constructed solely for the purpose of demonstrating a principle, was a little model, constructed in an empty cigar box which, when full, had contained fifty small Havana cigars. I made this model myself, and to sat that it was a crude affair would be only to describe it accurately." [1, page 265]

(C) "I then approached several machine makers asking for an estimate of the cost of making my machine, and from not one of them could I get anything approaching a firm bid, everyone of them was vague, and the best I could get by way of an estimate was that it would not be less than $5,000 and might run to $10,000 or more; ..." [1, page 267]

(D) "As to the principle of the machine, it is undoubtedly a most ingenious and effective device ..." [1, page 273, letter from Colonel Parker Hitt]

(E) "When I read Colonel Hitt's letter, it was clear to me that he had not at all fully comprehended the principle of my "machine", as he called it." [1, page 273]

(F) "And let me add that devices far more operable than my crude model could be mass-produced to sell at ten dollars each." [1, 282]

(G) "... but he did not bring the cipher machine 'explaining that it was too heavy and cumbersome.'" [2, page 194].

(B) "The first device, or machine, which I constructed solely for the purpose of demonstrating a principle, was a little model, constructed in an empty cigar box which, when full, had contained fifty small Havana cigars. I made this model myself, and to sat that it was a crude affair would be only to describe it accurately." [1, page 265]

(C) "I then approached several machine makers asking for an estimate of the cost of making my machine, and from not one of them could I get anything approaching a firm bid, everyone of them was vague, and the best I could get by way of an estimate was that it would not be less than $5,000 and might run to $10,000 or more; ..." [1, page 267]

(D) "As to the principle of the machine, it is undoubtedly a most ingenious and effective device ..." [1, page 273, letter from Colonel Parker Hitt]

(E) "When I read Colonel Hitt's letter, it was clear to me that he had not at all fully comprehended the principle of my "machine", as he called it." [1, page 273]

(F) "And let me add that devices far more operable than my crude model could be mass-produced to sell at ten dollars each." [1, 282]

(G) "... but he did not bring the cipher machine 'explaining that it was too heavy and cumbersome.'" [2, page 194].

Here are some thoughts:

- Quote (A) hints that the principle is a mathematical one (rather than a physical one) and could be inferred from Hero of Alexandria's mathematical treatise. I checked out Michael Lahanas's site "Heron of Alexander" for ideas of Heron's writings. The treatise on mathematics Byrne refers to was probably "Metrica", a collection of three books for the determination of areas and volume of objects. The first two books deal with areas and volumes. Book three deals with, among other things, his square root extraction formula (here's a summary of his mathematical ideas).
- Heron invented gear-related machines (for example, his taximeter, odometer, and Baroulkos (here and here)) which could certainly have been the basis of a cigar-based machine. Nonetheless, Byrne states clearly that the principle is deducible from a mathematical treatise, not a mechanical treatise.
- See Jeff Calof's e-mail above with his idea that Byrne may be referring to the square root extraction method.
- Quote (B) hints that the machine was a small, simple one, capable of fitting into a small cigar box. On the other hand, quote (C) tell us that the machine makers quoted a relatively huge sum to produce it. This would infer that the machine was not as simple as, say, two concentric disks on a spindle.
- It looks like we're talking about some mechanical feature.
- In quote (D), Parker Hitt is impressed with the concept (which Byrne believes he didn't 'get'). If it were a simple cipher disk, Hitt would not have been so lavish in his praise.
- In quote (E) Byrne puts the word "machine" in double quotes. As if to tell us that he did not consider his principle worthy of being called a "machine".
- In quote (F) Byrne suggests his device could be mass-produced for no more than ten dollars. Does this imply that the mechanism is not a very complicated one?j Having said that, quote (G) seems to imply that the final device was way to large and expensive than Byrne thought at first.

- He shows the diagram of the Baroulkos by Heron but says "I can think of no way to use it in a serious cryptographic system".
- He dismisses the use of continued fractions and "Egyptian Fractions".
- He says that irrational and transcendental numbers offer other possibilities (see Jeff Calof's comment above), but would be surprised if Byrne knew how to derive the functions, and how he could implement the function in a cigar box.
- In 1938 Byrne built a second model for demonstrating to the Navy. "We are not told if it was in a cigar box, but it was a 'working model on which I could do extended encipherments and decipherments, and on which I could with some freedom put my principles into operation'".
- From "extended" Mellen inferred that Byrne's machine can be built in different "sizes", possibly involving blocks of plaintext, the length of the block being determined by the size of the machine.
- "With some freedom" may imply that a small machine does not permit full exercise of the cipher algorithm.
- "Working model" may imply that Byrne's initial device was non-functional, useful for clarifying his concept but only a "paper model".
- In the sentence [1, page 277] "Working through the summer and fall of 1937, I made my model and prepared on and by it, a document I intended for submission to the Navy department ...", Mellen understands the phrase "on and by it" to mean that there is interaction of some kind between the machine's enciphering operation and the key or the plaintext, or both.
- He uses this, and the phrase "readily operable machine" [1, page 267]", to explain that Byrne could not apply for a patent years earlier. Mellen assumes that Byrne had no realizable embodiment of his principle, only a crude model.
- I'm not convinced that "on and by it" implies some special type of interaction. Byrne's narrative tends to be precise but sometimes florid. The phrase might imply that he was able to fully encipher his text on it without too much manual intervention.
- Mellen write [3, page 141] "I made paper models of rotors machines which I thought I could (and hence Byrne could) build in a cigar box. I ruled out complex gearings, or gearing of any sort, as being incompatible with "bits of string and odds and ends." I hope to expand at a later time on sophisticated uses of a standard cipher disk.

- A mathematical concept that could be used to produce a multiplicity of alphabets or a quasi-random keying sequence, e.g., decimal expansion of irrational or transcendental numbers, etc.
- Sophisticated usage of a standard cipher disk
- A gearing system (as per Hero). This brings to mind the Wheatstone, Pletts, or Wadsworth devices [5] (which was not used by Byrne in their classic form).

1]
hwmdgyceinzvspjfluorxqtakb 2] sbpuiagwlczfredohxmqytvnjk hwmdgyceinzvspjfluorxqtakb -------------------------- sbpuiagwlczfredohxmqytvnjk bpuiagwlczfredohxmqytvnjks puiagwlczfredohxmqytvnjksb uiagwlczfredohxmqytvnjksbp iagwlczfredohxmqytvnjksbpu agwlczfredohxmqytvnjksbpui gwlczfredohxmqytvnjksbpuia wlczfredohxmqytvnjksbpuiag lczfredohxmqytvnjksbpuiagw czfredohxmqytvnjksbpuiagwl zfredohxmqytvnjksbpuiagwlc fredohxmqytvnjksbpuiagwlcz redohxmqytvnjksbpuiagwlczf edohxmqytvnjksbpuiagwlczfr dohxmqytvnjksbpuiagwlczfre ohxmqytvnjksbpuiagwlczfred hxmqytvnjksbpuiagwlczfredo xmqytvnjksbpuiagwlczfredoh mqytvnjksbpuiagwlczfredohx qytvnjksbpuiagwlczfredohxm ytvnjksbpuiagwlczfredohxmq tvnjksbpuiagwlczfredohxmqy vnjksbpuiagwlczfredohxmqyt njksbpuiagwlczfredohxmqytv jksbpuiagwlczfredohxmqytvn ksbpuiagwlczfredohxmqytvnj (ancgilhsrqtvfompewbkjduxy)
(z) (bselxtnzfh) (ajoqvryg) (cwpdi) (mu) (k) (clmizrtjhpoywuqnfxve) (aksd) (b) (g) (asotkbphuylqjxnrvdgwifm) (ez) (c) (abutshirnefqkpxjmglyczdw) (ov) (apmwgcfyzondltbierjqsxkuvh) (aunojyftpqb) (crkid) (lvxsm) (gzh) (e) (w) (bgfvmcedzx) (pyrsq) (aiok) (hwln) (jtu) (bwcdfnxptihljvqukgr) (eosy) (mz) (a) (agehcoblkwzqixust) (drpvy) (fjnm) (awfklsvtgdeximrubchzyopnq) (j) (albztwriqgoupjkcx) (fsnyh) (em) (d) (v) (acmdoiyxghr) (kzvntlp) (eqw) (bf) (js) (u) (aznvjbrgxwdheymo) (cqluit) (fpsk) (dxlivkrwogmh) (betzjp) (afu) (cyq) (n) (s) (bdmxctfinjugqzkevs) (arl) (how) (p) (y) (bolgytrcv) (aenkdqf) (puwxzs) (ij) (h) (m) (adyvpikocnsulwmqrzbhxfgtej) (bxrfwqekhmyn) (cjgvu) (aozp) (dt) (is) (l) (ahqdvibmtoflckxes) (gnp) (jwy) (uz) (r) (axdnufcsgjlzipwthykmv) (bqore) (amniurdjcbyswvgkqhtxoeplfz) (btmjzgslrodky) (aqxhvwn) (cp) (eu) (f) (i) (aypzwjfrhngbvlei) (cuds) (ktqm) (o) (x) (atyuohjrxmszldbnwkvcigpfe) (q) (bjeguhknloxqyiwsfdprm) (avzc) (t) |

You can see the following:

- two generated pseudo-random alphabets
- a tableau consisting of one alphabet as the stator and the other alphabet shifting on position at a time, producing 26 enciphering alphabets
- the cycle notation for each of the 26 alphabets mentioned in the previous bullet

[2] John Byrne, Cipher A. Deavours and Louis Kruh. Chaocipher enters the computer age when its method is disclosed to Cryptologia editors. Cryptologia, 14(3): 193-197.

[3] Mellen, Greg. 1979. J. F. Byrne and the Chaocipher, Work in Progress. Cryptologia, 3(3): 136-154.

[4] Kahn, David. 1967. The Codebreakers: The Story of Secret Writing. Macmillan.

[5] Louis Kruh. The Mystery of Colonel Decius Wadsworth's Cipher Device. Cryptologia, 6(3): 238-247.

[6] Bauer, Friedrich. L. 2000. Decrypted Secrets: Methods and Maxims of Cryptology (2nd ed.). Berlin: Springer.

[7] Friedman, William F. Several Machine Ciphers and Methods for their Solution. Riverbank Publication Volume 2, No. 20. 1918. Reprinted by Aegean Park Press, 1979.

Copyright (c) 2009 Moshe Rubin

Created: 3 April 2009

Last Updated: 19 November 2009