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I'm working on one of those unsolved puzzles, and one theory I have is that one or more sections of it are using digits of pi, e, or some other irrational number as a one-time pad.

(The obvious approach would just be to try well-known irrational numbers, but I've tried that without success and if it is that there must be some other complication.)

Are there any cryptanalysis methods that that allows for that wouldn't apply to a one-time pad composed of letters? One thing that occurred to me is that since letters can't be shifted by more than 9 places, it ought to show very small peaks in its frequency distribution, one-tenth the size of ones for plain text and displaced about five letters to the right along the alphabet. But I don't know how much practical use that would be.

The question is probably equivalent to whether a Gronsfeld cipher has any weaknesses that a Vigenere cipher doesn't.

A. B.
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  • Other things aside, does https://media.sciencephoto.com/c0/01/27/10/c0012710-400px-wm.jpg help? – Paul Uszak Jan 27 '21 at 02:10
  • What am I looking at here? Can't make it out, except, obviously, that it's a scheme for coding letters as numbers. – A. B. Jan 28 '21 at 01:03
  • Infinite character sets can be fractionated to 0-9 decimals :-) – Paul Uszak Jan 28 '21 at 01:32
  • Ah, I see now. Well, these puzzle makers generally favoured mathematically pretty methods, but they did also like cryptographic history, so it's not impossible they might have used such an existing system for auld lang syne! – A. B. Jan 28 '21 at 01:53
  • I know, I mean the people who made the puzzle I'm working on, not the people who made that chart! :-D – A. B. Jan 28 '21 at 05:36

1 Answers1

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The one-time pad can be made of digits 0-9, and you can read more about that here. One could also use binary digits, 0 and 1, or letters such as A-Z. Those three alphabets are the most common.

Are there any cryptanalysis methods that that allows for that wouldn't apply to a one-time pad composed of letters?

No, there are not. A properly generated one-time pad is going to have equal strength no matter whether it is composed of letters A-Z, numbers 0-9, or the binary digits 0 and 1. The CIA, for example, was fond of using one-time pads composed of numbers 0-9 during the Cold War.

However, in practice, using numbers or binary could possibly introduce errors because of fractionation. In other words, using letters might be more straightforward.

One thing that occurred to me is that since letters can't be shifted by more than 9 places, it ought to show very small peaks in its frequency distribution...

Don't worry about that. It is not the case.

The question is probably equivalent to whether a Gronsfeld cipher has any weaknesses that a Vigenere cipher doesn't.

One can see what you mean by this, but Gronsfeld and Vigenere are a very different kind of cipher from the one-time pad.

Patriot
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  • "Don't worry about that. It is not the case." - why? If a computer is used to guess portions of the key that result in valid words, there will be many candidates per position, but probably not too many to process by a human. "Gronsfeld and Vigenere are a very different kind of cipher from the one-time pad." - the only difference I know about is key size. – the default. Jan 27 '21 at 02:58
  • One time pad has a random key equal to the size of the message, applied via a bijective transformation (it's reversible and 1:1). All plaintexts are thus equally likely for any given ciphertext: no statistical information about the plaintext is present. That's not the case for Vigenere or Gronsfeld, both leak statistical information about the plaintext. – SAI Peregrinus Jan 27 '21 at 03:09
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    @the default One could guess portions of the key all day long and guess plaintext constantly, but that does not matter because there is not way to distinguish what guessed plaintext is correct from guessed plaintext that is incorrect. The key does not repeat like Vigenere. The key of the one-time pad was generated at random and it must be at least as long as the plaintext, something like "HWOXYEPPJNBGS...", etc. The Vigenere key is very different. It is usually a dictionary word(s) such as "apple", etc., and it repeats. – Patriot Jan 27 '21 at 03:14
  • @Patriot the Vigenere key doesn't have to be a word. "but that does not matter because there is not way to distinguish what guessed plaintext is correct from guessed plaintext that is incorrect." - there is a way: let a human choose results that fit the context the best. – the default. Jan 27 '21 at 03:19
  • @the default On your first point, correct. I did say "usually". On the second point, I see what you are getting at, but in practice it does not work. One reason is that codewords could have been used – Patriot Jan 27 '21 at 03:36
  • Surely if you're using a one-time pad with only digits 0-9, then it's no longer true that any plaintext is equally possible? For instance, no possible 0-9 sequence could turn a ciphertext BNG into a plaintext THE. – A. B. Jan 28 '21 at 01:05
  • @A.B. I see how you are thinking about it. We use fractionation. Check out the "TAPIR" that was used by the STASI, then you will understand this more clearly. Good question though. – Patriot Jan 28 '21 at 01:12
  • Ah, I see what you're getting at. Yes, certainly you COULD use a scheme like that to avoid any such weakness. But this is a rather different scenario - it's not a "working" cipher but a puzzle, a cipher intended to be broken. And the puzzle makers have a history of liking mathematically elegant methods. So it's entirely possible that they might have used the digits as is. Given that, is there any weakness such a cipher would have? – A. B. Jan 28 '21 at 01:50
  • @A.B. Using a checkerboard like the TAPIR is not encipherment. That comes from the OTP. The STASI knew exactly what they were doing, no weaknesses that I can think of. The use of codewords in the message denied a firm understanding of the message's true length in normal language. The one-time pad offers perfect security if used correctly, and it was used by the most serious people for their most important purposes. You might enjoy reading this: http://users.telenet.be/d.rijmenants/en/table.htm – Patriot Jan 28 '21 at 07:21