How to decode and what is the use?(Math!)

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@Otienimous said (#21):

Now I wholly agree, if you use a sequence of the Pi digits from a given
position, to the best of my knowledge the sole method of breaking the
cipher should be breaking its user

I beg to - slightly - disagree. The first thing to mention is using pi digits is not a that polyalphabetic encryption any more but effectively a "one-time pad".

Polyalphabetic encryption (as like the monoalphabetic one) still preserves a 1:1 relationship of letters. As long as you don't encode empty space too, it is therefore possible to guess some information about the structure of the encoded message (i.e identify word boundaries) and from that work your way backwards to the encryption method being employed.

This, basically, is similar to how Alan Turing cracked the Enigma machines encryption: he established, that the first part of the message always was a certain number combination (i can't remember: it was either the clock time or the position of the submarine) and so had a lever to reduce the number of combinations to brute-force considerably.

Of course, such "contextual analysis" is only possible with texts of sufficient length (or many ecrypted texts using the same method, which amounts to the same). Luckily for Turing the commanding officer of the submarines liked to poll them many times a day for their status so he had a lod of texts to work with.

krasnaya

@krasnaya

Of course, you are quite right in your observations. I agree with most of them, so I'm sorry if they couldn't be derived clearly enough from my previous posts.

It is true that the method of using pi digits is a "one-time pad", as we already mentioned together with Tromeus. It is also true that polyalphabetic encryption - even with a very long key, for example a book key - is breakable by mathematical analysis, unlike a one-time pad. I tried to explain it further in posts #16 and #23. Finally, it is very true that encoding empty spaces usually makes the cipher significantly stronger.

About Enigma... as far as I know, the Germans were so sure of its safety that they fell into some serious malpractices. Not only time and position, but also mandatory "Long live the Chancellor" (in German, but translating it could get my account blocked...) on the end of the message. It made the analysis of the cipher much easier. By the way, with whole due respect to the Alan Turing's brilliancy and tragic fate, we should remember that his success with Enigma was mostly based on previous works of Rejewski, Różycki and Zygalski, who broke its earlier version already in 1932.

PS Why did you dislike your own message? It seems very good and precise, after all.

@krasnaya

Sometimes Google is your friend.

"A polyalphabetic substitution cipher involves the use of two or more cipher alphabets. Instead of there being a one-to-one relationship between each letter and its substitute, there is a one-to-many relationship between each letter and its substitutes."

So, no: "Polyalphabetic encryption (as like the monoalphabetic one) still preserves a 1:1 relationship of letters", you write in #32 is not true.

Also, one-time pad is polyalphabetic, too!

Best wishes, friends.

@tromeus

It looks like a misunderstanding. I think that by writing "Polyalphabetic encryption (as like the monoalphabetic one) still preserves a 1:1 relationship of letters" Krasnaya meant that the original message and the encrypted one will always have the same length, which is true. Of course, you are also right that "Instead of there being a one-to-one relationship between each letter and its substitute, there is a one-to-many relationship between each letter and its substitutes", but the two sentences, as you see, do not contradict each other.

And yes, one-time pad is a very particular case of a polyalphabetic substitution cipher with additional features. I think that I have already mentioned it, but I may be wrong. Best wishes to you, as well!

@Otienimous said (#33):

unlike a one-time pad. I tried to explain it further in posts #16 and #23.

You are right, my bad. I have somehow overread that part.

Not only time and position, but also mandatory "Long live the Chancellor"
(in German, but translating it could get my account blocked...)

I think context - as always - matters. In fact the "Heil Hitler" is addressing him personally, with name, rather than his function as chancellor. This is more in line with the person cult of the Nazis about the "Führer" (=leader, commander). Not that it matters here but i suppose for a non-native german speaker it is difficult to grasp the intricacies.

What we can take away from that is: the more (known) structure a message has the easier it is to decipher and the better it lends to "contextual analysis" of some sort. If i know the cipher to represent an encrypted digit i only have 1 out of 10 check whereas otherwise i would have 1 out of 36 (or maybe even more if punctuation is included) possibilities. If i know a message to be in a certain language i can go hunting for common words. And so on. Therefore: for maximum protection avoid any recurring structure and avoid any recognizable structure (like word boundaries).

krasnaya

#33 Yes the Poles did a lot of work cracking Enigma. It also helped when the British (with Ian Fleming, later James Bond author) tricked a German submarine with a false distress signal so as to seize its Enigma encoder.
Also the Japanese code was cracked. Admiral Yamamoto must have suspected that, as he ordered his Kido Butai task force of 6 aircraft carriers for the attack on Pearl Harbor to observe strict radio silence.

@Otienimous

#35 "And yes, one-time pad is a very particular case of a polyalphabetic substitution cipher with additional features. I think that I have already mentioned it, but I may be wrong"

Correct! The only special feature about one-time pad is that you change the key every time you're sending the message. The saying goes: "if you communicate a different key each time, why don't you communicate the un-encrypted message instead?"

That's why RSA is so powerful today. Once, you've published your prime number you're done with! No further action required. It remains on the hands of the enemy to factorize it. But very soon all this will change, I am afraid...

@krasnaya

I speak German only a little bit and I thought that mentioning the function is a proper way to avoid the name (which, I was afraid, could get my account blocked by some stupid bot).

@tromeus

Actually, you should publish a semi-prime number (such that is a product of two primes). And you are right, once quantum computers become able to factorize efficiently, another ciphers may be used, but nevertheless many past secrets will get revealed and many leading figures will be prone to blackmailing.

@tromeus said (#38):

The only special feature about one-time pad is that you change
the key every time you're sending the message.

No. The special feature about the one-time pad is that you use a key length equivalent to the length of the message (or even longer, but that wouldn't change the outcome). This is what sets apart "digits of pi" (it is, for practical purposes, a key of infinite length) from other polyalphabetical ciphers.

And further - i know i am wrong, as you have already pointed out, but i'd like to stick to my erroneous beliefs - the weakness of every polyalphabetic cipher is that the relationship between unencoded and encoded signs is 1:1. That means if you encode one character the result will always be one (other) character (or, as @Otienimous correctly mentioned in #35: the length of the message is preserved).

There are other ciphers (i.e. algorithmic encoding), which don't preserve message length and which are therefore much harder to crack (if even possible).

@Otienimous said (#33):

success with Enigma was mostly based on previous works of Rejewski, Różycki and Zygalski

True! Yes, i should have mentioned them. In fact the Poles had a lot of influential mathematicians at the early 20th century. Leśniewski, Teitelbaum, Sierpiński, ... i could go on.

krasnaya