Introduction

The fundamental structures in science (physics, chemistry, biology, etc.) sometimes elude us. They can appear mysterious, withholding some of their secrets and refusing the rules we try to impose upon them. Throughout history, conflicts between new theories, experimental data, and established beliefs have often been ignored or dismissed because they did not fit the accepted way of thinking. This attitude is frequently reflected in phrases such as “everyone knows” or “everyone should know.”

Discarding new ideas prematurely can be costly in the long run and, in our view, is almost always a mistake.

Of course, new theories must be tested rigorously, but the same standard must also apply continuously to existing theories. If we fail to do this, we risk leaving the path of science and moving instead toward complacency.

In physics, for example, we know about the coexistence of Albert Einstein’s relativity and Werner Heisenberg’s quantum mechanics. Both are experimentally successful in their domains, but the problem is that they are difficult to reconcile completely under one unified framework.

The question emerging for us was: "Can cryptography hold possibilities which would allow different theorems to co-exist, generating ciphers in compliance with Shannon's mathematical proof of Perfect Secrecy, I(P;C)=0." For this reason we put a question in front of 6 AI systems and 5 academics (mathematicians teaching at universities). The question will lead us back to Claude Shannon and his mathematical-theoretical proof of Perfect Secrecy and also the three axioms he formulated for his entropy function H:

Let's assume we have a plaintext string of random characters and a key string that only consists of 26 characters. The plaintext uses the same alphabet as the key string, but is 10.000 random characters in length. How would an attacker try to crack a cipher created, using modular arithmetic and what would be the result of this attempt?

The answers we received from the academics were without exception that the cipher would provide no reasonable information for an attacker. The AI systems answered to the contrary, insisting that Claude Shannon's mathematical theoretical framework and his entropy H function would be enough to decrypt the cipher and produce the correct random string of random (don't forget random means unpredictable) characters. However, after the additional question to show us the pattern and frequencies that would emerge which they claimed would appear, they joined the consensus that it was impossible to solve the problem we presented with our question.

This moves us back to Claude Shannon and the three axioms he postulated for his evaluation of the Vernam Cipher (re-branded OTP later) and the four key requirements which you will not find in his papers. These requirements were a later addition by mathematicians and cryptographers for pedagogical or educational teaching purposes.

1. A key that is truly random
2. A Key as long as "All" messages to be sent
3. Each part of the Key is only used once
4. The Key is kept completely secret

Shannon's entropy function H (3 axioms) explaining function H to be a measure of a message A and the information it contains:

1. It should be continuous in the P(ai), where P is a probability.
2. It increases monotonically with N if P(ai) = 1/N
3. It satisfies the decomposition rule for joint entropy of two discrete random variables.

One sentence we found missing in Shannon's papers, which have been re-published and interpreted on different websites to explain his entropy H function and its connection with the Vernam system and the required Markov process:
*You will find it in "A Mathematical Theory of Cryptography, Claude E. Shannon — Published September 1945", which was one precursor to the final version published in the Bell System Technical Journal in October 1949.

"With N = 1 any letter is clearly possible and has the same a posteriori probability as its a priori probability. For one letter the system is perfect."

The reasons for its removal are unknown to us, but the sentence triggers for us an important and genuinely thought-provoking question:

"If each plaintext character is transformed independently through its own permutation space, without statistical dependence on neighboring characters, does the system still behave like one large Vernam system — or like many isolated one-symbol Vernam systems? And how does it affect the key requirements?"

Let's go back to Shannon and what he said:

11 EQUIVOCATION (Page 683 -- A Mathematical Theory of Cryptography)

Let us suppose that a simple substitution cipher has been used on English text and that we intercept a certain amount, N letters, of the enciphered text. For N fairly large, more than say 50 letters, there is nearly always a unique solution to the cipher; i.e., a single good English sequence which transforms into the intercepted material by a simple substitution. With a smaller N, however, the chance of more than one solution is greater: with N = 15 there will generally be quite a number of possible fragments of text that would fit, while with N = 8 a good fraction (of the order of 1 8) of all reasonable English sequences of that length are possible, since there is seldom more than one repeated letter in the 8. With N = 1 any letter is clearly possible and has the same a posteriori probability as its a priori probability. For one letter the system is perfect.

Does the paragraph not point to Shannon’s concern about language statistics and Markov dependencies? In ordinary language, letters are correlated, words are correlated and structure leaks information. His paper rigorously defines perfect secrecy as the condition where a ciphertext provides no information about the plaintext, which is mathematically true for a single character encrypted with a unique, random key (like in a One-Time Pad), but also true for a single transposed character using a random permutation of the alphabet. The paper also discusses the use of a "suitable Markoff process" to model the statistical properties of the message source and the key source.

Our argument for our transposition system is, if every symbol transformation is isolated, and no transition information propagates between neighboring symbols, then higher-order statistical structure will collapse. That is a serious conceptual point one has to explore.

Here is a more conceptual answer to our question we formulated above.

In an isolated one-symbol system with each character encrypted independently with its own key (permutation), the security of each character is identical to that of a single-character Vernam cipher. The ciphertext for one character reveals nothing about that specific plain character, just as in a perfect one-symbol system.

In a large Vernam system, when we combine all these independent encryptions, the overall system is mathematically equivalent to a standard Vernam system (OTP) applied to the entire message. The "key" for a large system is simply the concatenation of all the individual keys (or permutations) used for each character. The requirement for perfect secrecy, that the key space is at least as large as the message space and is used only once, is fully satisfied.

Therefore, the system achieves perfect secrecy as defined by Shannon. The independence of the permutations ensures that the statistical properties of the plaintext (like letter frequency) are completely masked, and an attacker, even with infinite computational power, cannot determine the original message from the ciphertext alone, as every possible plaintext of the same length remains equally probable.