Are encryption algorithms with fixed-point free permutations inherently flawed? Flaw in Enigma
One of the Enigma machine's flaw was the derangement (fixed-point free permutation) of the produced ciphertext, or simply put: No plaintext-letter can be enciphered to itself. See this example from Wikipedia of how this text (in German) &quot;Keine besonderen Ereignisse&quot; can or can't be encrypted:

More modern encryption algorithms used during WW2, such as the Enigma-alike Typex machine used by the British, eliminated this flaw, meaning that a plaintext-letter sometimes could indeed be encrypted to itself.
And as far as I know most modern encryption algorithms (i.e. AES) also allow for the possibility of plaintext-letters being encrypted to themselves.
Questions
My questions are:

Are all encryption algorithms with fixed-point free permutations inherently flawed?
Can it be mathematically proven that all fixed-point free permutation algorithms can be broken with a &quot;faster-than-brute-force&quot; attack?
Are all encryption algorithms with fixed-point free permutations inherently flawed?
  


Yes - when fixed points, or the lack of them, is knowable and detectable.

This is a violation of multiple modern semantic security definitions. For example, this means that plaintexts with repeating symbols are distinguishable with high probability from plaintexts that has none. It means you with a high probability can determine what a plaintext does not include, especially if you can observe multiple communications.

This violates security notions like chosen plaintext attacks, indistinguishability from random, and more.


  
  Can it be mathematically proven that all fixed-point free permutation algorithms can be broken with a "faster-than-brute-force" attack?
  


It always makes it easier to guess the plaintext (as according to the answer of question 1), but it does not necessarily make it easier to derive the original key (what bruteforce normally refers to). You can often not say with certainty what the original plaintext is without the key, but often you can settle for an educated guess.