How much would removing enigmas biggest flaw improve it?

Question

Enigma's biggest flaw was that a letter could never be encrypted as itself. How much would enigma's security increase if it were possible that a letter could be encrypted as itself? I know that would void the method used by Turing and co.

Please give as much detail as possible in answers.

EDIT: clarifying improvement

The improvement is, in the simplest terms:

Once the signal has passed through the rotors and into the reflector, the rotors shift forward before allowing the signal back through, meaning that each encryption operation would use two shifts instead of one.

EDIT 2: Also, with enigmas flaws removed, how comparable to modern algorithms would it's security be? And as another note, I'm not talking about modifying enigma back in WWII, I'm talking about a virtual software based mod now.

Answer 1

with enigmas flaws removed, how comparable to modern algorithms would it's security be?

With the specific change you specified, it would still (by modern standards) be considered "broken".

Here is how you can perform a distinguishing attack, and do a partial key recovery (recover the first two rotor settings with a good probability of recovering the third rotor setting) with about 400 characters of known plaintext.

I'll be assuming a 26 character rotor, however the attack scales to other sizes.

Here's how to recover the first rotor setting (and if you do recover a setting that works, that's a distinguishing attack):

• In the encryption operation, the note that:
  • The plaintext character is sent through the first rotor: $A := R_i({Pt})$
  • The ciphertext character is sent through the first rotor after stepping it one position: $B = R_{i+1}(Ct)$
  • The mapped plaintext $A$ is different than the mapped ciphertext $B$

This holds because the internal rotors and the reflector do not step during the operation, and hence will never map a character to itself.

Note that this might not be true if the second rotor is stepped during the encryption operation; however that will occur only once every 13 character encryptions.

So, if we take a guess at an initial rotor setting, we can determine how all the plaintext and ciphertext characters are mapped. Then, if we find two instances where the plaintext and ciphertext characters are mapped to identical $A, B$ values (and those locations are not multiples of 13 apart), we know that that initial rotor setting is impossible.

After 400 characters, the probability that an incorrect rotor setting will show as possible is approximately $2^{-13}$; as there are significantly fewer than $2^{13}$ possible settings for the first rotor, then with high likelihood, only the correct setting will remain as possible (and if no settings remain, then this isn't modified Enigma).

Now, that we've recovered the first rotor setting (and position), we can peel off the first rotor operation (by applying the now-know first rotor operation to the known plaintext/ciphertext), and use a similar attack to recover the second rotor (which is actually easier; we have a lot more information per rotor step). And, if the third rotor steps during the 400 character encryption but not the fourth (probability >50%), then we can recover the third rotor (using the same trick)

Bottom line: this would be considered (by modern standards) totally broken

Answer 2

Remember this was a mechanical device. Holding the signal in the reflector, rotating the device, and then releasing the signal, that would have been very difficult.

But also remember that what made Enigma easy to use (and easy to use is very important) is the fact that if two enigmas are set up in an identical way, and I enter the clear text into one and the encrypted text into the second, then the second machine outputs the clear text. And since this works with two enigmas set up identically, then it works with a single enigma.

So “fixing” the flaw totally changes the way enigma is used.

Answer 3

I'm new to this, however. I do not believe the stated flaw, is really a large contributing factor to enigma being broken as is it made out to be. Information about the process was the biggest flaw. For example, if a cryptologist didn't know that the device will not encrypt an input character to itself, where could they start? It would have to be discovered first, and then made into a flaw during the analysis. How would one know (or calculate) that fact first?

The Germans fist mistake was using a known starting point that was easily available, the pre-war consumer version. If the device was created without a created starting point, the fact that the device did not encrypt an input character to itself would not be in play.

Think of ASE encryption. Everyone knows how it works. But what they don't know is the starting point. The key.