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Mathematically, why was the Enigma machine so hard to crack? In laymen terms, what was it exactly that made cracking the Enigma machine such a formidable task? Everything I have seen about the Enigma machine, from a to information about, is quite lengthy, and it appears to be difficult to pinpoint exactly the most salient mathematical difficulty facing the codebreakers other than the sheer number of possible settings (159 million million million according to this ) that changed every single day. Or was this really it? There seems to be much more to it than that. I was hoping someone with a fair amount of knowledge about the mathematics behind the Enigma and its breaking might be able to provide a condensed, simplified reason for what made cracking this machine such a monumental undertaking.
It was a hard wired machine with a gear based permutation group. It had a clock permutation system: It permutes the letters in a hardwired fashion and each key click shifts that set permutation one space. After 26 shifts in the first gear, it does one turn of the second gear. After 26 turns of the second gear, it turns the third gear. This means that every key press we change the permutation group, every 26^2 we change it in an additional way, and every 26^3 we change it in an additional way.
Added to that, characters were swapped at the end, and then it ran back through the gears again. – user142198 Apr 18 '15 at 7:51. Reading back, I do see I didn't mention the contributions of Turing and other Bletchley staff, and I didn't mean to imply they did not contribute greatly. As you say, the statistical method used on the later Enigma machines eliminated possible starting positions, which to me sounds like an attack on the quantity of possibilities, and not an attack on any kind of clever cryptographic scheme. We are still saying it was hard to crack because of so many possibilities, not German cypher innovation.
– Apr 19 '15 at 5:21. In modern computer cryptography, large numbers are one of the most important factors. That's why 40-bit encryption used to be considered security back in 1995, but today (with 512 bit encryption available on almost any security device) would be considered a joke. My reading of the history of cracking Enigma is that failures in use and implementation of Enigma by the Germans, combined with effective intelligence gathering were the most critical factors in enabling the codebreakers to succeed.
There wasn't much to the algorithm itself, it just had a huge number of combinations. Like the Rubik's Cube. It's just a finite non-Abelian group, but it's the huge order of that group that makes it so you can't just write down the entire Cayley table and find the minimum number of operations to a solution from there. The reason the large 'bombas' were constructed in the cracking of Enigma was to speed up the process, because there were so many possible combinations. Which again brings us back to modern computer cryptography, where the computing power available determines the length of time it would take to brute force decryption of a given key size, so using the longest practical key is critical. This quote from Marian Rejewski, one of the Polish codebreakers who worked on Enigma, basically says (to me), 'the Germans increased the number of combinations which makes our job a lot harder': we quickly found the wirings within the new rotors, but their introduction.
raised the number of possible sequences of drums from 6 to 60. and hence also raised tenfold the work of finding the keys. Thus the change was not qualitative but quantitative. We would have had to markedly increase the personnel to operate the bombs, to produce the perforated sheets (60 series of 26 sheets each were now needed, whereas up to the meeting on July 25, 1939, we had only two such series ready) and to manipulate the sheets. (emphasis mine). Again, the Rubik's cube analogy is only meant to show how sheer numbers can make a problem much more difficult. Imagine a Rubik's style cube with only eight pieces instead of 27.
Most people would be able to solve it very quickly. And figuring out the rules for solving a Rubik's Cube is what I consider analogous, not memorizing them after someone else has determined them. That would be like decrypting an Enigma coded message when you already know the starting position and plugboard settings. – Apr 19 '15 at 5:27. The answer to the question 'Mathematically, why was the Enigma machine so easy to crack?' : The first major weakness was the fact that the same settings were used for a whole day.
There are always messages that are easier to crack and others that are harder to crack; using the same settings for a day meant only one exceptionally easy to crack (and likely very unimportant) message needed to be cracked to crack all the messages of a day. The second major weakness was the fact that in every state, the Enigma machine produced an Enigma permutation (13 cycles of length 2), which made it accessible to mathematical attacks. The third major weakness was the fact that the huge number of possible settings could be separated into separate and easier problem. It was possible to first crack the rotor settings, and then the plugboard settings separately. (Same mistake was made in DVD encryption, where a 40 bit code could be separated into a 25 bit and a 16 bit code that could be cracked separately). In every possible state, an Enigma machine produces a permutation of the letters A to Z. And not just any permutation, but one that exchanges pairs of letters, for example in a certain setting it might exchange A and Q, B and F, C and R and so on.
Such a permutation, which consists entirely of cycles of length 2, is called an 'Enigma Permutation'. After transmitting a letter, the machine state would be changed in a deterministic way, so a different Enigma permutation was used. Importantly, a code cracker can be assumed to know all encrypted messages, because they were just sent over radio.
Rejewski's theorem says: 'The composite of any two Enigma permutations consists of disjunct cycles in pairs of equal lengths'. Since the total cycle length is 26, you might have for example two cycles of length 1, two cycles of length 5, and two cycles of length 7. Another quite simple theorem says that a permutation M and a permutation S M S^-1 have the same cycle characteristics. For the first few years, every transmission started by setting the machine into a fixed start state (known to sender and receiver, but not known to the code cracker), then the sender would pick a random three letter code and transmit it twice, then sender and receiver would use that three letter code to change the machine settings. After that, each message was sent with different machine settings. All machines sent the first six letters with the same permutations P1, P2, P3, P4, P5, P6. If the sender transmitted ABC ABC, and the receiver receives RST XYZ, then permutation P1 exchanges A and R, P4 exchanges A and X.
The cracker knows R and X, but not A. But the cracker knows that the permutation P1 P4 maps R to X, because P1 maps the known R to an unknown A, and P4 maps the unknown A to the known X. If you receive enough messages, with different random letters ABC, you gather enough information to find the complete permutation P1 P4. Same for P2 P5 and P3 P6. Now each of these permutations consists by Rejewski's theorem of cycles in pairs of equal lengths, with the lengths adding up to 26 or the lengths of one half of each pair adding up to 13. That gives a pattern; in the example above the pattern would be (1, 5, 7).
And this pattern only depends on the initial rotor settings, it is independent of the plugboard (the second of the two theorems mentioned). With the initial 3 rotors which could be installed in 3! = 6 orders, and 26 x 26 x 26 initial rotor rotations, there were 105,456 possible initial settings, each of which would produce 3 patterns for the permutations P1P4, P2P5, and P3P6. What the polish mathematicians did was create an index: For each of the 105,456 initial positions they found over months work the 3 patterns associated with each position. And with that it was easy to find the initial rotor settings for a day after intercepting about 100 messages.
Not all settings created unique patterns, but usually a pattern would be produced by one or very few initial settings. Some rotor settings are bad news; for example there are 313 rotor settings producing three pairs of cycles of length 13.
The plugboard settings could also be discovered: Knowing the initial rotor settings, we can determine the permutation P1' P4' that would have happened without the plugboard. We also have the permutation P1 P4 that was produced with the plugboard. These can be compared, and we have the same information for P2' P5' vs. P2 P5 and P3' P6' vs P3 P6. That was usually enough to determine the plugboard settings.
So initially the Polish were able to decipher messages by hand. This stopped working when the transmission method changed (no 3 letters transmitted twice) and when 3 rotors were replaced by 5, with 60 possible rotor choices. Later methods were substantially based on guessing messages or parts of messages. Since the same initial settings were used over a whole day, if just one message was cracked, every single message for the day was cracked (if not, all the messages for the day were unreadable). A good source of messages that could be guessed were weather reports: Since they were not very secret, they would be transmitted through the country with an easily cracked code, so the exact weather reports could be recovered. The exact same messages were sent to submarines with the enigma code, so that was a good source for known message texts. Later a fourth rotor was introduced for top secret messages.
That would have been uncrackable at the time. Except the settings for the four rotor machines were the same as the three rotor machines with another ring in 26 possible positions. So after cracking the three rotor code, just 26 attempts were needed to crack the four rotor machine.
CIPHER MACHINES AND CRYPTOLOGY Enigma Simulator v7.0 This software is an exact simulation of the 3-rotor Wehrmacht (Heer and Luftwaffe) Enigma, the 3-rotor Kriegsmarine M3, also called Funkschlussel M, and the famous 4-rotor Kriegmarine M4 Enigma cipher machine, used during World War II from 1939 until 1945. The sim has a very authentic feeling with its hands-on approach: you can select between the three models, actually lift out and insert different rotors, adjust their ring setting and set up the plugboard. The internal wiring of all rotors is identical to those that were used by the Wehrmacht and Kriegsmarine. This simulator is therefore fully compatible with the various real Enigma models and you can decrypt authentic wartime messages or encrypt and decrypt your own messages.
The program comes with a very complete 22 page helpfile, containing the manual, some original messages, the history of Enigma and all technical details of the machine. The simulator also has a picture gallery of Enigma machines. With this software you will finally be able to work with the most intriguing machine in military cryptology and examine how it works and how it was operated.
A true reference to Enigma, and an educational must! Check out the to discover all the nuts and bolts of the software. All questions or feedback are welcome or by visiting the. Download the Simulator Please check the readme file before installation. Runs on Windows™ and with WINE on Linux or Parallels Desktop on MAC. Please uninstall any previous version through configuration screen/software Discover also my, and cipher machine Simulators! Wehrmacht Enigma I Kriegsmarine M3 Kriegsmarine M4 Enigma I - M3 - M4 open Useful adds.
Create and print your own code books. More on Enigma on this site. The history of Enigma, its developement, use during the ware and cryptanalysis. Join the contest to decrypt 10 messages in a wartime context. Breaking an authentic U-boart message and the story of U-264 Off-Site. Bruce Culp's excellent website, designed to join Enigma enthousiasts and globally exchange Enigma encrypted messages.
Provides clear and simple instructions on how to encrypt and decrypt messages. Code books provided. showing the main features of my Enigma simulator. Mitchel Thomas and Indiana Popovich made a website where you can decrypt enigma messages (keys provided) to unveil the story of a fascinating 1938 intelligence operation in a shadowy pre-war atmosphere. More info about how and why © Copyright 2004 - 2018 Dirk Rijmenants.