Enigma Strength

Introduction

The Enigma Machine had more configurations than the number of conceivable atoms in the known universe and thus the Germans had extremely high confidence in it. However, as the Germans found out, having a very large number of arrangements exclusively doesn't ensure enough security to carry out missions. Security is primarily a people problem. Anyway, the following are the parts that contribute toward the total number of possibilities that the Enigma Machine had.

5-Components.

1. Plugboard

The plugboard consisted of 26 lettered sockets of which a total of 13 wires could connect any one given socket to any other one socket. So the number of possible ways the sockets could have been chosen is 26 C 2n, where n is the total number of cables such that 0 ≤ n ≤ 13. 26 C 2n is read "twenty-six choose two n" as in combinations.

At the start there are 2n number of sockets to choose from. So, for example, if you're using 10 cables then there are a total of 20 different sockets that will get used. Upon inserting one end in a socket from the first cable being used there are now 2n - 1 total sockets to choose from. Next, after the second cable's front end is inserted there are 2n - 3 possible sockets to choose from. Thus the total number of ways n cables can be inserted into 2n sockets is (2n - 1)(2n - 3)(2n - 5)(2n - 7) ... 1. So, the total number of connections that can be made is (26!)/((26 - n)!n!2ⁿ).

Finally, each value of the number of cables used, n, yields 13 possible mutually exclusive combinations. Therefore, we have Σ n = 0 to 13 [26!/((26 - 2n)!p!2^p)] which comes out to a total of 532,985,208,200,576 possible plugboard combinations.

2. Three-Rotors

Each rotor can be wired one of 26! ways. So, having 3 rotors the total number of possible combinations is 26!(26! - 1)(26! - 2).

3. Three-Rotor Setting

Initially, each rotor can be set to 1 position out of 26. Thus the total possible combinations that the rotors could be set by the operator is 26³ = 17,576.

4. Movable Ring on Rotors

A notch was located on each ring that caused a rotation of the next ring (going from left to right) once every 26 keystrokes. Subsequently, the leftmost ring out of the 3 rotated every 26² or 676 keystrokes. Of course, at the far left was the reflector that did not rotate.

5. Reflector

The reflector is similar to the plugboard in that there were only 26 points of contact where an incoming signal would go through one point and be mapped via a wire to any other point going back out. Because each wire consumed 2 points of contact the total number of unique reflector settings could be (26 - 1)(26 - 3)(26 - 5) ... 1 or 26/(13! * 213). This equals 7,905,853,580,625 ways.

Possible Configurations

So the absolute total number of ways the Enigma Machine could be configured is 3 * 10¹¹⁴.

German Navy Enigma.

The German navy version used 4 rotors. The only changes out of the 5 components listed above regarding the 3-rotor enigma are components 2,3 and 4. Components 1 and 5 had the same number of possibilities because the plugboard did not change and the reflector remained the same, each with 26 options.

For component 2 there were 4 rotors which yielded 26!(26! - 1)(26! - 2)26!.

For component 3, since there are 4 rotors, the total possible combinations that the rotors could be set by the operator is 26⁴ = 456,976.

For component 4, because a ring could have 2 notches, the total possible arrangements are 26 * 25. So, adding the rings with 1 notch would yield 26 + (26 * 25) which is 26². For some reason, the fourth rotor was immobile. It had no ratchets nor stepping lever. Thus, we have 26⁴ = 456,979 total possible positions for the middle two and rightmost single or dual notched rotors.

Possible Configurations

Finally, the total number of possible configurations is again the product of all 5 components above. 2 * 10¹⁴⁵, larger than the 3 rotor Enigma of course.

Actual Possibilities.

The Allies actually had some information that could help reduce the number of possibilities.

For component 1, the plugboard normally used 10 cables, which yields 150,738,274,937,250 possibilities.

For component 2, the number of rotors varied from 3 all the way up to 10. The fairest assumption was that 3 out of 5 discs were used and that the wiring of the discs were known. Also, the order of the discs had to be determine thus yielding a 5 P 3 (permutation) which is 5!/(5 - 2)! = 60.

For component 3, the initial setting of the rotor positions was an unknown key. So, the number of possibilities here was 26³ or 17,576.

For component 4, single notched rings were assumed. So, it was 26² or 676 possibilities.

For component 5, the wiring of the reflector was usually known so this equated to 1.

Possible Configurations

So in the "real" situation, the Allies had to deal with roughly 10²³ possibilities, which is the product of all 5 parts. Furthermore, the Enigma machine was at minimum able to thwart letter frequency counts due to its complex arrangements.