Editing Cryptanalysis of the Enigma (section)

===Rejewski's characteristics method===
Marian Rejewski quickly spotted the Germans' major procedural weaknesses of specifying a single indicator setting (''Grundstellung'') for all messages on a network for a day, and repeating the operator's chosen ''message key'' in the enciphered 6-letter indicator. Those procedural mistakes allowed Rejewski to decipher the message keys without knowing any of the machine's wirings. In the above example of ''DQYQQT'' being the enciphered indicator, it is known that the first letter ''D'' and the fourth letter ''Q'' represent the same letter, enciphered three positions apart in the scrambler sequence. Similarly with ''Q'' and ''Q'' in the second and fifth positions, and ''Y'' and ''T'' in the third and sixth. Rejewski exploited this fact by collecting a sufficient set of messages enciphered with the same indicator setting, and assembling three tables for the 1,4, the 2,5, and the 3,6 pairings. Each of these tables might look something like the following:
{| class="wikitable" | border=1 style="margin: 1em auto 1em auto"
|-
! First letter
| A||B||C||D||E||F||G||H||I||J||K||L||M||N||O||P||Q||R||S||T||U||V||W||X||Y||Z
|-
! Fourth letter
| N||S||Y||Q||T||I||C||H||A||F||E||X||J||P||U||L||W||R||Z||K||G||O||V||M||D||B
|}
A path from one first letter to the corresponding fourth letter, then from that letter as the first letter to its corresponding fourth letter, and so on until the first letter recurs, traces out a [[cycle graph (algebra)|cycle group]].<ref>Also referred to as a ''box shape'' or a ''chain''. See {{Harvnb|Alexander|c. 1945}} Ch. II Para. 4</ref> The following table contains six cycle groups.<!-- (ydqwvougc)(ket)(nplxmjfia)(zbs)(r)(h) -->
{| class="wikitable" style="margin: 1em auto 1em auto"
|-
! Cycle group starting at A (9 links)
| (A, N, P, L, X, M, J, F, I, A)
|-
! Cycle group starting at B (3 links)
| (B, S, Z, B)
|-
! Cycle group starting at C (9 links)
| (C, Y, D, Q, W, V, O, U, G, C)
|-
! Cycle group starting at E (3 links)
| (E, T, K, E)
|-
! Cycle group starting at H (1 link)
| (H, H)
|-
! Cycle group starting at R (1 link)
| (R, R)
|-
|}

Rejewski recognised that a cycle group must pair with another group of the same length. Even though Rejewski did not know the rotor wirings or the plugboard permutation, the German mistake allowed him to reduce the number of possible substitution ciphers to a small number. For the 1,4 pairing above, there are only {{math|1=1×3×9=27}} possibilities for the substitution ciphers at positions 1 and 4.

Rejewski also exploited cipher clerk laziness. Scores of messages would be enciphered by several cipher clerks, but some of those messages would have the same encrypted indicator. That meant that both clerks happened to choose the same three letter starting position. Such a collision should be rare with randomly selected starting positions, but lazy cipher clerks often chose starting positions such as "AAA", "BBB", or "CCC". Those security mistakes allowed Rejewski to solve each of the six permutations used to encipher the indicator.

That solution was an extraordinary feat. Rejewski did it without knowing the plugboard permutation or the rotor wirings. Even after solving for the six permutations, Rejewski did not know how the plugboard was set or the positions of the rotors. Knowing the six permutations also did not allow Rejewski to read any messages.