Rejewski’s equations: Solving for the entry permutation

In 1932, Marian Rejewski, who was a young mathematician working at the Polish Cipher Bureau, brilliantly recovered the internal wiring of the military Enigma. His initial efforts were unsuccessful because he assumed that the entry permutation was the same as in the commercial machine. Luckily he tried the identity permutation as an alternative and that proved to be correct. This note describes how Rejewski’s equations may be used to deduce the entry permutation without any guesswork, a technique that was later rediscovered by Alan Turing and by Lieutenant Andrew Gleason.     

Recovering the military Enigma using permutations—filling in the details of Rejewski's solution

During the last few months of 1932, the Polish mathematician Marian Rejewski solved the problem of finding the internal connections of the rotors and reflector of the Enigma cipher machine used by the German army at that time. This allowed the Polish Cipher Bureau to construct an analogue of the machine, and subsequently to find effective methods for deciphering secret messages. Rejewski performed this feat virtually alone using cryptographic material provided by the Polish secret services. His knowledge of the theory of permutation groups was essential in solving this problem.

This article describes in detail how to find the complete wiring of the rotors and reflector of Enigma, as well as other specifics, using data that Rejewski had at his disposal, by systematically presenting the resolution of all cases that could have been encountered. Similarly, we complete those stages of the procedure that were only outlined by Rejewski.     

Modern breaking of Enigma ciphertexts

“Breaking German Army Ciphers” is the title of a Cryptologia article from 2005, describing the lucky survival of several hundred authentic Enigma messages of World War II, and an account of a ciphertext-only cryptanalysis of a large number of these messages, leaving only a few (mostly short messages) unbroken. After reviewing the work done, and investigating the reasons for both lucky breaks and close misses, the modern ciphertext-only attack on Enigma messages is improved, especially on genuine ones with short lengths and/or many garbles. The difficulties of a proper measure for the candidate’s closeness to a plaintext are clarified. The influence on the decryption process of an empty plugboard and one with only a few correct plugs is examined. The method is extended by a partial exhaustion of the plugboard combined with an optimized hillclimbing strategy. The newly designed software succeeds in breaking formerly unbroken messages.     

KRIEGSMARINE SIGNAL INDICATORS

Indicators used by the Kriegsmarine in its signals traffic in World War II are explained.