On Rejewski’s Solution of Enigma Cipher

In this contribution I want to present the extract of the article about Rejewski’s solution of Enigma cipher, which is being prepared. The story of Enigma is a part of modern cryptology. However, there still remain some unclear facts about Enigma. The reason is that the information were for a long time classified, now there is a lack of reliable sources. Moreover, some of the reports are even conflicting. Main subject is an analysis of the method developed by Rejewski to reveal the internal connections in the machine. His description is not complete, it is impossible to calculate the internal wirings of all three rotors by the method he developed. I present here one of the two new methods how to calculate the connections in the third rotor together with the probability of having suitable data for this method. In original article there is also a detailed analysis of the complexity of Rejewski’s method. Introduction Rejewski’s calculation of the internal wiring of the military Enigma machine in December 1932 is one of the most remarkable events in the history of cryptology. The importance of Enigma and its solution before and during the WWII was described e.g. in [Bauer, 2000; Kozaczuk, 1984]. These sources emphasize the significance of contribution by Rejewski and also by his colleagues from the Polish military cryptanalytical service during the period between 1932 and 1939. Subsequent history of breaking the Enigma cipher in Bletchley Park during the WWII can be found e.g. in [Singh, 1999] Rejewski himself described his solution in two different manuscripts [Rejewski, 1940; Rejewski, 1967] (available only in Polish). Only long after the end of the WWII, in 1980, two papers prepared by Rejewski were published (only posthumously) in English. Probably due to historical circumstances (the first monograph was prepared in France in 1940, the second one in Poland in 1967), Rejewski’s description of his work is rather brief in some aspects. Moreover, long breaks between writing different papers (after the WWII Rejewski lived in the then Communist Poland and had in fact to hide his work for the prewar Polish Government as well as for the Polish Government in exile in London in the years 1940-1945) probably contributed to the fact that some moments in the history of solving the Enigma cipher were described in different manuscripts in a different way. In this paper we summarize the basic steps of Rejewski’s reconstruction of the Enigma machine and attempt to fill in some missing points. Some of the information described in Rejewski’s Polish manuscripts appear here for the first time in English. We shall assume familiarity with the operation of the military Enigma machine and its basic operations. Details can be found e.g.in [Kruh, 1985]. Summary of Rejewski’s Method The Begining of The Work The first Rejewski’s task was to uncover the internal wirings in the three rotors and in the reflector. He had at his disposal several dozens enciphered messages from each day, later the daily keys from September and October 1932 (obtained through espionage) and also the operation manual (also through espionage) describing the structure of the daily and message keys. Probably already before obtaining the operation manual he discovered that the first six letters of each enciphered message was the message key enciphered twice as one can conclude from [Rejewski, 1967], which is probably the source of corresponding information also in [Bauer, 2000]. Let us first denote by the letters A, . . . , F , six permutations defined by the first six successive positions of the rotors set up according to the daily key. These six permutations were used to double WDS'06 Proceedings of Contributed Papers, Part I, 124–129, 2006. ISBN 80-86732-84-3 © MATFYZPRESS