Cryptanalysis of Enigma double indicators with hill climbing

Abstract

The Enigma machines were a series of electromechanical rotor cipher machines developed in Germany and used in the first half of the twentieth century to protect commercial, diplomatic, and military communications. Until 1938, the German Army used the so-called double-indicator procedure to transmit Enigma-encoded messages. It was replaced in September 1938 by a new procedure also involving double indicators. Both procedures enabled a team of mathematicians from the Polish Cipher Bureau to recover the wiring of the rotors and to develop cryptanalytic methods for the recovery of the daily keys. The double-indicator procedure was discontinued by the German Army in May 1940, and new methods were developed by the British at Bletchley Park, who were assisted by the knowledge transferred to them by the Polish cryptanalysts. In this article, the authors introduce two new algorithms that build on the historical cryptanalytic attacks on the two variants of the double-indicator procedures. Those attacks are based on hill climbing, divide-and-conquer, and specialized scoring functions, and they can recover the daily key using a number of indicators significantly smaller than the number of indicators required for the historical methods. Unlike the historical methods, the new algorithms produce unique and unambiguous results, including for scenarios with turnover of the middle rotor, and they also fully recover the plugboard settings. With these algorithms we won an international Enigma contest organized in 2015 by the City of Poznan, in memory of the Polish Cipher Bureau mathematicians.