On latin squares, invariant differentials, random permutations and historical Enigma rotors

Abstract In this article we study the quality of permutations in historical cipher machines from Germany, Spain, Italy, Norway, Switzerland, Japan, Hungary, Croatia, Poland, Czechoslovakia, Russia and the United States. We show that numerous real-life rotors have been made in order to imitate or tend to a certain ideal property related to latin squares. Rotors from the same source and the same period have consistent properties deeply rooted in classical cryptography of polyalphabetical ciphers. We demonstrate this based on probabilities: random occurrence of permutations having such features is unlikely, or would amount to winning in a lottery several times in row. We put all this in the context of known historical sources on how cipher machines and cryptanalysis have developed on both German and Allied sides. We also exhibit strong linear and differential properties. The same occurs in Fialka cipher machines. Finally, a stronger property holds for the historical block cipher T-310.