Original scientific paper
https://doi.org/10.21857/y26kec4k89
On weak rotors, Latin squares, linear algebraic representations, invariant differentials and cryptanalysis of Enigma
Nicolas T. Courtois
; University College London, Gower Street, London, UK
Marek Grajek
; Independent expert on crypto history, Grodzisk Mazowiecki, Poland
Michal Rams
; Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
Abstract
Since the 1920s until today it was assumed that rotors in Enigma cipher machines do not have a particular weakness or structure. A curious situation compared to hundreds of papers about S-boxes and weak setup in block ciphers. In this paper we reflect on what is normal and what is not normal for a cipher machine rotor, with a reference point being a truly random permutation. Our research shows that most original wartime Enigma rotors ever made are not at all random permutations and conceal strong differential properties invariant by rotor rotation. We also exhibit linear/algebraic properties pertaining to the ring of integers modulo 26. Some rotors are imitating a certain construction of a perfect quasigroup which however only works when N is odd. Most other rotors are simply trying to approximate the ideal situation. To the best of our knowledge these facts are new and were not studied before 2020.
Keywords
Enigma; block ciphers; linear cryptanalysis; differential cryptanalysis; weak keys; Latin squares; algebraic representation; quasigroups; Turing-Welchman attack
Hrčak ID:
261467
URI
Publication date:
25.8.2021.
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