Formalization and Systematization of Differential Cryptanalysis Notions

Although differential cryptanalysis is widely employed for iterative block ciphers security analysis, it´s basic notions are not properly formalized and relations between them are not systematized (especially related to truncated differentials). We present a formalization of these notions using special binary masks, and use it for their systematization, which allows to show strictly that truncated differential characteristic is the most general notion and non-truncated characteristics, non-truncated differentials and truncated differentials are in fact truncated characteristics under certain conditions. We also formalize the procedure of truncated characteristics concatenation and show that for a Markov cipher the probability of the resultant characteristic is estimated as their product (as it was shown by Lai and Massey for non-truncated characteristics).