This paper includes some relalions between differential cryptanalysis and group theory. The main result is the following: If the one-round functions of :UI r-round iterated, cipher generate the rdternating or the symmetric group, then for all corresponding Markov ciphers the chains of differences are irreducible and aperiodic. As an application it will be shown that if the hypothesis of stochastic equivalence holds for any of these corresponding Uvkov ciphers, then the DES and the IDEA(32) 'we secure against a differential cryptanalysis attack after sufficiently m,my rounds for these Mmkov ciphers. The section about IDEA(32) includes the result that the one-round functions of this algorithm generate the altemqting group.The theoretic foundations in group theory and Markov chains are described for instance in [Wie 641 and [Fel 581. Properties of Markov Chains and Markov CiphersLet us recall some definitions and properties of Markov chains. The definitions follow the notations of [LMM 911. In this section we will briefly review parts of this paper.A sequence of discrete random variables vo,vl ,. .., v, is a Markov chain if for 0 I i < r (where r = -is allowed):A Markov chain is called homogeneous if P ( V~+~ = PI vi = a) is independent of i for all pairs (a$).Let I 7 = ipu[ denote the transition probability matrix of a finite homogeneous Markov chain with M states and p;i the transition probabilities.
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