Modeling Linear Characteristics of Substitution-Permutation Networks

In this paper we present a model for the bias values associated with linear characteristics of substitution-permutation networks (SPN's). The rst iteration of the model is based on our observation that for suuciently large s-boxes, the best linear characteristic usually involves one active s-box per round. We obtain a result which allows us to compute an upper bound on the probability that linear cryptanalysis using such a characteristic is feasible, as a function of the number of rounds. We then generalize this result, upper bounding the probability that linear cryptanalysis is feasible when any linear characteristic may be used (no restriction on the number of active s-boxes). The work of this paper indicates that the basic SPN structure provides good security against linear cryptanalysis based on linear characteristics after a reasonably small number of rounds.