Various authors have previously presented different approaches how to exploit multiple linear approximations to enhance linear cryptanalysis. In this paper we present a new truly multidimensional approach to generalise Matsui's Algorithm 1. We derive the statistical framework for it and show how to calculate multidimensional probability distributions based on correlations of one-dimensional linear approximations. The main advantage is that the assumption about statistical independence of linear approximations can be removed. Then we apply these new techniques to four rounds of the block cipher Serpent and show that the multidimensional approach is more effective in recovering key bits correctly than the previous methods that use a multiple of one-dimensional linear approximations.
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