Fast Computation of Large Distributions and Its Cryptographic Applications

LetX1,X2, . . . ,Xk be independent n bit random variables. If they have arbitrary distributions, we show how to compute distributions like Pr{X1 ⊕X2 ⊕ · · · ⊕Xk} and Pr{X1 X2 · · · Xk} in complexity O(kn2). Furthermore, if X1,X2, . . . , Xk are uniformly distributed we demonstrate a large class of functions F (X1, X2, . . . ,Xk), for which we can compute their distributions efficiently. These results have applications in linear cryptanalysis of stream ciphers as well as block ciphers. A typical example is the approximation obtained when additions modulo 2 are replaced by bitwise addition. The efficiency of such an approach is given by the bias of a distribution of the above kind. As an example, we give a new improved distinguishing attack on the stream cipher SNOW 2.0.