Distinguishing Error of Nonlinear Invariant Attacks

Linear cryptanalysis considers correlations between linear input and output combiners for block ciphers and stream ciphers. Daeman and Rijmen (2007) had obtained the distributions of the correlations between linear input and output combiners of uniform random functions and uniform random permutations. Our first contribution is to generalise these results to obtain the distributions of the correlations between arbitrary input and output combiners of uniform random functions and uniform random permutations. Recently, Todo et al. (2018) have proposed nonlinear invariant attacks which consider correlations between nonlinear input and output combiners for a key-alternating block cipher. In its basic form, a nonlinear invariant attack is a distinguishing attack. The second and the main contribution of this paper is to obtain precise expressions for the errors of nonlinear invariant attacks in distinguishing a key-alternating cipher from either a uniform random function or a uniform random permutation