The hypothesis of fixed-key equivalence for one-round of encryption

Matsui's (1993) linear cryptanalysis has been generalised by Harpes, Kramer and Massey (see Advances in Cryptology-Eurocrypt'95, LNCS 921, p.24-38, 1995) who make use of balanced functions (which take on each value an equal number of times) instead of linear expressions. In their attack, the "measure of effectiveness" is the average-key imbalance. This article shows that the hypothesis holds for almost all combinations of ciphers and balanced functions. Perhaps more importantly, the results give a reference for the effectiveness of (linear or more general) expressions.