On quadratic approximations in block ciphers

We consider quadratic approximations (of Boolean functions) of a special form and their potential applications in block cipher cryptanalysis. We show that the use of k-bent functions as ciphering functions extremely increases the resistance of ciphers to such approximations. We consider examples of 4-bit permutations recommended for use in S-boxes of the algorithms GOST 28147-89, DES, and s ³DES; we show that in almost all cases there exist more probable (than linear) quadratic relations of a special form on input and output bits of these permutations.