Quadratic approximations of the special type for the 4-bit permutations in S-boxes

We consider quadratic approximations (for Boolean functions) of the special type and study the ability of application them in nonlinear cryptanalysis of block ciphers. For 4-bit permutations with the most high nonlinearity recommended for using in S-boxes of GOST 28147-89, DES, s$^3$DES we show that for the all of them (excepting only one) there are quadratic equalities of the special type on input and output bits with probability more high than any linear equality has.