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 $k$-bit permutations recommended for use in S-boxes of the algorithms GOST 28147-89, DES, and $s^3\mathrm{DES}$; 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.