On the Complexity of Computing Two Nonlinearity Measures

We study the computational complexity of two Boolean non-linearity measures: the nonlinearity and the multiplicative complexity. We show that if one-way functions exist, no algorithm can compute the multiplicative complexity in time 2 O(n) given the truth table of length 2 n , in fact under the same assumption it is impossible to approximate the multiplicative complexity within a factor of (2 − ǫ) n/2. When given a circuit, the problem of determining the multiplicative complexity is in the second level of the polynomial hierarchy. For nonlinearity, we show that it is #P hard to compute given a function represented by a circuit.