On derivatives of boolean bent functions

Bent function can be defined as a Boolean function f(x) in n variables (n is even) such that for any nonzero vector y its derivative Dyf(x) = f(x) ф f(x ф y) is balanced, that is, it takes values 0 and 1 equally often. Whether every balanced function is a derivative of some bent function or not is an open problem. In this paper, special case of this problem is studied. It is proven that every non-constant affine function in n variables, n > 4, n is even, is a derivative of (2^n_1 - 1)|Bn_2|² bent functions, where |Bn_2| is the number of bent functions in n - 2 variables. New iterative lower bounds for the number of bent functions are presented.

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