Analysis on Boolean Function in a Restricted (Biased) Domain

A note on generalization of bent boolean functions

“…Thus, any symmetric function f ∈ B n have at most n + 1 distinct values of µ p -Walsh-Hadamard transform. The similar result also derived in [16].…”

“…We know that any non-constant affine function is balanced when X ∼ µ 1 2 , but it need not be so when the inputs are not uniformly distributed. Maitra et al [16] proved that a Boolean function is balanced if and only…”

“…if and only if p [16] derived the results of Walsh-Hadamard transforms of Boolean functions with input X ∼ µ p , where p = 1 2 , and proved that there exist at most n + 1 distinct values of bias Walsh-Hadamard transform of an n-variable symmetric function. The µ p -Walsh-Hadamard transform of any symmetric function f ∈ F n 2 at u ∈ F n 2 can be written as…”

“…In [16,Theorem 5] the biased Walsh-Hadamard transform of monomial Boolean functions in n variables of degree n is derived. We also derive an expression of µ p -Walsh-Hadamard transform of any monomial Boolean function in n variables with degree r using Proposition 1.…”

“…For example, Maitra et al [15,16] analyzed the bias of Walsh-Hadamard transform of the filter function of FLIP [1,2] over a restricted domain where input vectors are not independent and identically distributed. They showed in Table IV [16,Section IV.A] that the bounds provided by the biased Walsh-Hadamard transforms are better than those obtained by Carlet et al [2]. Further, in the same paper they proved that many cryptographic properties are different when we define the Boolean functions over a non-uniform domain.…”