Linear decomposition of Boolean functions into a sum or a product of components | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2018. № 40. DOI: 10.17223/20710410/40/2

Let / : GF(2)^ra ->• GF(2) be a Boolean function, n ^ 2, and U_s be a set of Boolean functions / of degree deg f ^ s. Here is a consideration of the disjunctive decomposition of / into sum and products modulo U_s of Boolean functions after a linear substitution on arguments. The main result is the following: if all arguments of the functions f(xA) under linear substitutions A of the vector space GF(2)^ra are essential modulo U_s and / may be represented as disjunctive sum / = f\ ® ... ® f_m (mod U_s), where m is maximal, then subsequent direct sum of subspaces GF(2)^ra = V⁽-¹-⁾ + ... + V⁽-^m-⁾ is unique and invariant under stabilizer group of the function / in general linear group. The article contains analogous result describing sufficient uniqueness condition for disjunctive products f = f\... f_m (mod U_s), namely, every function fi has no affine multipliers and the set {a e Vi : fi(x фа)ф /г(ж) has affine multipliers} generates the whole subspace Vi, i = 1,..., m. For instance, this class of functions contains a nondegenerated quadratic forms.