Boolean functions and their Möbius transforms are involved in logical calculation, digital communications, coding theory and modern cryptography. In this paper, three methods to compute the Möbius transform of a Boolean function are suggested: using matrices, using polynomials and using recursive formulas, and new properties of Möbius transforms are deduced according to the relations between the truth table and the algebraic normal form (ANF) of a Boolean function. A Boolean function which is identical to its Möbius transform is called coincident. The paper presents characterizations and constructions of coincident functions by matrices, by polynomials, and by recursive formulas, operations of coincident functions, a classification and enumeration of coincident functions, properties of coincident functions and coincident functions with high nonlinearity and high degree. Finally, other related classes of Boolean functions are defined: $h$-non-coincident functions and anti-coincident functions, and some properties of these functions are deduced, e.g., there are precisely $2^{2^{n-1}}$ coincident functions and the same number of anti-coincident functions of $n$ variables, and the union of these sets of functions is a $(2^{n-1}+1)$-dimensional subspace of the set of all Boolean functions of $n$ variables.
Reviewer: Ioan Tomescu (Bucureşti)