Balanced Boolean functions

Many common logic circuits such as adders, parity checkers and multiplexers realise Boolean functions that are true for exactly half their input combinations, and false for the other half; we refer to such functions as balanced. Recently, these functions have been shown to be very useful for testing logic circuits, and for data encryption in cryptography. Here, we present a general theory of balanced Boolean functions. We derive a necessary and sufficient condition for balance by establishing an equivalence between a balanced function f(X) and a bijection from X to itself. We then analyse the conditions under which functional compositions preserve balance, and examine some specific balance-preserving decompositions. A new characterisation of functional completeness in terms of balance is presented. Finally, we address the problem of counting equivalence classes of balanced functions