Concrete Multiplicative Complexity of Symmetric Functions

The multiplicative complexity of a Boolean function f is defined as the minimum number of binary conjunction (AND) gates required to construct a circuit representing f, when only exclusive-or, conjunction and negation gates may be used. This article explores in detail the multiplicative complexity of symmetric Boolean functions. New techniques that allow such exploration are introduced. They are powerful enough to give exact multiplicative complexities for several classes of symmetric functions. In particular, the multiplicative complexity of computing the Hamming weight of n bits is shown to be exactly n − H^ℕ(n), where H^ℕ(n) is the Hamming weight of the binary representation of n. We also show a close relationship between the complexity of symmetric functions and fractals derived from the parity of binomial coefficients.