The Multiplicative Complexity of Boolean Functions on Four and Five Variables

A generic way to design lightweight cryptographic primitives is to construct simple rounds using small nonlinear components such as 4x4 S-boxes and use these iteratively (e.g., PRESENT [1] and SPONGENT [2]). In order to efficiently implement the primitive, efficient implementations of its internal components are needed. Multiplicative complexity of a function is the minimum number of AND gates required to implement it by a circuit over the basis (AND, XOR, NOT). It is known that multiplicative complexity is exponential in the number of input bits n. Thus it came as a surprise that circuits for all 65 536 functions on four bits were found which used at most three AND gates [3]. In this paper, we verify this result and extend it to five-variable Boolean functions. We show that the multiplicative complexity of a Boolean function with five variables is at most four.