Computing all monomials of degree n-1 using 2n-3 AND gates

06/06/2023

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by   Thomas Häner, et al.

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We consider the vector-valued Boolean function f:{0,1}^n→{0,1}^n that outputs all n monomials of degree n-1, i.e., f_i(x)=⋀_j≠ ix_j, for n≥ 3. Boyar and Find have shown that the multiplicative complexity of this function is between 2n-3 and 3n-6. Determining its exact value has been an open problem that we address in this paper. We present an AND-optimal implementation of f over the gate set {AND,XOR,NOT}, thus establishing that the multiplicative complexity of f is exactly 2n-3.