# On the multiplicative complexity of some Boolean functions

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## Abstract

In this paper, we study the multiplicative complexity of Boolean functions. The multiplicative complexity of a Boolean function *f* is the smallest number of &-gates in circuits in the basis {*x* & *y, x* ⊕ *y*, 1} such that each such circuit computes the function *f*. We consider Boolean functions which are represented in the form *x* _{1}, *x* _{2}⋯*x* _{ n } ⊕ *q*(*x* _{1}, ⋯, *x* _{ n }), where the degree of the function *q*(*x* _{1}, ⋯, *x* _{ n }) is 2. We prove that the multiplicative complexity of each such function is equal to (*n* − 1). We also prove that the multiplicative complexity of Boolean functions which are represented in the form *x* _{1} ⋯ *x* _{ n } ⊕ *r*(*x* _{1}, ⋯, *x* _{ n }), where *r*(*x* _{1}, ⋯, *x* _{ n }) is a multi-affine function, is, in some cases, equal to (*n* − 1).

## Keywords

Boolean function circuit complexity multiplicative complexity upper bound## Preview

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