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).
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Original Russian Text © S.N. Selezneva, 2015, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2015, Vol. 55, No. 4, pp. 730–736.
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Selezneva, S.N. On the multiplicative complexity of some Boolean functions. Comput. Math. and Math. Phys. 55, 724–730 (2015). https://doi.org/10.1134/S0965542515040119
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DOI: https://doi.org/10.1134/S0965542515040119