On the multiplicative complexity of quasi-quadratic Boolean functionsстатья
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Дата последнего поиска статьи во внешних источниках: 9 июня 2015 г.
Аннотация:The multiplicative complexity μ(f) of Boolean function f(x 1, …, x n ) is the smallest number of AND gates in the circuits of basis {&, ⊕, 1} such that each circuit executes function f. Boolean function f(x 1, …, x n ) is quasi-quadratic if it can be presented as φ(x 1, …, x k ) ⊕ q(x 1, …, x n ), where φ is an arbitrary function, q is a quadratic function (i.e., a function of degree), k ≤ n. This work is concerned with the multiplicative complexity of quasi-quadratic functions with k = 3 and arbitrary n. We prove that if f(x 1, …, x n ) is a quasi-quadratic Boolean function where k = 3, n ≥ k, then μ(f) ≤ ⌈(n + 1)/2⌉, where ⌈a⌉ denotes the smallest integer not less than a. In addition, we describe the family of quasi-quadratic Boolean functions f n (x 1, …, x n ), k = 3, n = 5, 6, …, for which we prove that μ(f n ) = ⌈(n + 1)/2⌉.