Multiplicative complexity (MC) is defined as the minimum number of AND gates required to implement a function with a circuit over the basis (AND, XOR, NOT). Boolean functions with MC 1 and 2 have been characterized in Fisher and Peralta (2002), and Find et al. (IJICoT 4(4), 222–236, 2017), respectively. In this work, we identify the affine equivalence classes for functions with MC 3 and 4. In order to achieve this, we utilize the notion of the dimension dim(f) of a Boolean function in relation to its linearity dimension, and provide a new lower bound suggesting that the multiplicative complexity of f is at least ⌈dim(f)/2⌉. For MC 3, this implies that there are no equivalence classes other than those 24 identified in Çalık et al. (2018). Using the techniques from Çalık et al. and the new relation between the dimension and MC, we identify all 1277 equivalence classes having MC 4. We also provide a closed formula for the number of n-variable functions with MC 3 and 4. These results allow us to construct AND-optimal circuits for Boolean functions that have MC 4 or less, independent of the number of variables they are defined on.

中文翻译：

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具有乘法复杂性3和4的布尔函数

乘法复杂度（MC）定义为在基础上通过电路实现功能所需的最小AND门（AND，XOR，NOT）。Fisher和Peralta（2002）和Find等人（2002年）对具有MC 1和2的布尔函数进行了描述。（IJICoT 4（4），222–236，2017）。在这项工作中，我们为带有MC 3和4的函数确定仿射等价类。为了实现这一点，我们利用布尔函数的维度d i m（f）与其线性维度有关的概念，并提供一个新的下界表明的乘法复杂˚F是至少⌈ ð我中号（˚F）/2⌉。对于MC 3，这意味着除Çalık等人中确定的24类之外，没有其他等效类。（2018）。使用Çalık等人的技术。以及维度与MC之间的新关系，我们确定了所有1277个等价类均具有MC4。我们还为MC 3和4的n变量函数的数量提供了一个封闭式。这些结果使我们能够构建AND最优电路对于MC等于或小于4的布尔函数，与定义它们的变量数量无关。