Boolean functions with multiplicative complexity 3 and 4 - Çalık, Çağdaş; Turan, Meltem Sönmez; Peralta, René (Cryptography and communications, 2020)
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 d i m ( 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 ⌈ d i m ( 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.