Semi-bent Boolean functions are interesting from a cryptographic standpoint, since they possess several desirable properties such as having a low and flat Walsh spectrum, which is useful to resist linear cryptanalysis. In this paper, we consider the search of semi-bent functions through a construction based on cellular automata (CA). In particular, the construction defines a Boolean function by computing the XOR of all output cells in the CA. Since the resulting Boolean functions have the same algebraic degree of the CA local rule, we devise a combinatorial algorithm to enumerate all quadratic Boolean functions. We then apply this algorithm to exhaustively explore the space of quadratic rules of up to $6$ variables, selecting only those for which our CA-based construction always yields semi-bent functions of up to $20$ variables. Finally, we filter the obtained rules with respect to their balancedness, and remark that the semi-bent functions generated through our construction by the remaining rules have a constant number of linear structures.
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