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An interesting thread in the research of Boolean functions for cryptographyand coding theory is the study of secondary constructions: given a knownfunction with a good cryptographic profile, the aim is to extend it to a(usually larger) function possessing analogous properties. In this work, wecontinue the investigation of a secondary construction based on cellularautomata, focusing on the classes of bent and semi-bent functions. We provethat our construction preserves the algebraic degree of the local rule, and wenarrow our attention to the subclass of quadratic functions, performing severalexperiments based on exhaustive combinatorial search and heuristic optimizationthrough Evolutionary Strategies (ES). Finally, we classify the obtained resultsup to permutation equivalence, remarking that the number of equivalence classesthat our CA-XOR construction can successfully extend grows very quickly withrespect to the CA diameter.

Heuristic Search of (Semi-)Bent Functions based on Cellular Automata