Heuristic Search of (Semi-)Bent Functions based on Cellular Automata
This work addresses incremental improvements in constructing cryptographic Boolean functions for coding theory and cryptography.
The authors tackled the problem of extending bent and semi-bent Boolean functions using a cellular automata-based secondary construction, proving it preserves algebraic degree and experimentally showing that the number of extendable equivalence classes grows quickly with CA diameter.
An interesting thread in the research of Boolean functions for cryptography and coding theory is the study of secondary constructions: given a known function with a good cryptographic profile, the aim is to extend it to a (usually larger) function possessing analogous properties. In this work, we continue the investigation of a secondary construction based on cellular automata, focusing on the classes of bent and semi-bent functions. We prove that our construction preserves the algebraic degree of the local rule, and we narrow our attention to the subclass of quadratic functions, performing several experiments based on exhaustive combinatorial search and heuristic optimization through Evolutionary Strategies (ES). Finally, we classify the obtained results up to permutation equivalence, remarking that the number of equivalence classes that our CA-XOR construction can successfully extend grows very quickly with respect to the CA diameter.