The evolutionary design of trace form bent functions in cryptography

Based on the trace representation of Boolean functions, we devise an evolutionary algorithm to design bent functions. Using this algorithm, we then construct many bent functions and perform some analyses. First, we observe that each of the four affinely inequivalent bent functions in six variables can be written as the linear sum of two or three monomial trace functions. We draw the conclusion that the affine transformation can be used to change the linear span of the Boolean functions and thereby change the trace representation of our obtained bent functions. Second, we find that certain exponents are more suitable for constructing bent functions than others. From this observation, we assign each exponent a cost function, which makes our algorithm more efficient than an exhaustive search algorithm or a random algorithm. Third, we classify the obtained bent functions into affinely inequivalent classes, and the number of classes is presented.