9-variable Boolean Functions with Nonlinearity 242 in the Generalized Rotation Class

In 2006, 9-variable Boolean functions having nonlinearity 241, which is strictly greater than the bent concatenation bound of 240, have been discovered in the class of Rotation Symmetric Boolean Functions (RSBFs) by Kavut, Maitra and Yücel. To improve this nonlinearity result, we have firstly defined some subsets of the n-variable Boolean functions as the “generalized classes of k-RSBFs and k-DSBFs (k-Dihedral Symmetric Boolean Functions)”, where k is a positive integer dividing n and k-RSBFs is a subset of l-RSBFs if k < l. Secondly, utilizing the steepest-descent like iterative heuristic search algorithm used previously to identify the 9-variable RSBFs with nonlinearity 241, we have made a search within the classes of 3-RSBFs and 3-DSBFs. The search has accomplished to find 9-variable Boolean functions with nonlinearity 242 in both of these classes. It should be emphasized that although the class of 3RSBFs contains functions with nonlinearity 242; 1-RSBFs or simply RSBFs, which is a subset of 3-RSBFs, does not contain any. This result also shows that the covering radius of the first order Reed-Muller code R(1, 9) is at least equal to 242. Thirdly, motivated by the fact that RSBFs are invariant under a special permutation of the input vector, we have classified all possible permutations up to the linear equivalence of Boolean functions that are invariant under those permutations. Specifically, for 9-variable Boolean functions, 9! possible permutations are classified into 30 classes; and the search algorithm identifies some of these classes as rich. The rich classes yield new Boolean functions with nonlinearity 242 having different autocorrelation spectra from those of the functions found in the generalized 3-RSBF and 3-DSBF classes. However, there is no zero value in the Walsh spectra of these functions; hence, none of them can be linearly transformed to 9-variable balanced functions with nonlinearity 242.