Boolean functions with excellent cryptographic properties in autocorrelation and walsh spectra

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2008
Kavut, Selçuk
We introduce a steepest-descent-like search algorithm for the design of Boolean functions, yielding multiple desirable cryptographic properties in their Walsh and autocorrelation spectra together. The algorithm finds some Boolean functions on 9, 10, 11, 13 variables with very good cryptographic properties unattained in the literature. More specifically, we have discovered 9-variable rotation symmetric Boolean functions (RSBFs) having nonlinearity of 241, which exceeds the bent concatenation bound and has remained as an open question in the literature for almost three decades. We have then shown that there is no RSBF having nonlinearity greater than 241, and that there are 8x189 many RSBFs having nonlinearity of 241, such that, among them there are only two that are different up to the affine equivalence. We also propose a generalization to RSBFs and dihedral symmetric Boolean functions (DSBFs), which improves the nonlinearity result of 9-variable Boolean functions to 242. Further, we classify all possible permutations (362, 880) on the input variables of 9-variable Boolean functions and find that there are only 30 classes, which are different with respect to the linear equivalence of invariant Boolean functions under some permutations. Some of these classes and their subsets yield new 9-variable Boolean functions having the nonlinearity of 242 with different autocorrelation spectra from those of the Boolean functions found in generalized RSBF and DSBF classes. Moreover, we have attained 13-variable balanced Boolean functions having nonlinearity of 4036 which is greater than the bent concatenation bound of 4032, and improves the recent result of 4034.

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Citation Formats
S. Kavut, “Boolean functions with excellent cryptographic properties in autocorrelation and walsh spectra,” Ph.D. - Doctoral Program, Middle East Technical University, 2008.