In this paper we study the lower bounds of second-order nonlinearities of bent functions in the class D_0 constructed by modifying certain cubicMaiorana-McFarland (MMF) type bent functions. We also obtain improvements on existing results on second-order nonlinearities of some cubic MMF type bent functions having no affine derivative.
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The paper discusses the problem of recognizing the Boolean function linearity. A spectral method of the analysis of Boolean functions using the Walsh transform is described. Linearity and nonlinearity play important roles in the design of digital circuits. The analysis of the distribution of spectral coefficients allows us to determine various combinatorial properties of Boolean functions, such as redundancy, monotonicity, self-duality, correcting capability, etc., which seems more difficult be performed by means of other methods. In particular, the basic synthesis method described in the paper allows us to compute the spectral coefficients in an iterative manner. The method can be easily used in investigations of large Boolean functions (of many variables), which seems very attractive for modern digital technologies. Experimental results demonstrate the efficiency of the approach.
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