The Study Of Walsh Spectrum And Nonlinearity Of Boolean Functions

Boolean functions play important roles in the design of cryptography algorithm,and rotation symmetric Boolean functions have attracted widely attention from many researchers because of their simple representations and good cryptographic properties.The nonlinearity of Boolean functions is a significant criterion for measuring the safety-performance of cryptosystem and how effective to resist linear attack.As a powerful tool of studying cryptographic properties of Boolean functions,Walsh spectrum has the direct relation to many cryptographic properties,such as Hamming weight,nonlinearity and so on.In this paper,we study the Walsh spectrum and nonlinearity of Boolean functions under certain conditions by using the equivalence classifications of Boolean functions.Firstly,according to the property of the affine invariant of Walsh spectrum,the absolute value distribution of Walsh spectrums of 6-variable Boolean functions has been studied,and the relevant conclusions of their Walsh spectrum support and the number of non-zero values have been drawn.Moreover,based on the Walsh spectrum transform and the decomposition of vector space IF2n,we have proved that the Hamming weight of a special class of rotational symmetry Boolean functions with arbitrary orders is the same as their nonlinearity.Finally,the covering radius of Reed-Muller code RM(3,7)has been studied.We design a recursive algorithm of calculating the r-order nonlinearity of Boolean function.Applying the algorithm and the heuristic search strategy,we draw conclusions that the covering radii of Reed-Muller code RM(3,7)both in rotation symmetric Boolean functions and on the subset RM(5,7)are 20.