Study On Constructions Of Bent Functions From Rothaus Construction

A bent function is a Boolean function with optimal nonlinearity,which plays an important role in the fields of cryptography,coding,and combinatorial mathematics.So far,a complete classification of bent functions is still elusive and looks hopeless.The known bent functions only account for a small portion of all bent functions.Even it is a difficult task to prove whether the constructed bent functions are included in the completed Maiorana-McFarland(M-M)class.Thus,this thesis studies the the construction and classification of bent functions,and the main research results are as follows:(1)The “Rothaus construction” of bent functions was studied,and then a special construction of the “Rothaus construction” was given.Combining the the D class and C class bent functions,a large number of new bent functions are given by using the new given construction of bent functions.On this basis,the affine equivalence relation between the constructed functions and “the completed M-M class functions” is studied,and the sufficient conditions that the constructed bent functions do not belong to the completed M-M class are given.Compared with the conditions proposed by Zhang et al.,these sufficient conditions given in this thesis are easier to satisfy.(2)By analyzing the interrelationships among three permutations,some methods for constructing the triples of permutations and the corresponding algorithms are given.Compared to the construction method proposed by Coulter et al.,more triples of permutations can be obtained by using our methods in this thesis.Combining the M-M construction of bent functions with the triples of permutations,a large number of bent functions with optimal algebraic degree can be obtained by using “Rothaus construction”.