Research On A Class Of Generalized Bent Functions And Its Application In Error Correction Code

Bent functions play a very important role in designing sequence ciphers,block ciphers,and hash functions.In recent years,the generalized bent functions which inherit many features of the bent functions have been used in many fields such as multiple access communication,cryptography and combined design.Bent functions and the generalized bent functions have been extensively studied.In view of this,this paper studies the generalized bent function and obtains the following results:?1?After studying the construction of the generalized bent function,we have analyzed the necessary and sufficient conditions for n-ary generalized Boolean function which can belong to the generalized bent function.On this basis,a?8,n?-GBF construction method which based on Maiorana-McFarland constructive method is proposed.So,lots of generalized bent functions can be constructed by this method.On the other hand,the construction of?4,n?-Generalized Bent Functions has been studied.Masses of new?4,n?-Generalized Bent Functions are constructed by using the different constructive methods for the bent functions,such as direct sum construction method,non-direct sum construction method,psclass and so on.?2?We have investigated the constructions of permutation polynomials of the shape G?X????yTr?H?X??overIF_2n.A mapping function which transforms a Boolean function on n variables to a univariate function overIF_2n has been provided.On basis of the mapping function,we put forward two methods for constructing two classes of univariate functions overIF_2n.Further,two classes of permutation polynomials of the shape G?X????yTr?H?X??which have linear structure can be obtained using the two classes of univariate functions.