Constructions Of Semibent-negabent Functions With Odd Variables

Boolean functions are widely used in cryptography and error-correcting codes.They should satisfy several properties when used in the cryptosystems.One of the most important requirements for a Boolean function is nonlinearity.A function in even number of variables that has the maximum nonlinearity is called bent function,and its spectrum with respect to the Walsh-Hadamard transform is flat.Some scholars extend-ed the concept of bent functions,and introduced the concepts of semibent functions and negabent functions.Semibent functions exist for odd variables and have high non-linearity and balancedness.While negabent functions exists for any variables and have flat nega-Hadamard spectrums.In recent years,some constructions of bent-negabent functions have been proposed.Although a bent-negabent function has two kinds of flat spectrum,it is not balanced and only exists for even number of variables.To make up for this deficiency,in this paper,we will construct functions which are simultane-ously semibent and negabent,i.e.which have balancedness,high nonlinearity and flat nega-Hadamard spectrums.In this paper,semibent-negabent functions in any odd number of variables and any algebraic degrees are firstly constructed.According to the relationship between quadratic functions and matrixes over the binary finite field,we construct quadratic semibent-negabent functions in any odd number n>3 of variables.Further,by s-plitting negabent functions in n + 1 variables,we derive some important results of the nega-Hadamard transform between a negabent function and its subfunctions.Final-ly,two constructions of semibent-negabent functions are presented,and by using the constructions of bent-negabent functions with optimal algebraic degree,we construct semibent-negabent functions in any odd number n>3 of variables with suboptimal algebraic degree.