The Construction Of Bent Function And Its Application

Bent functions have important applications in cryptography,coding theory,sequences,combinatorics,and graph theory.Constructing new bent functions and 1-resilient functions with higher nonlinearity and lower absolute indicator from bent functions is significant in both theory and practice.This thesis further studies the construction of bent functions and their applications under some previous results.The main fruits are listed as follows:(1)We obtain some new bent functions in bivariate form,and prove the EA-inequivalence between those bent functions and other known bent functions.Moreover,we get a new cubic bent function in 10 variables,which neither has affine derivative nor belongs to the completed (?) class,which is also different from all cubic bent functions in 8 or 6 variables since they are inside the completed (?) class.This is the first time to prove the existence of cubic functions in 10 variables which are outside the completed (?) class.(2)We derive a generic construction of bent functions in univariate form,which covers many previous constructions of bent functions,gives an unification for some different kinds of constructions of bent functions,and provides a computing method of duals for some known bent functions.Based on that,we give a characterization for the bent functions associated with indicators and a univariate trace representation of (?) bent functions.In addition,we obtain many idempotent bent functions,many bent functions in univariate trace form and many self-dual bent functions with any possible algebraic degree.(3)We get a generic construction of vectorial bent functions and obtain the duals of three famous classes of bent functions,and then obtain three classes of vectorial bent functions of optimal algebraic degree.(4)We solve an open problem on bent functions proposed by Mesnager in IEEE T.I.T.(5)We give the compositional inverse of a kind of permutation polynomials.Using that,we determine the explicit dual of monomial bent function with Leader exponent.(6)We obtain two classes of SAO 1-resilient functions with the currently lowest absolute indicator,from bent functions and plateaued functions,respectively.