Constructions Of Bent And Negabent Functions

Boolean functions have widespread applications in cryptography,coding theory,design of sequences and graph theory.Those Boolean functions with flat absolute spectrum under the Walsh-Hadamard and nega-Hadamard transforms,are called bent and negabent functions,respectively.They have important applications in cryptography for constructing cryptographically strong functions and error correcting codes for better performance.This dissertation is devoted to the construction of bent and negabent functions.The main results and contributions are given as follows.(1)We investigate the bentness of the sum of some characteristic functions of affine subspaces and a Maiorana-McFarland bent function,and give two general constructions of bent functions,which cover eight known constructions proposed by Su et al.2017 IEEE Transactions on Information Theory,2020 IEEE Transactions on Information Theory,2022 Designs,Codes and Cryptography and Zhang et al.2020 Advances in Mathematics of Communications.(2)We prove that the condition of Rothaus’s construction is also necessary,which answers a question proposed by Zhang and Pasalic et al.2017 IEEE Transactions on Information Theory.We show that the construciton of bent functions by using Rothaus’construction iteratively works only if three initial functions are the same.In this case,those produced bent functions can be expressed by the direct sum of a bent function and a quadratic bent function.Hence,it cannot make a substantial contribution to the construction of bent functions.This solve an open problem proposed by Zhang and Pasalic et al.2017 IEEE Transactions on Information Theory.(3)By modifying two classes of quadratic bent-negabent functions,we propose constructions of bent-negabent functions on arbitrary even numbers of variables.We then derive some necessary and sufficient condition under which these functions have the maximum algebraic degree,and determine duals of these bent-negabent functions.By modifying one class of quadratic 2-rotation symmetric bent-negabent functions,we give a construction of 2-rotation symmetric bent-negabent functions with any possible algebraic degrees.It is the first important attempt for constructing bent-negabent functions in the generalized rotation symmetric class.(4)We consider constructions of bent-negabent functions under three frameworks of secondary constructions:indirect sum(Carlet 2004 Coding,Cryptography and Combinatorics),modified indirect sum(Hodzic et al.2020 Designs,Codes and Cryptography)and Rothaus’constructions.We first give a method to construct bent-negabent functions by using the indirect sum construction,and prove that those functions are outside the completed Maiorana-McFarland class under certain conditions.Here we correct two results on bent-negabent functions in a paper IEEE Transactions on Information Theory in 2015.We then investigate the nega-Hadamard transform of composition functions.Using this tool,we analyze and find out some necessary and sufficient conditions for the modified indirect sum and Rothaus’constructions to generate bent-negabent functions.Furthermore,we propose several specified constructions which satisfy the required conditions.(5)We present a general new construction of quadratic and cubic negabent functions over finite fields.Those functions can be represented as the sum of the three components:one is the trace function of the monomial term λx3 or it multiplying by the trace function of x;the second,the sum of the all quadratic monomial functions except for the term of exponent 3;and the third,the product of two linear functions,where the parameter λ and variable x belong to an arbitrary binary finite field of 2n elements for n odd.