Research On Properties And Constructions Of Plateaued Functions

Plateaued functions include bent functions and partially bent functions as their subset. They can be highly nonlinear and satisfy correlation immune and balanceness. Also, they can have not nonzero linear structures. This dissertation discusses the cryptographic properties and constructions of Plateaued functions. The main results and creations are as the following:1. Proof the results of linear dimension and the order of Plateaued functions when it has linear structure; the relation between Plateaued functions and partially-Bent functions or and Bent functions are studied. Points out that each partially-Bent function can be divided into two Plateaued functions.2. The propagation of Plateaued functions is studied. A definition of second autocorrelation is proposed. The character of second autocorrelation is explored. The upper bound of the rank and its element's number is obtained, which is composed by all nonzero autocorrelation points .And the lower bound of the dimension of its linear subspace is also achieved.3. The properties and structure of spectrum support of Plateaued function are explored. Proposes the concept of Walsh spectrum index and analyses its properties. The autocorrelation character of low order Plateaued functions is given; pointed out the relation between spectrum support of Plateaued function and the autocorrelation coefficient.4. The cryptographic properties of multioutput Plateaued functions are studied. The difference characterization of the multi-output Plateaued functions is given with the characteristic function of multi-output functions and its Walsh spectrum. The links among different parameter of the nonbalanced multi-output Plateaued functions are explored. These results provide the guidance on designing S-box in block cipher.5. The constructions of Plateaued functions are discussed. The way of constructing new Plateaued functions by Bent functions and known Plateaued functions are put forward. And also present the recursive construction way on order, dimension and algebraic degree of Plateaued function. Furthermore, the idea of constructing Plateaued functions by resilient functions is first present. It provides a method of constructing Plateaued functions which satisfying correlation immunity, non-existence of non-zero linear structures and balancedness, which is based on the extended Maiorana- McFarland construction. Later, a construction method of multi-output Plateaued functions on finite field is given. 6. The links between Plateaued functions and the problem of studying the cross-correlation between a binary maximum-long sequence and its decimation sequence are established. Proof some Boolean functions is plateaued function on finite field if and only if the associate sequence has a three valued cross-correlation function.7. The definition of Plateaued on finite field GF(q) is proposed. And its properties and constructions are studied.