Construction Of Bent Functions Research Articles

Characteristics of highly nonlinear functions

The highly nonlinear odd-dimensional Boolean-functions have many applications in the cryptographic practice, that is why the research of that function-classes and construction of such functions have a great importance. This study focuses on some types of functions having special characteristics in the class of highly nonlinear odd-dimensional Boolean-functions. Upper bound can be given for the number of non-zero linear structures of such functions and regarding them as mappings some functional-relations can be proved. From the results one can gain two algorithms. By the help of the first one special highly nonlinear odd dimensional Boolean-functions can be constructed by using functions having the same characteristics, the second one renders possible the construction of bent functions of a one-level higher dimension by the use of special highly nonlinear odd-dimensional Boolean-functions. The paper shows a relation between bent functions in even dimensional Boolean-space and odd dimensional highly nonlinear Boolean functions.

Maximally Nonlinear Functions and Bent Functions

We give a construction of bent functions in 2n variables, here n is odd, by using maximally nonlinear functions on \GF.

Generalized bent functions and their properties

Jet J q m denote the set of m-tuples over the integers modulo q and set i= −1 , w = e i( 2π q ) . As an extension of Rothaus' notion of a bent function, a function f, f: J q m → J q 1 is called bent if all the Fourier coefficients of w f have unit magnitude. An important feature of these functions is that their out-of-phase autocorrelation value is identically zero. The nature of the Fourier coefficients of a bent function is examined and a proof for the non-existence of bent functions over J q m , m odd, is given for many values of q of the form q = 2 (mod 4). For every possible value of q and m (other than m odd and q = 2 (mod 4)), constructions of bent functions are provided.