On the value distributions of Walsh spectrums of quadratic Plateaued functions

Plateaued functions have many cryptographically desirable properties, and have been used in cryptography and coding theory. However the properties of Plateaued functions have not been completely determined yet. Since many cryptographic properties can be estimated or evaluated by the value distribution of Walsh spectrum, it is essential to determine the value distribution of Walsh spectrum of a given Plateaued function.Based on the properties of trace functions and quadratic forms, this paper investigates the value distributions of Walsh spectrums of quadratic Plateaued functions of the form Tr(R(x)) with n variables. Firstly, we give all possible value distributions of Walsh spectrums of the functions. Furthermore, we proceed to determine the value distributions of Walsh spectrums of the functions on condition that the coefficients of R(x) belong to some given sets. Our results can be used to estimate the nonlinearities of these functions and their resiliency orders.

All possible value distributions of Walsh spectrums of the functions h(x)’s are derived. And then, the exact value distributions of Walsh spectrums of four subclasses of these functions are determined, among them the subclass of functions in Section 4.1 contain three types of functions.Figure optionsDownload as PowerPoint slideHighlights
► We give all possible value distributions of Walsh spectrums of Plateaued functions.
► We obtain some functions with determined value distributions of Walsh spectrums.
► These functions consist of one or two trace terms.
► We can give their nonlinearities and numbers of non-zero Walsh spectrum values.
► We can also estimate their resiliency orders.