Generalized partially bent functions

TL;DR

This paper explores the properties of generalized partially bent functions over ring Z N, establishing their relationship with generalized bent and affine functions, and demonstrating a decomposition when N is prime, thus extending prior work in the field.

Contribution

It introduces a new relationship among generalized partially bent, bent, and affine functions over Z N, and proves a decomposition theorem for prime N, generalizing earlier results.

Findings

Generalized partially bent functions can be decomposed into a generalized bent function and an affine function when N is prime.

The work extends the theory of partially bent functions to a broader class over ring Z N.

The relationship among different classes of functions over Z N is clarified using linear transformation theory.

Abstract

Based on the definition of generalized partially bent functions, using the theory of linear transformation, the relationship among generalized partially bent functions over ring Z N, generalized bent functions over ring Z N and affine functions is discussed. When N is a prime number, it is proved that a generalized partially bent function can be decomposed as the addition of a generalized bent function and an affine function. The result obtained here generalizes the main works concerning partially bent functions by Claud Carlet in [1].