[Paper Review] A general framework for secondary constructions of bent and plateaued functions

Samir Hodžić, Enes Pašalić+1 more|arXiv (Cornell University)|Sep 19, 2018

Coding theory and cryptography32 references3 citations

TL;DR

This paper introduces a general framework for secondary constructions of bent and plateaued Boolean functions using composite representation—expressing a function as a composition of a Boolean function and a vectorial function. By leveraging this structure, the authors unify and extend existing methods like indirect sums and Rothaus' construction, enabling new, efficient constructions without restrictive conditions on initial functions, and solving open problems in the field.

ABSTRACT

In this work, we employ the concept of {\em composite representation} of Boolean functions, which represents an arbitrary Boolean function as a composition of one Boolean function and one vectorial function, for the purpose of specifying new secondary constructions of bent/plateaued functions. This representation gives a better understanding of the existing secondary constructions and it also allows us to provide a general construction framework of these objects. This framework essentially gives rise to an {\em infinite number} of possibilities to specify such secondary construction methods (with some induced sufficient conditions imposed on initial functions) and in particular we solve several open problems in this context. We provide several explicit methods for specifying new classes of bent/plateaued functions and demonstrate through examples that the imposed initial conditions can be easily satisfied. Our approach is especially efficient when defining new bent/plateaued functions on larger variable spaces than initial functions. For instance, it is shown that the indirect sum methods and Rothaus' construction are just special cases of this general framework and some explicit extensions of these methods are given. In particular, similarly to the basic indirect sum method of Carlet, we show that it is possible to derive (many) secondary constructions of bent functions without any additional condition on initial functions apart from the requirement that these are bent functions. In another direction, a few construction methods that generalize the secondary constructions which do not extend the variable space of the employed initial functions are also proposed.

Motivation & Objective

• To develop a unified, general framework for secondary constructions of bent and plateaued functions.
• To address open problems in secondary constructions, particularly those involving minimal or no conditions on initial functions.
• To demonstrate that existing constructions like indirect sums and Rothaus' method are special cases of the proposed framework.
• To enable the construction of bent/plateaued functions on larger or the same variable space with flexible, easily satisfied conditions.
• To explore whether such constructions yield functions outside known primary classes.

Proposed method

• Representing a Boolean function as a composition of a Boolean function f and a vectorial function H, i.e., f(H(x)), enabling flexible manipulation of coordinate functions.
• Using the composite representation to derive sufficient conditions under which the composed function remains bent or plateaued.
• Applying Walsh-Hadamard transform (WHT) analysis to characterize the spectral behavior of the composed function.
• Employing subspace decomposition and duality (e.g., U⊥) to control the Walsh spectrum and ensure plateaued or bent properties.
• Introducing a generic method (Proposition 5.3) combining multiple plateaued functions via direct sum of their Walsh supports to generate new plateaued or bent functions.
• Utilizing the structure of affine subspaces and spectral partitioning to ensure non-zero Walsh coefficients and desired amplitude values.

Experimental results

Research questions

• RQ1 Can a general framework be established that unifies diverse secondary constructions of bent and plateaued functions?
• RQ2 Can secondary constructions be designed without imposing restrictive conditions on the initial bent or plateaued functions?
• RQ3 Are well-known constructions like indirect sums and Rothaus’ construction special cases of a broader, unified framework?
• RQ4 Can new classes of bent functions be generated that are not equivalent to known primary classes?
• RQ5 Can the framework efficiently produce bent/plateaued functions on the same or larger variable space with minimal constraints?

Key findings

• The composite representation framework generalizes and subsumes existing constructions such as indirect sums and Rothaus’ construction as special cases.
• The framework enables the construction of bent functions without additional constraints beyond the initial functions being bent, solving an open problem in secondary constructions.
• Explicit construction methods are provided where the required initial conditions (e.g., on Walsh supports and subspaces) can be easily satisfied, as demonstrated in examples.
• The method allows for efficient generation of new bent or plateaued functions on larger variable spaces, with spectral properties controlled via subspace decomposition.
• A generic method is proposed (Proposition 5.3) that combines r plateaued functions via direct sum of their Walsh supports to yield a t-plateaued function, with t=0 implying bentness.
• The framework provides a systematic way to generate infinitely many secondary constructions by varying the choice of coordinate functions and outer functions under defined sufficient conditions.