A Computational Search for Cubic-Like Bent Functions

Boolean functions are a central topic in computer science. A subset of Boolean functions, Bent Boolean functions, provide optimal resistance to various cryptographical attack vectors, making them an interesting subject for cryptography, as well as many other branches of mathematics and computer science. In this work, we search for cubic Bent Boolean functions using a novel characterization presented by Carlet & Villa in [CV23]. We implement a tool for the search of Bent Boolean functions and cubic-like Bent Boolean functions, allowing for constraints to be set on the form of the ANF of Boolean functions generated by the tool; reducing the search space required for an exhaustive search. The tool guarantees efficient traversal of the search space without redundancies. We use this tool to perform an exhaustive search for cubic-like Bent Boolean functions in dimension 6. This search proves unfeasible for dimension 8 and higher. We further attempt to find novel instances of Bent functions that are not Maioarana-McFarland in dimension 10 but fail to find any interesting results. We conclude that the proposed characterization does not yield a significant enough reduction of the search space to make the classification of cubic Bent Boolean functions of dimensions 8 or higher viable; nor could we use it to produce new instances of cubic Bent Boolean functions in 10 variables.Masteroppgave i informatikkINF399MAMN-PROGMAMN-IN

Construction of cubic homogeneous boolean bent functions

We prove that cubic homogeneous bent functions f : V2n → GF(2) exist for all n ≥ 3 except for n = 4

On bent and hyper-bent functions

Bent functions are Boolean functions which have maximum possible nonlinearity i.e. maximal distance to the set of affine functions. They were introduced by Rothaus in 1976. In the last two decades, they have been studied widely due to their interesting combinatorial properties and their applications in cryptography. However the complete classification of bent functions has not been achieved yet. In 2001 Youssef and Gong introduced a subclass of bent functions which they called hyper-bent functions. The construction of hyper-bent functions is generally more difficult than the construction of bent functions. In this thesis we give a survey of recent constructions of infinite classes of bent and hyper-bent functions where the classification is obtained through the use of Kloosterman and cubic sums and Dickson polynomials

Algebraic normal form of a bent function: properties and restrictions

Maximally nonlinear Boolean functions in $n$ variables, where n is even, are called bent functions. There are several ways to represent Boolean functions. One of the most useful is via algebraic normal form (ANF). What can we say about ANF of a bent function? We try to collect all known and new facts related to ANF of a bent function. A new problem in bent functions is stated and studied: is it true that a linear, quadratic, cubic, etc. part of ANF of a bent function can be arbitrary? The case of linear part is well studied before. In this paper we prove that a quadratic part of a bent function can be arbitrary too

On the Systematic Constructions of Rotation Symmetric Bent Functions with Any Possible Algebraic Degrees

In the literature, few constructions of $n$-variable rotation symmetric bent functions have been presented, which either have restriction on $n$ or have algebraic degree no more than $4$. In this paper, for any even integer $n=2m\ge2$, a first systemic construction of $n$-variable rotation symmetric bent functions, with any possible algebraic degrees ranging from $2$ to $m$, is proposed