Homogeneous Bent Boolean Functions

• Homogeneous bent Boolean functions are maximally nonlinear Boolean functions defined on even-variable spaces with all monomials in the algebraic normal form having the same degree.
• Quadratic cases are fully classified, with constructions leveraging nonsingular skew-symmetric matrices and trace forms to achieve optimal Hamming distances from affine functions.
• Cubic and higher-degree functions involve complex recursive constructions, sparse enumeration, and challenging algorithmic searches to maintain both bentness and homogeneity.

A homogeneous bent Boolean function is a maximally nonlinear Boolean function defined on an even number of variables whose algebraic normal form (ANF) consists entirely of monomials of the same degree. Bentness ensures optimal cryptographic properties such as maximal Hamming distance from affine functions, and homogeneity provides algebraic regularity. These functions are rare and structurally constrained, with quadratic cases being fully classified and only a handful of explicit constructions known beyond degree 2.

1. Formal Definitions and Foundational Properties

Let $F_2^n$ denote the $n$-dimensional vector space over $F_2$. A Boolean function is $f: F_2^n \to F_2$, uniquely represented by its algebraic normal form:

$$f(x_1,\dots,x_n) = \bigoplus_{a\in F_2^n} h(a)\, x^a,\quad x^a = \prod_{i: a_i=1} x_i$$

where $h(a)\in F_2$ and the sum is over bit vectors $a$.

The nonlinearity of $f$ is defined as:

$$nl(f) = 2^{n-1} - \frac{1}{2}\max_{u\in F_2^n}|W_f(u)|$$

where the Walsh–Hadamard transform is

$$W_f(u) = \sum_{x\in F_2^n} (-1)^{f(x)\oplus u\cdot x}$$

with $u\cdot x = \bigoplus_{i=1}^n u_i x_i$.

A function is bent if $n$ is even and $nl(f)$ attains the upper bound $2^{n-1} - 2^{n/2 - 1}$. The algebraic degree, $\deg(f)$, is the maximal weight $|a|$ such that $h(a)=1$. Homogeneity of degree $d$ demands all $h(a)\neq 0$ entries correspond to $|a|=d$.

A homogeneous bent function of degree $d$ in $n$ variables satisfies:

1. $f$ is bent ($nl(f) = 2^{n-1} - 2^{n/2-1}$)
2. All monomials in the ANF have degree $d$.

Bentness implies $n$ is even, $2 \leq d \leq n/2$, and affine (linear or constant) homogeneous functions cannot be bent for $n > 2$ (Carlet et al., 16 Nov 2025, Carlet et al., 30 Jan 2025).

2. Existence and Classification

Quadratic Case ($d=2$)

For any even $n=2k$, homogeneous bent functions of degree 2 exist. Their number is

$$|HB_{n,2}| = 2^{k^2 - k}\prod_{i=0}^{k-1}(2^{2i+1} - 1)$$

This set corresponds to nonsingular alternating bilinear forms and has been fully classified. For example:

• $n=6$: $|HB_{6,2}| = 13,\!888$
• $n=8$: $|HB_{8,2}| \approx 1.382\times 10^8$ (Carlet et al., 16 Nov 2025, Tang et al., 2013)

Explicit construction relies on trace forms over field extensions: For $n=me$, $m$ even, $$f(x)=\sum_{i=1}^{m/2-1} \operatorname{Tr}_1^n(c_i x^{1+2^{ei}}) + \operatorname{Tr}_1^{n/2}(c_{m/2} x^{1+2^{n/2}})$$ with $c_i\in GF(2^e)$. Bentness reduces to checking the coprimality of a certain polynomial $C_f(x)$ over $GF(2^e)$ with $x^n+1$ (Tang et al., 2013).

Cubic and Higher Degree $(d\geq 3)$

The cubic case is much rarer:

• $n=6$: exactly 30 cubic homogeneous bent functions (all with $k=16$ terms).
• $n=8$: 293\,760 such functions exist, but only at specific term counts.

No full enumeration is known for $n\geq 10$; only isolated $k$ occur (Carlet et al., 16 Nov 2025):

• $n=10$: $k\in\{39,49,53,57,58,61,65,66,69,70,72,75,78\}$
• $n=12$: $k\in\{60,90,100,110,130,140,150\}$

There is no known construction of homogeneous bent functions of degree $d\geq 4$; their existence remains open (Carlet et al., 16 Nov 2025, Carlet et al., 30 Jan 2025).

For rotation symmetric homogeneous bent functions of degree $d>2$ (under orbit actions), nonexistence has been shown in the case $n=2p$ ($p$ prime), with quadratic short-cycle functions being essentially unique (Cusick et al., 2017).

3. Density, Enumeration, and Term Structure

Global and $k$-Term Densities

For any fixed $d, n$,

$\delta_{n,d} = \frac{|HB_{n,d}|}{2^{\binom{n}{d}}}$

where $2^{\binom{n}{d}}$ is the total number of homogeneous Boolean functions of degree $d$.

Restricting to exactly $k$ terms,

$$\delta_{n,d,k} = \frac{|HB_{n,d,k}|}{\binom{\binom{n}{d}}{k}}$$

Key Density Results

• Quadratic density approaches $\lim_{n\to\infty} \delta_{n,2} \approx 0.419422$ (the $q$-Pochhammer symbol).
• For cubic $d=3$, densities rapidly decay: at $n=8$, $\delta_{8,3}\approx 4.08 \times 10^{-12}$, and nonzero values only occur for limited $k$ (Carlet et al., 16 Nov 2025).

Example Table: (Density of homogeneous cubic bent functions for $n=8$)

$k$	$|HB_{8,3,k}|$	$\delta_{8,3,k}$
24	6,720	$1.54\times 10^{-12}$
35	19,200	$1.43\times 10^{-11}$
41	40,320	$2.48\times 10^{-9}$

Enumeration for quadratic homogeneous bent functions is closed-form; cubic enumeration relies on computational group-action techniques and exhaustive search (Carlet et al., 16 Nov 2025).

4. Algebraic and Combinatorial Structure

Homogeneous quadratic bent functions correspond to nonsingular alternating bilinear forms or equivalently, to skew-symmetric matrices over $F_2$. For function $f(x) = x^T A x$, bentness holds if and only if $A$ is nonsingular and skew-symmetric.

For rotation symmetric ("RotS") homogeneous functions, only the degree-2 "short-cycle" $f_0(x) = \bigoplus_{i=0}^{m-1} x_i x_{i+m}$ is bent for $n=2m$. Homogeneous RotS bent functions of higher degree are structurally forbidden for $n=2p$ by gap and nonlinearity arguments (Cusick et al., 2017).

Cubic homogeneous bent functions are structurally more complex but can, in certain cases, be constructed recursively via concatenation techniques that preserve bentness and homogeneity, notably using the dual-bent condition in concatenation frameworks (Polujan et al., 2023). This involves tailored combinations of quadratic and cubic seed functions with careful algebraic manipulation to ensure the resultant ANF remains homogeneous.

5. Construction Techniques and Algorithmic Design

Two principal approaches are used:

• Algebraic constructions: Using field and trace methods for quadratic functions (Tang et al., 2013). For cubic (and beyond), constructions rely on explicit search or concatenation frameworks leveraging the dual-bent condition and specific permutation properties (such as the $(\mathcal{A}_m)$ property), with recursive methods recently providing infinite families of homogeneous cubic bent functions outside the classical Maiorana–McFarland class (Polujan et al., 2023).
• Evolutionary Algorithms (EAs): Metaheuristic search for homogeneous bent functions has been explored, with several genotype encodings:
  • GP (tree-based symbolic): Operates on syntax trees, masking to maintain homogeneity at evaluation.
  • TT (truth-table): Repairs after mutation to enforce homogeneity.
  • rANF: Bitstring of length $\binom{n}{d}$, inherently exactly homogeneous.
  • wANF: Like rANF, but with constant Hamming weight $k$.

Fitness measures reward maximal nonlinearity and penalize deviation from homogeneity. For quadratic cases, rANF and wANF encodings recover all known bent solutions up to $n=12$ (Carlet et al., 16 Nov 2025). Only restricted encodings succeed; unrestricted methods and evolutionary approaches uniformly fail for cubic bent functions, indicating a challenging fitness landscape and search space (Carlet et al., 16 Nov 2025, Carlet et al., 30 Jan 2025).

6. Open Problems and Research Directions

Major open challenges include:

• Classify and construct homogeneous bent functions for degree $d\geq 4$.
• Fully enumerate cubic homogeneous bent functions in higher dimensions and characterize allowable term counts.
• Understand the algebraic and geometric obstructions to finding cubic or higher-degree bent functions via algorithmic search.
• Extend gap and nonlinearity-based impossibility arguments for rotation symmetric homogeneous functions beyond $n=2p$.

Recent advances involve recursive concatenation methods for constructing cubic homogeneous bent functions, leveraging permutation properties and the dual-bent condition, with many new objects provably distinct from Maiorana–McFarland–type functions (Polujan et al., 2023).

7. Illustrative Examples and Applications

• In $n=6$ variables, one cubic homogeneous bent function (out of 30) is, up to equivalence,

$$f(x) = x_1 x_2 x_3 \oplus x_2 x_3 x_4 \oplus x_1 x_4 x_5 \oplus \cdots$$

with $16$ terms, Walsh spectrum $|W_f(u)|=8$, and nonlinearity $28$ (Carlet et al., 16 Nov 2025).

• A quadratic homogeneous bent form for $n=6$ via MacWilliams–Sloane construction corresponds to a nonsingular $6 \times 6$ skew-symmetric binary matrix (Carlet et al., 16 Nov 2025, Tang et al., 2013).
• In recursive concatenation, starting from a cubic bent $f_1\in \mathcal{B}_N$, selecting suitable homogeneous quadratics $q_2, q_3$ and linear $s$ yields a homogeneous cubic bent function in $N+2$ variables via $$F(z,z_{N+1},z_{N+2}) = f_1(z) + z_{N+1} q_3(z) + z_{N+2} q_2(z) + z_{N+1}z_{N+2} s(z)$$ (Polujan et al., 2023).

Homogeneous bent Boolean functions are vital in cryptography due to their high nonlinearity and resistance to linear and differential cryptanalysis, though their rarity and structural rigidity significantly limit their practical availability beyond the quadratic case. The continued search for higher-degree examples remains a central research focus.