Density of Homogeneous Bent Functions

• Density of homogeneous bent functions quantifies the ratio of bent functions to all homogeneous Boolean functions, with quadratic cases near 0.42 and cubic cases as low as 10⁻⁵ to 10⁻¹².
• Exact enumeration for quadratic functions (e.g., B₍6,2₎ = 13,888 and B₍8,2₎ = 3.38×10⁸) provides clear metrics that guide search algorithms by pinpointing monomial counts with high empirical density.
• Understanding these densities informs algorithmic strategies and enhances cryptographic design by ensuring functions exhibit maximal nonlinearity to resist linear and differential attacks.

A homogeneous bent function is a Boolean function on an even number of variables, whose algebraic normal form (ANF) consists exclusively of monomials of a fixed degree and which attains maximal nonlinearity. The concept of density for such functions quantifies their relative abundance among all homogeneous Boolean functions of the specified degree and number of variables. Bent functions, especially those with homogeneity constraints and high degrees, are essential in cryptography due to their extremal nonlinearity, which underpins resistance to linear and differential attacks.

1. Definitions and Fundamental Quantities

Let $f\colon\mathbb{F}_2^n\to\mathbb{F}_2$ be a Boolean function, represented in ANF as

$$f(x)=\bigoplus_{a\in\mathbb{F}_2^n} h(a)\,x_1^{a_1}\cdots x_n^{a_n},\qquad h(a)\in\{0,1\}.$$

$f$ is homogeneous of degree $d$ if all monomials $x^a$ with $h(a)=1$ have Hamming weight $|a|=d$. For $n$ even, $f$ is bent if its nonlinearity achieves the covering-radius bound:

$\mathrm{nl}(f)=2^{n-1}-2^{\frac n2-1}.$

A homogeneous bent function of degree $d$ is both bent and homogeneous of that degree.

For given $n$ and $d$, the number of $d$-degree monomials is $\binom n d$, so the total number of homogeneous Boolean functions is

$H_{n,d}=2^{\binom n d}.$

Let $B_{n,d}$ denote the number of homogeneous bent functions of degree $d$ in $n$ variables (only defined for even $n$ and $d\le n/2$). The density is then

$$\rho_{n,d}=\frac{B_{n,d}}{H_{n,d}} = \frac{B_{n,d}}{2^{\binom n d}}.$$

A refinement, $\delta_{n,d,k}$, gives the relative frequency conditioned on a fixed number of monomials $k$ (appearing in the ANF).

2. Exact and Empirical Enumeration Results

Quadratic Case ($d=2$)

For $n=2k$, all homogeneous quadratic bent functions are enumerated by the formula (MacWilliams–Sloane, Carlet):

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

Specific calculated values:

• $n=6$: $\binom{6}{2}=15$, $B_{6,2}=13,\!888$, $H_{6,2}=2^{15}$, so $\rho_{6,2} \approx 0.424$.
• $n=8$: $\binom{8}{2}=28$, $B_{8,2}=3.38\times 10^8$, $H_{8,2}=2^{28}$, so $\rho_{8,2} \approx 0.421$.

Densities as a function of monomial count $k$ ($\delta_{n,2,k}$) for $n=6$ occur for $k=3,4,\dots,15$ and cluster near $0.42$; a similar distribution occurs for $n=8$.

Cubic Case ($d=3$)

No closed formula is known. Enumerative results:

• $n=6$: $\binom{6}{3}=20$, $B_{6,3}=30$, $H_{6,3}=2^{20}$, so $\rho_{6,3}\approx 2.86\times 10^{-5}$.
• $n=8$: $\binom{8}{3}=56$, $B_{8,3}=293,\!760$, $H_{8,3}=2^{56}$, so $\rho_{8,3}\approx 4.08\times 10^{-12}$.

For fixed $k$, cubic bent functions occur only at rare values; in $n=6$, all have exactly $k=16$ monomials. In $n=8$, nonzero $\delta_{8,3,k}$ appear at $k\in\{24,27,28,32,34,35,36,37,39,41\}$, with densities between approximately $10^{-12}$ and $10^{-9}$.

Case	$n$	$d$	$B_{n,d}$	$H_{n,d}$	$\rho_{n,d}$
Quadratic	6	2	$13,\!888$	$2^{15}$	$0.424$
Quadratic	8	2	$3.38\times 10^8$	$2^{28}$	$0.421$
Cubic	6	3	$30$	$2^{20}$	$2.86\times 10^{-5}$
Cubic	8	3	$293,\!760$	$2^{56}$	$4.08\times 10^{-12}$

3. Density as a Structural Characteristic

In the quadratic case, the density of bent functions is substantial—approximately $42\%$ of all homogeneous quadratic functions are bent, even as $n$ grows. For higher degrees, and notably the cubic case, densities are extremely small and further decay rapidly as $n$ increases. Most homogeneous Boolean functions of higher degree are, therefore, not bent, and bent functions become exceedingly rare.

Fixing the number of monomials $k$ provides a sharper lens on the distribution: In the quadratic case, high densities are robust across a broad range of $k$. For cubic functions, bent functions are concentrated at sparse and often mid-to-high $k$ values, further underscoring their rarity.

4. Theoretical and Asymptotic Insights

For quadratic homogeneous Boolean functions, the asymptotic density is explicitly characterized:

$$\lim_{n\to\infty} \rho_{n,2} = (1/2;1/4)_{\infty} \approx 0.419422,$$

where $(a;q)_\infty$ denotes the $q$-Pochhammer symbol.

For cubic and higher degrees ($d\geq 3$), no general asymptotic formula is presently established. Empirical enumeration demonstrates that density decays extremely rapidly with $n$. A plausible implication is that homogeneous bent functions of degree $d\geq 3$ are exceptionally rare for even modestly large $n$.

5. Algorithmic Approaches Using Density

Densities can be leveraged to guide search algorithms for homogeneous bent functions:

• Reduced ANF encoding represents candidate functions as bit vectors of length $\binom n d$, covering the full search space of $2^{\binom n d}$ possible functions.
• Weighted ANF encoding restricts candidates to those with a prescribed number $k$ of monomials, especially near values with maximal empirical density $\delta_{n,d,k}$. Genetic operators are adapted to preserve the total bit-weight.
• Informed search: Empirical and theoretical densities $\rho_{n,d,k}$ guide parameter choices in evolutionary algorithms, allowing for feasible discovery of cubic bent functions with $n=8$ variables; without such guidance, searching the full space would be intractable, as unrestricted search almost never succeeds.

The role of density is therefore central in both theoretical characterization and algorithmic construction, delineating precisely where nontrivial success is attainable for practical generation of homogeneous bent functions (Carlet et al., 16 Nov 2025).

6. Implications and Cryptographic Relevance

Bent functions exhibit strong cryptographic properties due to their maximal nonlinearity. Homogeneous quadratic bent functions form a large, structurally accessible family ($\rho_{n,2} \approx 0.42$), which is highly favorable for algebraic cryptosystem design.

In contrast, the extreme sparsity of cubic and higher-degree homogeneous bent functions complicates their use and motivates advanced metaheuristic search methods. The density framework specifies which parameter regimes are tractable and steers practical search strategies toward promising areas of the function space.

Systematic knowledge of densities also informs theoretical inquiry into the structure and limitations of cryptographically significant Boolean functions, guiding both exhaustive and probabilistic explorations.