Bent–Negabent Boolean Functions

• Bent–negabent functions are Boolean functions that achieve maximal spectral flatness under both the Walsh–Hadamard and nega–Hadamard transforms, ensuring optimal nonlinearity and robustness.
• They are characterized by strict autocorrelation properties and elegant constructions via methods such as Maiorana–McFarland, quadratic switching, and field trace techniques.
• Their applications span resilient S-box design, coding schemes like Kerdock–Preparata codes, and the generation of mutually unbiased bases in quantum information.

A bent–negabent function is a Boolean function that exhibits maximal spectral flatness with respect to both the classical Walsh–Hadamard transform and the nega–Hadamard transform. Such functions play a central role in coding theory, cryptography, and combinatorial design due to their optimal resistance to both linear and nonlinear attacks, as well as their correspondence to fundamental mathematical objects, including relative difference sets and mutually unbiased bases.

1. Fundamentals: Bent, Negabent, and Bent–Negabent Functions

Let $f : \mathbb F_2^n \to \mathbb F_2$ be a Boolean function. The Walsh–Hadamard transform is defined as

$$W_f(u) = \sum_{x\in\mathbb F_2^n} (-1)^{f(x) + u\cdot x}, \quad u \in \mathbb F_2^n,$$

where $u \cdot x = \sum_{i=1}^{n} u_i x_i$. The function $f$ is called bent if $|W_f(u)| = 2^{n/2}$ for all $u$; this requires $n$ even and achieves maximal nonlinearity.

The nega–Hadamard transform of $f$ is

$$N_f(u) = \sum_{x\in\mathbb F_2^n} (-1)^{f(x) + u \cdot x} i^{\mathrm{wt}(x)}, \quad u \in \mathbb F_2^n,$$

where $\mathrm{wt}(x)$ is the Hamming weight of $x$ and $i^2 = -1$. A function is negabent if $|N_f(u)| = 2^{n/2}$ for all $u$. Negabent functions exist for both even and odd $n$, and notably every affine function is negabent.

A function is called bent–negabent if it is simultaneously bent and negabent. For even $n$, this requires both $|W_f(u)| = 2^{n/2}$ and $|N_f(u)| = 2^{n/2}$ for all $u$ (Su et al., 2012, Carlet et al., 31 Jan 2026).

2. Characterization via Spectral Conditions and Autocorrelations

Connections between the Walsh and nega–Hadamard spectra are revealed through spectral and autocorrelation identities:

• For even $n$, $f$ is negabent if and only if $f \oplus \sigma_2$ is bent, where $\sigma_2(x) = \sum_{1\leq i<j\leq n} x_i x_j$ is the all-pairwise quadratic symmetric function (Su et al., 2012, Carlet et al., 31 Jan 2026).
• For general $n$, the possible values of $N_f(u)$ for negabent functions are: $\{\pm 2^{n/2}, \pm i 2^{n/2}\}$ if $n$ is even, and $\{( \pm 1 \pm i ) 2^{(n-1)/2} / \sqrt{2}\}$ if $n$ is odd (Su et al., 2012).

In terms of autocorrelation, for a function $f$ interpreted as $f(x)=\mathrm{Tr}(F(x))$ over $\mathbb F_{2^n}$:

• $f$ is bent if $\sum_{x}( -1 )^{f(x) + f(x+a) } = 0$ for all $a\neq 0$,
• $f$ is negabent if $$\sum_{x}( -1 )^{f(x) + f(x+a) + \mathrm{Tr}(a x) } = 0$$ for all $a\neq 0$ (Sarkar, 2014).

The condition for bent–negabent can be elegantly rephrased: $f$ is bent–negabent if both $f$ and $f+\sigma_2$ are bent (Carlet et al., 31 Jan 2026, Su et al., 2012).

3. Explicit Constructions and Degree Bounds

3.1. Maiorana–McFarland and Quadratic "Switching" Constructions

For $n=2m$, classic bent functions are given by

$f(x, y) = x \cdot T(y) \oplus g(y),$

where $T$ is a permutation of $\mathbb F_2^m$ (Su et al., 2012). The quadratic form $h(x) = \sum_{i=1}^m x_i x_{m+i}$ serves as a "switch" to produce new classes:

If $f$ and $f \oplus h$ are both bent, then the function $f'(x) = f(xA \oplus b) \oplus \od_2(x)$, where $\od_2(x) = \sum_{1\leq i<j\leq n} x_i x_j$ and $A\in\operatorname{GL}(n,2)$, is bent–negabent (Su et al., 2012).

3.2. Field Trace and Complete Mapping Polynomial Approaches

Quadratic monomials can also be analyzed via field trace representation. For $f(x) = \mathrm{Tr}(\lambda x^{2^k+1})$ over $\mathbb F_{2^n}$:

• $f$ is bent if and only if $\lambda$ avoids the Gold-exponent class,
• $f$ is negabent according to a single-root criterion involving complete mapping polynomials (Sarkar, 2014).

Maximum degree for bent–negabent in this setting is $n/2$. Explicit infinite families achieving this bound arise by choosing $h(y)$ of degree $t = n/2$ in the Maiorana–McFarland construction and appropriate complete mappings for $T$ or $\pi$ (Su et al., 2012, Sarkar, 2014).

3.3. Modify–Truth–Table and Rotation-Symmetric Generalizations

Systematic constructions employ controlled modifications of the truth table of foundational bent–negabent seeds. For instance, in $4k$ or $8k$ variables, start from a quadratic seed $g_0(x,y)$, and flip values on affine subspaces or code cosets. Under precise conditions on fragmentary spectra, the resulting function remains bent–negabent and can reach degree $2k$ or $4k$ (Guo et al., 2022).

Bent–negabent functions invariant under 2-step rotations (but not full cyclic shift) are constructed by combining quadratic symmetric seeds with modifications over 2-rotation orbits, thus overcoming the nonexistence of fully rotation-symmetric bent–negabent functions (Guo et al., 2022).

4. Generalizations: Vectorial and $\mathbb Z_{2^k}$-Valued Bent–Negabent

Bent–negabent notions extend to vectorial and generalized Boolean mappings. Two main concepts for vectorial negabent/bent–negabent have been developed:

• PKPM: A mapping $F:\mathbb F_2^n\to\mathbb F_2^k$ is vectorial bent–negabent if every nonzero linear combination of coordinates is bent–negabent (for $k\leq m-1$). Maiorana–McFarland constructions yield such mappings (Anbar et al., 2024).
• AM ("vectorial bent$_4$"): $F:\mathbb F_{2^n}\to\mathbb F_{2^k}$ is vectorial negabent if a $k$-variate nega-spectrum is flat for all nonzero $c$, relating to non-splitting relative difference sets in $Z_2^{n-k}\times Z_4^k$.

In the generalized Boolean setting $f:\mathbb F_2^n \to \mathbb Z_{2^k}$, the nega–$\mathbb Z_{2^k}$–bent transform is defined on a non-abelian group constructed via a quadratic extension of the binary vector space. This generalization unifies the shift-based correspondence between standard bent and negabent functions, and supports constructions via bent partitions and inverse permutations (Anbar et al., 2024).

5. Evolutionary and Algorithmic Generation of Bent–Negabent Functions

Recent research demonstrates the efficacy of evolutionary computation, including steady-state genetic algorithms and symbolic genetic programming, in discovering bent–negabent functions up to $n=16$ variables. Fitness functions combine nonlinearity and flatness criteria, targeting both Walsh and nega-spectra extremality. Tree-based genetic programming encodings outperform truth-table encodings, especially in higher dimensions (Carlet et al., 31 Jan 2026). The evolved functions are highly structured and considerably expand the catalogue beyond previously known algebraic constructions.

6. Cryptographic, Coding, and Quantum Implications

Bent–negabent functions provide the highest resistance to both linear and differential cryptanalysis, as their outputs are maximally distant from all affine functions under both standard and nega-modulation. They are optimal for constructing resilient S-boxes, codebooks for Kerdock–Preparata codes, and correlation-immune sequences (Carlet et al., 31 Jan 2026, Guo et al., 2022). In quantum information, bent–negabent functions correspond to states invariant under both standard and nega–Hadamard local unitaries, forming maximal mutually unbiased bases.

The duals of bent–negabent functions remain within the class, ensuring stability under various cryptographic transformations. The algebraic degree upper bound of $n/2$ ensures immunity against low-degree algebraic attacks, while explicit constructions with degrees approaching this maximum are now available in both classical and generalized function domains (Su et al., 2012, Sarkar, 2014, Guo et al., 2022).

7. Open Problems and Future Directions

Several structural and combinatorial questions remain open, particularly the synthesis of vectorial functions whose Gray images are simultaneously vectorial bent–negabent in both PKPM and AM senses. Systematic enumeration and classification, especially in the generalized and vectorial settings, remain active research topics (Anbar et al., 2024). The recent success of evolutionary and algorithmic construction suggests substantial untapped potential for further extension of the known function classes and application domains.

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Key References

• (Su et al., 2012) Characterization of Negabent Functions and Construction of Bent-Negabent Functions with Maximum Algebraic Degree
• (Sarkar, 2014) Some Results on Bent-Negabent Boolean Functions over Finite Fields
• (Guo et al., 2022) Systematic Constructions of Bent-Negabent Functions, 2-Rotation Symmetric Bent-Negabent Functions and Their Duals
• (Anbar et al., 2024) Vectorial Negabent Concepts: Similarities, Differences, and Generalizations
• (Carlet et al., 31 Jan 2026) NegaBent, No Regrets: Evolving Spectrally Flat Boolean Functions