Negabent Boolean Functions in Cryptography

• Negabent Boolean functions are defined by a flat nega-Hadamard spectrum, ensuring optimal autocorrelation under nega‐periodic conditions.
• They are characterized via a quadratic twist that links them to ordinary bent functions, with distinct formulations for even and odd dimensions.
• Applications in cryptography and coding theory are enhanced by explicit constructions using Maiorana–McFarland methods and evolutionary computation.

Negabent Boolean functions are Boolean functions whose spectrum with respect to the nega‐Hadamard transform is flat; that is, all values of the spectrum have equal magnitude, a generalization of the bent property under the ordinary Walsh–Hadamard transform. Negabent functions exist for both even and odd numbers of variables and have become central in cryptography and coding theory due to their optimal autocorrelation properties under nega‐periodic boundary conditions. The subclass of bent–negabent functions—simultaneously bent and negabent—exhibits combined optimal resistance to both periodic and negaperiodic spectral attacks. Key results include the existence and explicit construction of negabent and bent–negabent functions of maximum possible algebraic degree and comprehensive characterizations in terms of trace expressions, permutation polynomials, and spectral criteria.

1. Definition and Spectral Characterization

A Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$ is negabent if its nega‐Walsh–Hadamard (or nega‐Hadamard) transform has constant magnitude at all points: $$N_f(u) = \sum_{x \in \{0,1\}^n} (-1)^{f(x) \oplus u \cdot x} \, i^{w(x)},$$ where $w(x)$ is the Hamming weight, $i^2 = -1$, and $u \cdot x$ is the usual inner product. The negabent property holds if and only if

$|N_f(u)| = 2^{n/2} \quad \forall u \in \{0,1\}^n.$

After normalization, the values lie on the unit circle, and the flatness of the nega spectrum is equivalent to the vanishing of all nontrivial nega-autocorrelations. The key difference from the ordinary bent property is that affine functions, including linear and constant functions, are always negabent, whereas only nonlinear (specifically, maximally nonlinear) functions can be bent (Su et al., 2012, Carlet et al., 31 Jan 2026).

2. Characterization in Even and Odd Dimensions

The characterization of negabent functions fundamentally relies on the relationship between the nega and ordinary Walsh–Hadamard spectra by means of a quadratic "twist." Let $o_2(x) = \sum_{1 \le i < j \le n} x_i x_j$ denote the elementary symmetric quadratic form.

Even $n$:

A function $f$ is negabent if and only if $f \oplus o_2$ is (ordinary) bent. The nega spectrum in this case takes at most four values (after normalization, $\pm 1, \pm i$), with explicit multiplicities (Su et al., 2012, Carlet et al., 31 Jan 2026). This equivalence allows all structural properties and constructions of bent functions to be transferred to negabent functions by twisting with this quadratic.

Odd $n$:

For odd $n$, $f$ is negabent if and only if $f \oplus o_2$ is semibent and its Walsh–Hadamard spectrum satisfies

$$\{ |W_{f \oplus o_2}(u)|, |W_{f \oplus o_2}(u \oplus 1^n)| \} = \{ 0, 2^{(n+1)/2} \}$$

for all $u$. There exists a decomposition via two bent $(n-1)$-variable functions $g$ and $h$ such that

$$f(x) = o_2(x) \oplus (1 \oplus x_n) g(x_1, ..., x_{n-1}) \oplus x_n h(x_1, ..., x_{n-1}).$$

Alternatively, $f$ is negabent if and only if the $(n+1)$-variable function $F(x_1,..., x_n,y) = f(x) \oplus o_2(x) y$ is bent (Su et al., 2012, Carlet et al., 31 Jan 2026).

Trace/Dual Characterization:

In the finite field setting ($\F_{2^n}$), the main criterion for negabentness is that for all nonzero $a$,

$\sum_{x \in \F_{2^n}} (-1)^{f(x) + f(x+a) + \Tr_1^n(ax)} = 0,$

i.e., the function $x \mapsto f(x) + f(x+a) + \Tr_1^n(ax)$ is balanced for all $a \neq 0$ (Wu et al., 2016, Sarkar, 2014).

3. Spectrum Structure and Explicit Value Distributions

The nega‐spectrum of a negabent function consists of at most four values. For even $n$, all negabent functions have their nega spectra distributed as follows (after normalization):

Value	Multiplicity (Case 1)	Multiplicity (Case 2)
$+1$	$2^{n-2} + 2^{n/2-1}$	$2^{n-2} - 2^{n/2-1}$
$-1$	$2^{n-2} - 2^{n/2-1}$	$2^{n-2} + 2^{n/2-1}$
$+i$	$2^{n-2}$	$2^{n-2}$
$-i$	$2^{n-2}$	$2^{n-2}$

For odd $n$, the spectrum takes values in $\{1, 1\pm i, -1\pm i\}$, with analogous closed-form multiplicity structures determined by dimension (Su et al., 2012). This four-value property is a tight constraint and facilitates both structural proofs and algorithmic searches for negabent functions.

4. Constructions and Infinite Families

Maiorana–McFarland and Complete Mapping Families

For even $n = 2t$, the construction of bent–negabent functions hinges on the combination of the Maiorana–McFarland class and a quadratic twist. Let $\pi(y)$ be a permutation (or more strongly, a complete mapping polynomial) of $\F_{2^t}$, and $h(y)$ any Boolean function of algebraic degree $t$: $f(x, y) = \Tr_1^t(x \pi(y)) \oplus \Tr_1^t(h(y)), \quad (x, y) \in \F_{2^t}^2.$ Then, $F(x, y) = f(x, y) \oplus G(x, y)$ with $G(x, y) = \Tr_1^t(x y)$ is bent–negabent if and only if $\pi(y)$ and $\pi(y) + y$ are permutations—i.e., $\pi$ is a complete mapping (Sarkar, 2014). By choosing $h$ of algebraic degree $t = n/2$, this method yields infinite explicit families of bent–negabent functions of maximal degree.

Trace-Monomial and Permutation Polynomial Criteria

For quadratic trace monomials $f(x) = \Tr_1^n(\lambda x^{2^k+1})$, negabentness requires that the associated linearized polynomial $L(x) = \lambda x^{2^k} + x$ (or its variants for related forms) is a permutation. The precise characterization of all quadratic negabent monomials over $\F_{2^n}$ is established by necessary and sufficient conditions on $\lambda$ (Sarkar, 2014).

Further constructions include binomial and trinomial forms involving traces—e.g., $f(x) = \Tr_1^n(\lambda x^{2^k+1}) + \Tr_1^n(u x) \Tr_1^n(v x)$—with explicit criteria for negabentness given in terms of parameter traces and properties of permutation linearized polynomials (Wu et al., 2016).

Cubic and Niho-Type Monomials:

Negabentness of monomials $f(x) = \Tr_1^n(\lambda x^d)$ with cubic or more general Niho exponents is tightly constrained; e.g., for cubic monomials with $d=2^k+3$, negabentness holds if and only if $\lambda \in \F_2$ and related combinatorial conditions are met. A conjectural classification for higher-degree Niho exponents relates to trace constraints and field structure, based on explicit computations and Kloosterman sum bounds (Wu et al., 2016).

5. Maximum Algebraic Degree and Structural Limits

No $n$-variable bent–negabent function can have algebraic degree exceeding $n/2$. For every even $n$, there exist explicit constructions of bent–negabent functions of degree exactly $n/2$, for example, via the trace-Maiorana–McFarland twist using degree-$t$ Boolean functions or suitable complete mapping polynomials (Su et al., 2012, Sarkar, 2014). These results resolve open questions posed by Parker & Pott and Stănică et al. regarding both maximal degree and the realization of bent–negabent functions for every algebraic degree up to this bound.

6. Evolutionary and Algorithmic Approaches

Evolutionary computation, particularly symbolic tree-based genetic programming (GP), has been shown to efficiently discover both negabent and bent–negabent functions. Two encodings are used: (1) bitstring encoding, representing the full truth table, and (2) symbolic (expression-tree) encoding over Boolean operators. GP outperforms raw bitstring evolution at higher dimensions because it exploits algebraic structure and allows effective navigation of the exponentially large Boolean function space (Carlet et al., 31 Jan 2026).

Empirical results confirm that symbolic GP reliably finds bent–negabent functions for all even $n$ up to 16, including explicit examples matching theoretical expectations for spectrum flatness and maximal nonlinearity. Observations suggest that integrating algebraic degree targets or spectrum constraints into the fitness functions enhances efficiency.

Table: Success of Evolutionary Approaches

Encoding	Range Succeeded	Comments
Bitstring	$n=6,7$	Fails for larger $n$
Symbolic (GP)	$n=6$–16	Finds bent–negabent for even, maximally nonlinear for odd $n$

A plausible implication is that future algorithmic searches for such functions, especially under additional cryptographic constraints, will benefit from symbolic representations and spectral (rather than semantic) fitness objectives.

7. Open Problems and Research Directions

Outstanding challenges include (1) full characterization of higher-degree negabent monomials beyond quadratics and cubics, (2) generalized construction techniques for bent–negabent functions outside the Maiorana–McFarland class, and (3) evolutionary search strategies for functions balancing negabentness with other cryptographic properties (balancedness, propagation criteria, etc.) (Su et al., 2012, Sarkar, 2014, Carlet et al., 31 Jan 2026). Further, the conjecture for Niho-type negabent exponents remains unresolved outside special cases.

Negabent Boolean functions, both by their deep algebraic structure and practical cryptographic relevance, continue to motivate research into spectrum theory, permutation polynomials, trace function techniques, and advanced heuristic algorithms for combinatorial Boolean spaces.