Almost Maiorana–McFarland Bent Functions

• Almost Maiorana–McFarland bent functions are Boolean functions on F₂^(2m) defined via a generalized split (F₂^(m-1)×F₂^(m+1)) that departs from the classical MM structure.
• They achieve bentness by partitioning F₂^(m+1) into affine 2-planes, enforcing a 4-to-1 mapping with each fiber’s h function restricted to odd weight values.
• Construction methods, including recursive partitioning and concatenation, yield exponentially many EA-inequivalent bent functions that often lie outside the completed MM class.

Almost Maiorana–McFarland bent functions are Boolean bent functions on $\mathbb F_2^{2m}$ of the form

$f(x,y)=x\cdot \phi(y)+h(y),$

with

$$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$

where $\phi:\mathbb F_2^{m+1}\to \mathbb F_2^{m-1}$ and $h:\mathbb F_2^{m+1}\to \mathbb F_2$. In the 2025 terminology, these functions belong to the generalized Maiorana–McFarland class $\mathcal{GMM}_{m+1}$ and are called “almost Maiorana–McFarland bent functions” because they are structurally one step away from the classical Maiorana–McFarland class: instead of the balanced split $\mathbb F_2^m\times \mathbb F_2^m$, they use the first nontrivial generalized split $\mathbb F_2^{m-1}\times \mathbb F_2^{m+1}$ (Kudin et al., 19 Aug 2025).

1. Classical MM background and the almost-MM setting

For even $n=2m$, a Boolean function $f:\mathbb F_2^{2m}\to \mathbb F_2$ is bent when its Walsh–Hadamard transform has constant magnitude $f(x,y)=x\cdot \phi(y)+h(y),$0, equivalently

$f(x,y)=x\cdot \phi(y)+h(y),$1

The classical Maiorana–McFarland class $f(x,y)=x\cdot \phi(y)+h(y),$2 consists of functions

$f(x,y)=x\cdot \phi(y)+h(y),$3

where $f(x,y)=x\cdot \phi(y)+h(y),$4 is a permutation of $f(x,y)=x\cdot \phi(y)+h(y),$5 and $f(x,y)=x\cdot \phi(y)+h(y),$6 is arbitrary. Its completed class is

$f(x,y)=x\cdot \phi(y)+h(y),$7

A central criterion used throughout the literature is Dillon’s characterization: a bent function lies in $f(x,y)=x\cdot \phi(y)+h(y),$8 if and only if it has an $f(x,y)=x\cdot \phi(y)+h(y),$9-subspace of dimension $$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$0, meaning a vector subspace $$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$1 such that

$$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$2

(Kudin et al., 19 Aug 2025).

Almost Maiorana–McFarland bent functions replace the classical $$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$3 split by the generalized $$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$4 split. In the broader generalized family one considers

$$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$5

and the 2025 paper isolates $$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$6 as the first nontrivial case. The resulting class $$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$7 is therefore the precise setting in which the phrase “almost Maiorana–McFarland bent functions” is used (Kudin et al., 19 Aug 2025).

2. Exact bentness criterion and the dual

The defining structural theorem is a complete characterization of bentness for

$$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$8

Such a function is bent if and only if two conditions hold. First, the fibers

$$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$9

form a partition of $\phi:\mathbb F_2^{m+1}\to \mathbb F_2^{m-1}$0 into $\phi:\mathbb F_2^{m+1}\to \mathbb F_2^{m-1}$1-dimensional affine subspaces. Second, for every $\phi:\mathbb F_2^{m+1}\to \mathbb F_2^{m-1}$2, the restriction of $\phi:\mathbb F_2^{m+1}\to \mathbb F_2^{m-1}$3 to $\phi:\mathbb F_2^{m+1}\to \mathbb F_2^{m-1}$4 has odd weight (Kudin et al., 19 Aug 2025).

Because

$\phi:\mathbb F_2^{m+1}\to \mathbb F_2^{m-1}$5

bentness forces $\phi:\mathbb F_2^{m+1}\to \mathbb F_2^{m-1}$6 to be $\phi:\mathbb F_2^{m+1}\to \mathbb F_2^{m-1}$7-to-$\phi:\mathbb F_2^{m+1}\to \mathbb F_2^{m-1}$8. Each fiber therefore has cardinality $\phi:\mathbb F_2^{m+1}\to \mathbb F_2^{m-1}$9, and the theorem strengthens this to the statement that every fiber must be an affine plane

$h:\mathbb F_2^{m+1}\to \mathbb F_2$0

Equivalently, if

$h:\mathbb F_2^{m+1}\to \mathbb F_2$1

then the fiber is an affine $h:\mathbb F_2^{m+1}\to \mathbb F_2$2-flat exactly when

$h:\mathbb F_2^{m+1}\to \mathbb F_2$3

On each such fiber the odd-weight requirement means that $h:\mathbb F_2^{m+1}\to \mathbb F_2$4 has weight $h:\mathbb F_2^{m+1}\to \mathbb F_2$5 or $h:\mathbb F_2^{m+1}\to \mathbb F_2$6. Consequently,

$h:\mathbb F_2^{m+1}\to \mathbb F_2$7

The Walsh analysis is fiberwise. For $h:\mathbb F_2^{m+1}\to \mathbb F_2$8,

$h:\mathbb F_2^{m+1}\to \mathbb F_2$9

Thus bentness is equivalent to requiring that for every fiber and every $\mathcal{GMM}_{m+1}$0, the corresponding $\mathcal{GMM}_{m+1}$1-term sign sum is $\mathcal{GMM}_{m+1}$2. This is what forces both the affine-plane geometry of the fibers and the odd-weight condition on $\mathcal{GMM}_{m+1}$3.

The same paper gives an explicit dual formula. Fix any total order $\mathcal{GMM}_{m+1}$4 on $\mathcal{GMM}_{m+1}$5. If $\mathcal{GMM}_{m+1}$6 is bent, then for all $\mathcal{GMM}_{m+1}$7 and $\mathcal{GMM}_{m+1}$8,

$\mathcal{GMM}_{m+1}$9

The dual is therefore determined fiberwise by the affine planes $\mathbb F_2^m\times \mathbb F_2^m$0 and the restriction of $\mathbb F_2^m\times \mathbb F_2^m$1 to those planes (Kudin et al., 19 Aug 2025).

3. Partition geometry and membership in $\mathbb F_2^m\times \mathbb F_2^m$2

The geometry of the partition induced by $\mathbb F_2^m\times \mathbb F_2^m$3 governs whether an almost MM bent function remains inside the completed MM class. The 2025 paper distinguishes between partitions that are “non-proper” and those that are “proper” (Kudin et al., 19 Aug 2025).

A sufficient condition for inclusion in $\mathbb F_2^m\times \mathbb F_2^m$4 is the existence of a nonzero direction common to all fiber-planes in a precise sense: there exists nonzero $\mathbb F_2^m\times \mathbb F_2^m$5 such that for every fiber $\mathbb F_2^m\times \mathbb F_2^m$6, one can choose $\mathbb F_2^m\times \mathbb F_2^m$7 with

$\mathbb F_2^m\times \mathbb F_2^m$8

When this happens, the function belongs to $\mathbb F_2^m\times \mathbb F_2^m$9. The proof constructs an $\mathbb F_2^{m-1}\times \mathbb F_2^{m+1}$0-dimensional $\mathbb F_2^{m-1}\times \mathbb F_2^{m+1}$1-subspace

$\mathbb F_2^{m-1}\times \mathbb F_2^{m+1}$2

A particularly transparent case is that of trivial partitions. If

$\mathbb F_2^{m-1}\times \mathbb F_2^{m+1}$3

for a fixed $\mathbb F_2^{m-1}\times \mathbb F_2^{m+1}$4-dimensional linear subspace $\mathbb F_2^{m-1}\times \mathbb F_2^{m+1}$5, then the resulting bent function has at least three different $\mathbb F_2^{m-1}\times \mathbb F_2^{m+1}$6-dimensional $\mathbb F_2^{m-1}\times \mathbb F_2^{m+1}$7-subspaces and therefore lies in $\mathbb F_2^{m-1}\times \mathbb F_2^{m+1}$8.

The converse is false. The same paper exhibits a bent function in $\mathbb F_2^{m-1}\times \mathbb F_2^{m+1}$9 whose partition does not satisfy the common-direction condition, yet the function still belongs to $n=2m$0, because after a variable permutation it becomes a strict MM function. Thus the partition criterion is sufficient rather than necessary.

For exclusion from $n=2m$1, the paper introduces an extended derivative property

$n=2m$2

For the associated generalized-MM function

$n=2m$3

Theorem 5.3 proves that any $n=2m$4-subspace has dimension at most $n=2m$5, and under an additional condition on $n=2m$6, the canonical subspace

$n=2m$7

is the unique maximal $n=2m$8-subspace. In particular, when $n=2m$9 and $f:\mathbb F_2^{2m}\to \mathbb F_2$0 is bent, these conditions imply $f:\mathbb F_2^{2m}\to \mathbb F_2$1 (Kudin et al., 19 Aug 2025).

4. Construction principles

Once a suitable partition into affine planes is fixed, the choice of $f:\mathbb F_2^{2m}\to \mathbb F_2$2 is simple. On each $f:\mathbb F_2^{2m}\to \mathbb F_2$3-point fiber $f:\mathbb F_2^{2m}\to \mathbb F_2$4, the restriction of $f:\mathbb F_2^{2m}\to \mathbb F_2$5 must have odd weight, so one may choose either a unique $f:\mathbb F_2^{2m}\to \mathbb F_2$6 or a unique $f:\mathbb F_2^{2m}\to \mathbb F_2$7 on that fiber. The paper counts $f:\mathbb F_2^{2m}\to \mathbb F_2$8 valid restrictions per fiber and states that for fixed $f:\mathbb F_2^{2m}\to \mathbb F_2$9 the number of admissible Boolean functions $f(x,y)=x\cdot \phi(y)+h(y),$00 is exactly

$f(x,y)=x\cdot \phi(y)+h(y),$01

(Kudin et al., 19 Aug 2025).

Several partition-construction methods are described. One is a product method, obtained by writing

$f(x,y)=x\cdot \phi(y)+h(y),$02

and taking products of partitions into $f(x,y)=x\cdot \phi(y)+h(y),$03-dimensional affine subspaces. Another is a clique method: for small $f(x,y)=x\cdot \phi(y)+h(y),$04, one forms the graph whose vertices are the $f(x,y)=x\cdot \phi(y)+h(y),$05-dimensional affine subspaces of $f(x,y)=x\cdot \phi(y)+h(y),$06, with adjacency defined by disjointness; a partition then corresponds to a clique of size $f(x,y)=x\cdot \phi(y)+h(y),$07.

The paper also gives a direct recursive algorithm. One begins with

$f(x,y)=x\cdot \phi(y)+h(y),$08

a $f(x,y)=x\cdot \phi(y)+h(y),$09-dimensional linear subspace. Then one chooses $f(x,y)=x\cdot \phi(y)+h(y),$10 and defines an affine $f(x,y)=x\cdot \phi(y)+h(y),$11-flat

$f(x,y)=x\cdot \phi(y)+h(y),$12

Recursively, after $f(x,y)=x\cdot \phi(y)+h(y),$13 have been chosen, one selects

$f(x,y)=x\cdot \phi(y)+h(y),$14

and sets

$f(x,y)=x\cdot \phi(y)+h(y),$15

When $f(x,y)=x\cdot \phi(y)+h(y),$16 disjoint affine $f(x,y)=x\cdot \phi(y)+h(y),$17-flats have been obtained, $f(x,y)=x\cdot \phi(y)+h(y),$18 is defined by assigning a distinct value of $f(x,y)=x\cdot \phi(y)+h(y),$19 to each $f(x,y)=x\cdot \phi(y)+h(y),$20 (Kudin et al., 19 Aug 2025).

5. The $f(x,y)=x\cdot \phi(y)+h(y),$21-variable case and concatenation phenomena

The case $f(x,y)=x\cdot \phi(y)+h(y),$22 is treated explicitly. Then

$f(x,y)=x\cdot \phi(y)+h(y),$23

The paper gives a concrete partition of $f(x,y)=x\cdot \phi(y)+h(y),$24 into eight affine planes $f(x,y)=x\cdot \phi(y)+h(y),$25, uses it to define $f(x,y)=x\cdot \phi(y)+h(y),$26, and shows that the associated function

$f(x,y)=x\cdot \phi(y)+h(y),$27

is bent and has a unique $f(x,y)=x\cdot \phi(y)+h(y),$28-dimensional $f(x,y)=x\cdot \phi(y)+h(y),$29-subspace, namely the canonical one

$f(x,y)=x\cdot \phi(y)+h(y),$30

so that $f(x,y)=x\cdot \phi(y)+h(y),$31 (Kudin et al., 19 Aug 2025).

Using the simple partition algorithm, the authors found $f(x,y)=x\cdot \phi(y)+h(y),$32 decompositions of $f(x,y)=x\cdot \phi(y)+h(y),$33 into disjoint $f(x,y)=x\cdot \phi(y)+h(y),$34-dimensional affine subspaces, of which $f(x,y)=x\cdot \phi(y)+h(y),$35 are proper. For each such proper partition $f(x,y)=x\cdot \phi(y)+h(y),$36, they defined $f(x,y)=x\cdot \phi(y)+h(y),$37 by $f(x,y)=x\cdot \phi(y)+h(y),$38, chose a random admissible $f(x,y)=x\cdot \phi(y)+h(y),$39, and obtained bent functions. All $f(x,y)=x\cdot \phi(y)+h(y),$40 tested bent functions were outside $f(x,y)=x\cdot \phi(y)+h(y),$41. Among them, at least $f(x,y)=x\cdot \phi(y)+h(y),$42 were verified to be EA-inequivalent, which yields at least

$f(x,y)=x\cdot \phi(y)+h(y),$43

distinct bent functions on $f(x,y)=x\cdot \phi(y)+h(y),$44 outside $f(x,y)=x\cdot \phi(y)+h(y),$45. The same paper notes that the total number of $f(x,y)=x\cdot \phi(y)+h(y),$46-variable bent functions in $f(x,y)=x\cdot \phi(y)+h(y),$47 is approximately

$f(x,y)=x\cdot \phi(y)+h(y),$48

so this construction alone produces more examples outside $f(x,y)=x\cdot \phi(y)+h(y),$49 than the entire completed MM class in dimension $f(x,y)=x\cdot \phi(y)+h(y),$50.

The paper also develops a concatenation theory specialized to almost MM functions with a common $f(x,y)=x\cdot \phi(y)+h(y),$51. If

$f(x,y)=x\cdot \phi(y)+h(y),$52

with the same affine-plane partition for all four components, then the concatenation

$f(x,y)=x\cdot \phi(y)+h(y),$53

is bent if and only if, for every $f(x,y)=x\cdot \phi(y)+h(y),$54,

$f(x,y)=x\cdot \phi(y)+h(y),$55

and either

$f(x,y)=x\cdot \phi(y)+h(y),$56

or

$f(x,y)=x\cdot \phi(y)+h(y),$57

Two consequences are especially notable. First, given one bent $f(x,y)=x\cdot \phi(y)+h(y),$58, there exist bent $f(x,y)=x\cdot \phi(y)+h(y),$59 such that

$f(x,y)=x\cdot \phi(y)+h(y),$60

lies in $f(x,y)=x\cdot \phi(y)+h(y),$61, even when $f(x,y)=x\cdot \phi(y)+h(y),$62. This answers the open problem of Kudin et al. by showing that a bent function in $f(x,y)=x\cdot \phi(y)+h(y),$63 can have all four $f(x,y)=x\cdot \phi(y)+h(y),$64-variable restrictions outside $f(x,y)=x\cdot \phi(y)+h(y),$65. Second, the converse phenomenon also occurs: starting with MM bent functions

$f(x,y)=x\cdot \phi(y)+h(y),$66

where $f(x,y)=x\cdot \phi(y)+h(y),$67 has property

$f(x,y)=x\cdot \phi(y)+h(y),$68

and applying a suitable linear permutation $f(x,y)=x\cdot \phi(y)+h(y),$69, one gets

$f(x,y)=x\cdot \phi(y)+h(y),$70

such that

$f(x,y)=x\cdot \phi(y)+h(y),$71

is bent and lies outside $f(x,y)=x\cdot \phi(y)+h(y),$72. After a suitable linear permutation $f(x,y)=x\cdot \phi(y)+h(y),$73, this function can again be represented in a generalized MM form in $f(x,y)=x\cdot \phi(y)+h(y),$74 (Kudin et al., 19 Aug 2025).

6. Broader MM-derived literature and neighboring notions

The exact phrase “almost Maiorana–McFarland” is recent and specific, but several neighboring strands of work study mathematically adjacent families. In asymptotic enumeration, Construction $f(x,y)=x\cdot \phi(y)+h(y),$75 of a “modification of the Maiorana–McFarland family” replaces singleton Walsh-support blocks by affine subspaces carrying plateaued components; it reduces to classical MM when $f(x,y)=x\cdot \phi(y)+h(y),$76, and its asymptotically optimal improvement occurs in the nearest nontrivial case $f(x,y)=x\cdot \phi(y)+h(y),$77, making it a precise near-MM construction even though the phrase is not used (Potapov et al., 2021).

A different broad sense appears in the study of quadratic symmetric bent functions: for even $f(x,y)=x\cdot \phi(y)+h(y),$78, the functions

$f(x,y)=x\cdot \phi(y)+h(y),$79

are proved not to be of Maiorana–McFarland type, yet are stated to be affine equivalent to it. In that equivalence-orbit sense they are MM-adjacent without being MM in their native coordinates (Rifà et al., 2012).

Concatenation theory supplies another route. Four MM bent functions

$f(x,y)=x\cdot \phi(y)+h(y),$80

can be concatenated into a bent function

$f(x,y)=x\cdot \phi(y)+h(y),$81

via the dual bent condition

$f(x,y)=x\cdot \phi(y)+h(y),$82

Under additional hypotheses on permutations with the $f(x,y)=x\cdot \phi(y)+h(y),$83 property, the resulting function can lie outside the completed MM class $f(x,y)=x\cdot \phi(y)+h(y),$84, even though each quarter of the truth table is MM (Polujan et al., 2023). Complementarily, complete necessary-and-sufficient criteria for when bent concatenations

$f(x,y)=x\cdot \phi(y)+h(y),$85

do not belong to $f(x,y)=x\cdot \phi(y)+h(y),$86 are given in terms of the $f(x,y)=x\cdot \phi(y)+h(y),$87-subspaces of the components; in the special family

$f(x,y)=x\cdot \phi(y)+h(y),$88

membership in $f(x,y)=x\cdot \phi(y)+h(y),$89 is equivalent to $f(x,y)=x\cdot \phi(y)+h(y),$90 and $f(x,y)=x\cdot \phi(y)+h(y),$91 having a common $f(x,y)=x\cdot \phi(y)+h(y),$92-dimensional $f(x,y)=x\cdot \phi(y)+h(y),$93-subspace (Kudin et al., 2024).

More recent work pushes the MM boundary in two opposite directions. One line studies permutations satisfying $f(x,y)=x\cdot \phi(y)+h(y),$94 or $f(x,y)=x\cdot \phi(y)+h(y),$95, which control uniqueness of the maximal MM-type subspace in functions

$f(x,y)=x\cdot \phi(y)+h(y),$96

These are then used to construct large families outside $f(x,y)=x\cdot \phi(y)+h(y),$97, as well as $f(x,y)=x\cdot \phi(y)+h(y),$98-optimal bent functions with the smallest possible linearity index, described as opposite to MM bent functions (Kudin et al., 19 Aug 2025). Another line formulates generalized Maiorana–McFarland functions

$f(x,y)=x\cdot \phi(y)+h(y),$99

and proves that suitable choices of the bent slices $$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$00, often themselves classical MM functions, yield generalized MM bent functions outside both the classical $$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$01 and $$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$02 classes and hence outside their completed classes (Alkan et al., 30 Mar 2026).

Within this broader landscape, almost Maiorana–McFarland bent functions in the strict 2025 sense occupy a particularly explicit position. They are not merely MM-derived or MM-equivalent; they are characterized exactly by affine-plane partitions of $$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$03, odd restrictions of $$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$04 on those planes, and a partition geometry that determines whether the resulting bent function remains in $$x\in \mathbb F_2^{m-1},\qquad y\in \mathbb F_2^{m+1},$$05 or escapes it (Kudin et al., 19 Aug 2025).