---
title: Almost Maiorana–McFarland Bent Functions
url: https://www.emergentmind.com/topics/almost-maiorana-mcfarland-bent-functions
type: topic
---

# Almost Maiorana–McFarland Bent Functions

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 \(\mathbb F_2^{m-1}\times \mathbb F_2^{m+1}\) [2508.14265].

## 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 \(2^m\), equivalently
\[
W_f(\omega)=\pm 2^m\qquad \text{for all }\omega\in\mathbb F_2^{2m}.
\]
The classical Maiorana–McFarland class \(\mathcal M\) consists of functions
\[
f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,
\]
where \(\pi\) is a permutation of \(\mathbb F_2^m\) and \(h\) is arbitrary. Its completed class is
\[
\mathcal M^\#=\{f(xA+b)+c\cdot x+d: f\in \mathcal M,\ A\in GL(n,\mathbb F_2),\ b,c\in \mathbb F_2^n,\ d\in \mathbb F_2\}.
\]
A central criterion used throughout the literature is Dillon’s characterization: a bent function lies in \(\mathcal M^\#\) if and only if it has an \(\mathcal M\)-subspace of dimension \(m\), meaning a vector subspace \(V\subseteq \mathbb F_2^{2m}\) such that
\[
D_aD_bf=0,\qquad \forall a,b\in V
\]
[2508.14265].

Almost Maiorana–McFarland bent functions replace the classical \(m\mid m\) split by the generalized \(m-1\mid m+1\) split. In the broader generalized family one considers
\[
f(x,y)=x\cdot \phi(y)+h(y),\qquad (x,y)\in \mathbb F_2^{m-k}\times \mathbb F_2^{m+k},
\]
and the 2025 paper isolates \(k=1\) as the first nontrivial case. The resulting class \(\mathcal{GMM}_{m+1}\) is therefore the precise setting in which the phrase “almost Maiorana–McFarland bent functions” is used [2508.14265].

## 2. Exact bentness criterion and the dual

The defining structural theorem is a complete characterization of bentness for
\[
f(x,y)=x\cdot \phi(y)+h(y), \qquad x\in \mathbb F_2^{m-1},\ y\in \mathbb F_2^{m+1}.
\]
Such a function is bent if and only if two conditions hold. First, the fibers
\[
\{\phi^{-1}(a)\mid a\in \mathbb F_2^{m-1}\}
\]
form a partition of \(\mathbb F_2^{m+1}\) into \(2\)-dimensional affine subspaces. Second, for every \(a\in \mathbb F_2^{m-1}\), the restriction of \(h\) to \(\phi^{-1}(a)\) has odd weight [2508.14265].

Because
\[
\frac{|\mathbb F_2^{m+1}|}{|\mathbb F_2^{m-1}|}=4,
\]
bentness forces \(\phi\) to be \(4\)-to-\(1\). Each fiber therefore has cardinality \(4\), and the theorem strengthens this to the statement that every fiber must be an affine plane
\[
u+U,\qquad \dim U=2.
\]
Equivalently, if
\[
\phi^{-1}(a)=\{y_1,y_2,y_3,y_4\},
\]
then the fiber is an affine \(2\)-flat exactly when
\[
y_1\oplus y_2\oplus y_3\oplus y_4=0_{m+1}.
\]
On each such fiber the odd-weight requirement means that \(h|_{\phi^{-1}(a)}\) has weight \(1\) or \(3\). Consequently,
\[
2^{m-1}\le \operatorname{wt}(h)\le 3\cdot 2^{m-1}.
\]

The Walsh analysis is fiberwise. For \((a,b)\in \mathbb F_2^{m-1}\times \mathbb F_2^{m+1}\),
\[
W_f(a,b)=2^{m-1}\sum_{y\in \phi^{-1}(a)}(-1)^{h(y)+y\cdot b}.
\]
Thus bentness is equivalent to requiring that for every fiber and every \(b\), the corresponding \(4\)-term sign sum is \(\pm 2\). This is what forces both the affine-plane geometry of the fibers and the odd-weight condition on \(h\).

The same paper gives an explicit dual formula. Fix any total order \(\prec\) on \(\mathbb F_2^{m+1}\). If \(f\) is bent, then for all \(a\in \mathbb F_2^{m-1}\) and \(b\in \mathbb F_2^{m+1}\),
\[
f^*(a,b)= \sum_{\substack{y,y' \in \phi^{-1}(a)\\ y \precneqq y'}}(h(y)+y \cdot b)(h(y')+y' \cdot b).
\]
The dual is therefore determined fiberwise by the affine planes \(\phi^{-1}(a)\) and the restriction of \(h\) to those planes [2508.14265].

## 3. Partition geometry and membership in \(\mathcal M^\#\)

The geometry of the partition induced by \(\phi\) 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” [2508.14265].

A sufficient condition for inclusion in \(\mathcal M^\#\) is the existence of a nonzero direction common to all fiber-planes in a precise sense: there exists nonzero \(v\in \mathbb F_2^{m+1}\) such that for every fiber \(\phi^{-1}(z)\), one can choose \(w_z\in \phi^{-1}(z)\) with
\[
v\in w_z+\phi^{-1}(z).
\]
When this happens, the function belongs to \(\mathcal M^\#\). The proof constructs an \(m\)-dimensional \(\mathcal M\)-subspace
\[
V=\langle \mathbb F_2^{m-1}\times \{0_{m+1}\},\ (0_{m-1},v)\rangle.
\]

A particularly transparent case is that of trivial partitions. If
\[
\{\phi^{-1}(a)\mid a\in \mathbb F_2^{m-1}\}=\{c+W\mid c\in Q\}
\]
for a fixed \(2\)-dimensional linear subspace \(W\), then the resulting bent function has at least three different \(m\)-dimensional \(\mathcal M\)-subspaces and therefore lies in \(\mathcal M^\#\).

The converse is false. The same paper exhibits a bent function in \(\mathcal{GMM}_{m+1}\) whose partition does not satisfy the common-direction condition, yet the function still belongs to \(\mathcal M^\#\), because after a variable permutation it becomes a strict MM function. Thus the partition criterion is sufficient rather than necessary.

For exclusion from \(\mathcal M^\#\), the paper introduces an extended derivative property
\[
(P_1^*):\qquad D_vD_w\phi\neq 0_{m-k}\quad\text{for all linearly independent }v,w\in \mathbb F_2^{m+k}.
\]
For the associated generalized-MM function
\[
f(x,y)=x\cdot \phi(y)+h(y),
\]
Theorem 5.3 proves that any \(\mathcal M\)-subspace has dimension at most \(m-k\), and under an additional condition on \(D_{b_2}\phi\), the canonical subspace
\[
\mathbb F_2^{m-k}\times \{0_{m+k}\}
\]
is the unique maximal \(\mathcal M\)-subspace. In particular, when \(k\ge 1\) and \(f\) is bent, these conditions imply \(f\notin \mathcal M^\#\) [2508.14265].

## 4. Construction principles

Once a suitable partition into affine planes is fixed, the choice of \(h\) is simple. On each \(4\)-point fiber \(\phi^{-1}(a)\), the restriction of \(h\) must have odd weight, so one may choose either a unique \(1\) or a unique \(0\) on that fiber. The paper counts \(8\) valid restrictions per fiber and states that for fixed \(\phi\) the number of admissible Boolean functions \(h\) is exactly
\[
8^{2^{m-1}}
\]
[2508.14265].

Several partition-construction methods are described. One is a product method, obtained by writing
\[
\mathbb F_2^{m+1}\cong \mathbb F_2^k\times \mathbb F_2^t
\]
and taking products of partitions into \(1\)-dimensional affine subspaces. Another is a clique method: for small \(n\), one forms the graph whose vertices are the \(2\)-dimensional affine subspaces of \(\mathbb F_2^n\), with adjacency defined by disjointness; a partition then corresponds to a clique of size \(2^{n-2}\).

The paper also gives a direct recursive algorithm. One begins with
\[
A_1=\{0_{m+1},a_0,b_0,a_0+b_0\},
\]
a \(2\)-dimensional linear subspace. Then one chooses \(a_1,b_1,c_1\notin A_1\) and defines an affine \(2\)-flat
\[
A_2=a_1+\{0_{m+1},b_1+a_1,c_1+a_1,b_1+c_1\}.
\]
Recursively, after \(A_1,\dots,A_i\) have been chosen, one selects
\[
a_i,b_i,c_i\in \mathbb F_2^{m+1}\setminus \bigcup_{j=1}^i A_j
\]
and sets
\[
A_{i+1}=\{a_i,b_i,c_i,a_i+b_i+c_i\}.
\]
When \(2^{m-1}\) disjoint affine \(2\)-flats have been obtained, \(\phi\) is defined by assigning a distinct value of \(\mathbb F_2^{m-1}\) to each \(A_i\) [2508.14265].

## 5. The \(8\)-variable case and concatenation phenomena

The case \(2m=8\) is treated explicitly. Then
\[
x\in \mathbb F_2^3,\qquad y\in \mathbb F_2^5,\qquad \phi:\mathbb F_2^5\to \mathbb F_2^3.
\]
The paper gives a concrete partition of \(\mathbb F_2^5\) into eight affine planes \(A_0,\dots,A_7\), uses it to define \(\phi\), and shows that the associated function
\[
f(x,y)=x\cdot \phi(y)
\]
is bent and has a unique \(3\)-dimensional \(\mathcal M\)-subspace, namely the canonical one
\[
\mathbb F_2^3\times\{0_5\},
\]
so that \(f\in \mathcal{GMM}_5\setminus \mathcal M^\#\) [2508.14265].

Using the simple partition algorithm, the authors found \(4960\) decompositions of \(\mathbb F_2^5\) into disjoint \(2\)-dimensional affine subspaces, of which \(3785\) are proper. For each such proper partition \(\Pi=\{A_1,\dots,A_8\}\), they defined \(\phi\) by \(\phi(A_i)=i-1\), chose a random admissible \(h\), and obtained bent functions. All \(3785\) tested bent functions were outside \(\mathcal M^\#\). Among them, at least \(588\) were verified to be EA-inequivalent, which yields at least
\[
2^{78}
\]
distinct bent functions on \(\mathbb F_2^8\) outside \(\mathcal M^\#\). The same paper notes that the total number of \(8\)-variable bent functions in \(\mathcal M^\#\) is approximately
\[
2^{77},
\]
so this construction alone produces more examples outside \(\mathcal M^\#\) than the entire completed MM class in dimension \(8\).

The paper also develops a concatenation theory specialized to almost MM functions with a common \(\phi\). If
\[
f_i(x,y)=x\cdot \phi(y)+h_i(y),
\]
with the same affine-plane partition for all four components, then the concatenation
\[
f=f_1\|\|f_2\|\|f_3\|\|f_4
\]
is bent if and only if, for every \(a\in \mathbb F_2^{m-1}\),
\[
\sum_{i=1}^4 \sum_{\substack{y,y' \in \phi^{-1}(a)\\ y \precneqq y'}}h_i(y)h_i(y')=1,
\]
and either
\[
\sum_{i=1}^4 h_i(y)=0 \quad\text{for all } y\in \phi^{-1}(a),
\]
or
\[
\sum_{i=1}^4 h_i(y)=1 \quad\text{for all } y\in \phi^{-1}(a).
\]

Two consequences are especially notable. First, given one bent \(f_1\in \mathcal{GMM}_{m+1}\), there exist bent \(f_2,f_3,f_4\) such that
\[
f=f_1\|\|f_2\|\|f_3\|\|f_4
\]
lies in \(\mathcal M^\#\), even when \(f_1\notin \mathcal M^\#\). This answers the open problem of Kudin et al. by showing that a bent function in \(\mathcal M^\#\) can have all four \(2m\)-variable restrictions outside \(\mathcal M^\#\). Second, the converse phenomenon also occurs: starting with MM bent functions
\[
q(x,y)=x\cdot \pi(y)
\]
where \(\pi\) has property
\[
D_vD_w\pi\neq 0_m\quad \text{for all linearly independent }v,w\in \mathbb F_2^m,
\]
and applying a suitable linear permutation \(A\), one gets
\[
g=q\circ A
\]
such that
\[
f=q\|\|g\|\|q\|\|(g+1)
\]
is bent and lies outside \(\mathcal M^\#\). After a suitable linear permutation \(B\), this function can again be represented in a generalized MM form in \(\mathcal{GMM}_{(2m+2)/2+1}\) [2508.14265].

## 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 \((K)\) 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 \(n_1=n_2=n/2\), and its asymptotically optimal improvement occurs in the nearest nontrivial case \(n_2-n_1=2\), making it a precise near-MM construction even though the phrase is not used [2108.00232].

A different broad sense appears in the study of quadratic symmetric bent functions: for even \(m\), the functions
\[
f_{i_1,i_2}(x)=1 \iff \operatorname{wt}(x)\equiv i_1 \text{ or } i_2 \pmod 4
\]
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 [1211.5257].

Concatenation theory supplies another route. Four MM bent functions
\[
f_i(x,y)=x\cdot \pi_i(y)+h_i(y)
\]
can be concatenated into a bent function
\[
f=f_1\|\|f_2\|\|f_3\|\|f_4
\]
via the dual bent condition
\[
f_1^*+f_2^*+f_3^*+f_4^*=1.
\]
Under additional hypotheses on permutations with the \((\mathcal A_m)\) property, the resulting function can lie outside the completed MM class \(\mathcal M^\#\), even though each quarter of the truth table is MM [2310.10162]. Complementarily, complete necessary-and-sufficient criteria for when bent concatenations
\[
f=f_1\|\|f_2 \quad\text{or}\quad f=f_1\|\|f_2\|\|f_3\|\|f_4
\]
do not belong to \(\mathcal M^\#\) are given in terms of the \(M\)-subspaces of the components; in the special family
\[
f=g\|\|h\|\|g\|\|(h+1),
\]
membership in \(\mathcal M^\#\) is equivalent to \(g\) and \(h\) having a common \(n/2\)-dimensional \(M\)-subspace [2404.16220].

More recent work pushes the MM boundary in two opposite directions. One line studies permutations satisfying \((P_1)\) or \((P_2)\), which control uniqueness of the maximal MM-type subspace in functions
\[
f(x,y)=x\cdot \pi(y)+h(y).
\]
These are then used to construct large families outside \(\mathcal M^\#\), as well as \(\ell\)-optimal bent functions with the smallest possible linearity index, described as opposite to MM bent functions [2508.14277]. Another line formulates generalized Maiorana–McFarland functions
\[
f(x,y,z)=f^{(z)}(x)+\,_1^k(yz)
\]
and proves that suitable choices of the bent slices \(f^{(z)}\), often themselves classical MM functions, yield generalized MM bent functions outside both the classical \(MM\) and \(\mathcal{PS}_{ap}\) classes and hence outside their completed classes [2603.28485].

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 \(\mathbb F_2^{m+1}\), odd restrictions of \(h\) on those planes, and a partition geometry that determines whether the resulting bent function remains in \(\mathcal M^\#\) or escapes it [2508.14265].

Source: https://www.emergentmind.com/topics/almost-maiorana-mcfarland-bent-functions