---
title: Completed Maiorana–McFarland Class
url: https://www.emergentmind.com/topics/completed-maiorana-mcfarland-class
type: topic
---

# Completed Maiorana–McFarland Class

The completed Maiorana–McFarland class is the EA-invariant enlargement of one of the central primary bent-function constructions. In the Boolean setting on \(n=2m\) variables, the classical Maiorana–McFarland class consists of functions
\[
f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,
\]
with \(\pi\) a permutation of \(\mathbb F_2^m\) and \(h\) an arbitrary Boolean function; the completed class, usually written \(\mathcal M^\#\), is the smallest EA-invariant class containing \(\mathcal M\) [2508.14277]. A standard equivalent characterization is Dillon’s second-derivative criterion: a bent \(f\in\mathcal B_{2m}\) belongs to \(\mathcal M^\#\) if and only if there exists an \(m\)-dimensional linear subspace \(V\subseteq \mathbb F_2^{2m}\) such that
\[
D_aD_bf(x)=0\qquad \text{for all }a,b\in V,\ \text{all }x\in \mathbb F_2^{2m}.
\]
In later literature the same phrase also appears in \(p\)-ary and vectorial settings, sometimes as an EA-completed class and sometimes as a secondary “completed/generalized” MM construction; the distinction is substantive rather than terminological [1508.05672].

## 1. Classical definition and EA-completion

In the strict Boolean form, the Maiorana–McFarland class is
\[
\mathcal M=\Bigl\{\,f(x,y)=x\cdot \pi(y)+h(y): x,y\in\mathbb F_2^m,\ \pi \text{ a permutation of }\mathbb F_2^m\,\Bigr\}.
\]
Its completed version is the extended-affine closure
\[
\mathcal M^\# =\{f(xA+b)+c\cdot x+d:\ f\in\mathcal M,\ A\in GL(n,\mathbb F_2),\ b,c\in\mathbb F_2^n,\ d\in\mathbb F_2\}.
\]
Thus \(\mathcal M^\#\) is not another explicit normal form; it is the equivalence-stable class generated by \(\mathcal M\) [2508.14265].

This distinction becomes important as soon as one leaves the balanced \((m,m)\) split. A recent “almost Maiorana–McFarland” family studies
\[
f(x,y)=x\cdot \phi(y)+h(y),\qquad x\in\mathbb F_2^{m-1},\ y\in\mathbb F_2^{m+1},
\]
as a strict subfamily of the generalized Maiorana–McFarland class \(\mathcal{GMM}_{m+1}\). These functions retain the MM pattern of affine restrictions in one block of variables, but they are not members of the strict MM class \(\mathcal M\) unless additional structure is present [2508.14265].

The completed class therefore plays two roles at once. It is an equivalence-stable replacement for the rigid explicit template \(x\cdot \pi(y)+h(y)\), and it is the benchmark against which broader MM-like constructions are tested. Much of the recent literature is concerned less with constructing members of \(\mathcal M\) than with deciding whether a bent function that “looks MM-like” is actually inside \(\mathcal M^\#\).

## 2. Derivative characterization and \(\mathcal M\)-subspaces

The derivative formulation of \(\mathcal M^\#\) is encoded by \(\mathcal M\)-subspaces. For a Boolean function \(f\in B_n\), the first derivative is
\[
D_af(x)=f(x+a)+f(x),
\]
and the second derivative is
\[
D_aD_bf(x)=f(x)+f(x+a)+f(x+b)+f(x+a+b).
\]
A vector subspace \(V\subseteq \mathbb F_2^n\) is an \(\mathcal M\)-subspace of \(f\) if
\[
D_aD_bf=0\qquad \text{for all }a,b\in V.
\]
If \(n=2m\), Dillon’s criterion becomes:
\[
f\in \mathcal{MM}^\# \iff \exists\,V\le \mathbb F_2^n,\ \dim(V)=m,\ \forall a,b\in V:\ D_aD_bf=0.
\]
For an MM function
\[
f(x,y)=x\cdot T(y)+h(y),
\]
the canonical example is the \(m\)-dimensional subspace
\[
\mathbb F_2^m\times\{0_m\},
\]
which is always an \(\mathcal M\)-subspace [2304.13432].

This viewpoint supplies an EA-invariant. The number of \(\mathcal M\)-subspaces of a Boolean function is invariant under equivalence, so two bent functions with different numbers of \(\mathcal M\)-subspaces are inequivalent. This turns second-order derivatives into a classification device, not merely a membership test [2304.13432].

A plausible implication is that \(\mathcal M^\#\) is best viewed geometrically. What matters is not only the explicit form \(x\cdot \pi(y)+h(y)\), but the existence, dimension, and arrangement of vanishing second-derivative subspaces. That is exactly why the same class reappears in concatenation arguments, exclusion criteria, and low-linearity constructions.

## 3. Internal structure of \(\mathcal M^\#\): uniqueness, \(P_1\), \(P_2\), and linearity index

A central refinement concerns how many maximal \(\mathcal M\)-subspaces an MM bent function possesses. For
\[
f(x,y)=x\cdot \pi(y)+h(y),
\]
every MM function has the canonical \(m\)-dimensional \(\mathcal M\)-subspace
\[
\mathbb F_2^m\times\{0_m\},
\]
but this need not be unique. The literature isolates two permutation properties controlling uniqueness. The stronger condition is
\[
D_vD_w\pi\neq 0_m \qquad \text{for all linearly independent } v,w\in F_2^m. \tag{P_1}
\]
If \(\pi\) satisfies \((P_1)\), then \(\pi\) has no linear structures and \(F_2^m\times\{0_m\}\) is the only \(m\)-dimensional \(\mathcal M\)-subspace of \(f\). A weaker condition \((P_2)\) is defined through subspaces \(S\le F_2^m\) with
\[
D_aD_b\pi=0_m \qquad \text{for all } a,b\in S,
\]
and excludes the existence of a complementary dual collapse
\[
v\cdot D_a\pi(y)=0 \qquad \text{for all } a\in S,\ y\in F_2^m,\ v\in V.
\]
For non-affine \(\pi\), \((P_2)\) is equivalent to uniqueness of the maximal \(\mathcal M\)-subspace for \(f(x,y)=x\cdot \pi(y)\). The paper also gives the equivalent reformulation
\[
\dim(V_S(\pi))>\dim(S),
\qquad
V_S(\pi)=\left\langle D_a\pi(y): a\in S,\ y\in F_2^m\right\rangle,
\]
and proves
\[
(P_1)\implies (P_2),
\]
with strict implication: among the 75 equivalence classes of quadratic permutations on \(F_2^5\), 34 satisfy \((P_2)\), while only 2 satisfy \((P_1)\) [2508.14277].

The same paper organizes this geometry by the linearity index
\[
\operatorname{ind}(f)=\max_{U\in \mathcal{MS}(f)} \dim(U).
\]
For bent functions on \(2m\) variables,
\[
1\le \operatorname{ind}(f)\le m,
\]
and
\[
\operatorname{ind}(f)=m \quad \Longleftrightarrow \quad f\in \mathcal M^\#.
\]
This makes \(\mathcal M^\#\) the maximal-linearity extreme. The opposite extreme is the class of \(\ell\)-optimal bent functions, defined by
\[
\operatorname{ind}(f)=1.
\]
These are described as opposite to Maiorana–McFarland bent functions because MM and \(\mathcal M^\#\) functions have maximal possible linearity index \(m\), whereas \(\ell\)-optimal functions have the minimum possible index [2508.14277].

## 4. Generalized and almost MM forms at the boundary of \(\mathcal M^\#\)

The almost-MM family
\[
f(x,y)=x\cdot \phi(y)+h(y),\qquad x\in\mathbb F_2^{m-1},\ y\in\mathbb F_2^{m+1},
\]
gives a precise test case for how far MM flavor can be pushed before one exits \(\mathcal M^\#\). Its bentness is completely characterized: \(f\) is bent if and only if the fibers
\[
\phi^{-1}(a),\qquad a\in\mathbb F_2^{m-1},
\]
form a partition of \(\mathbb F_2^{m+1}\) into \(2\)-dimensional affine subspaces, and the restriction of \(h\) to each fiber has odd Hamming weight, equivalently weight \(1\) or \(3\). For a fixed \(\phi\) with this partition property, the number of valid \(h\) is
\[
8^{2^{m-1}}=2^{3\cdot 2^{m-1}}.
\]
The dual is also determined fiberwise [2508.14265].

Membership in \(\mathcal M^\#\) is subtler. A sufficient condition is the existence of a nonzero \(v\in \mathbb F_2^{m+1}\) such that every affine plane \(\phi^{-1}(z)\) contains the common direction \(v\); then
\[
V=\langle \mathbb F_2^{m-1}\times\{0\},\ (0,v)\rangle
\]
is an \(m\)-dimensional \(\mathcal M\)-subspace, so \(f\in\mathcal M^\#\). Partitions with such a common direction are called non-proper, and those without it are called proper. The paper proves that non-proper partitions force \(f\in\mathcal M^\#\), but the converse fails: the condition is sufficient, not necessary [2508.14265].

The same work also identifies when an almost-MM bent function is actually in the strict MM class. A bent \(f\in\mathcal{GMM}_{m+1}\) belongs to \(\mathcal M\) if and only if it can be written in the almost-MM form with \(\phi\) depending only on the last \(m\) variables and
\[
h'(y):=h(0,y)+h(1,y)
\]
balanced in such a way that \((\phi_1(y),\dots,\phi_{m-1}(y),h'(y))\) extends to a permutation of \(\mathbb F_2^m\) [2508.14265].

The 8-variable case is especially explicit. The paper found exactly \(4960\) decompositions of \(\mathbb F_2^5\) into \(2\)-dimensional affine subspaces; among them \(3785\) are proper. For these \(3785\) proper partitions, all obtained bent functions were reported to be outside \(\mathcal M^\#\). The resulting lower bound is at least
\[
2^{78}
\]
distinct 8-variable bent functions outside \(\mathcal M^\#\), compared with an estimated total of approximately
\[
2^{77}
\]
8-variable bent functions inside \(\mathcal M^\#\) [2508.14265].

## 5. Outside \(\mathcal M^\#\): concatenation, cubic counterexamples, and asymptotic limits

The most systematic Boolean exclusion machinery uses \(\mathcal M\)-subspaces. For bent 4-concatenation
\[
f=f_1\|f_2\|f_3\|f_4,
\]
if \(f\in \mathcal{MM}^\#\), then there must exist a common \((n/2-1)\)-dimensional \(\mathcal M\)-subspace in the four constituents. This yields generic non-membership criteria. In particular, suitably chosen MM components can be concatenated to produce bent functions outside \(\mathcal{MM}^\#\) for any even \(n\ge 8\); explicit 8-variable examples constructed this way are also outside \(\mathcal{PS}^\#\) [2304.13432].

A complementary MM-based route uses the dual bent condition. For MM components
\[
f_i(x,y)=\operatorname{Tr}(x\pi_i(y))+h_i(y),
\]
if the permutations satisfy the \((\mathcal A_m)\) property and the offsets satisfy
\[
h_1(\pi_1^{-1}(x))+h_2(\pi_2^{-1}(x))+h_3(\pi_3^{-1}(x))
+h_4((\pi_1+\pi_2+\pi_3)^{-1}(x))=1,
\]
then the 4-concatenation is bent. Under generic hypotheses including property \((P_1)\) for \(\pi_1,\pi_2,\pi_3\) and \(\pi_1+\pi_2\), together with the absence of linear structures in the components of \(\pi_1+\pi_2\), the resulting concatenation is outside \(\mathcal M^\#\) [2310.10162].

The completed MM class is also known not to exhaust cubic bent functions. All cubic bent functions in 6 and 8 variables belong to \(\mathcal M^\#\), but this fails in larger even dimensions. There exist cubic bent functions outside \(\mathcal M^\#\) for all even \(n>10\); moreover, homogeneous cubic bent functions outside \(\mathcal M^\#\) exist for all \(n>26\), and homogeneous cubic bent functions without affine derivatives exist outside \(\mathcal M^\#\) for all \(n>50\) [1908.11271].

Not every proposed route away from MM is genuinely productive. A corrective note on Rothaus-based iteration proved that the condition
\[
AB+AC+BC=A+B+C
\]
has only the trivial solution
\[
A=B=C,
\]
so the iterated Rothaus construction collapses to
\[
f(x,x_{n+1},x_{n+2})=A(x)+x_{n+1}x_{n+2}.
\]
The paper concludes that this iterative Rothaus mechanism contributes no genuinely new bent functions and does not provide a nontrivial route beyond MM or its completion [2502.10192].

From a counting perspective, the completed MM family is asymptotically small relative to broader MM-inspired generalizations. A counting paper estimates the leading scale of the completed MM family by
\[
2\cdot 2^{n/2}(1+o(1))
\]
in \(\log_2\)-asymptotics, while a modification of the MM family based on ordered partitions of \(\mathbb F_2^{n_2}\) into affine \(2\)-subspaces yields
\[
\log_2 b_n \ge \frac{3n}{2}\,2^{n/2} - 2\log_2 e\,2^{n/2}+o(2^{n/2}).
\]
That paper does not identify its construction with \(\mathcal M^\#\), but it does show that affine-subspace generalizations of the MM philosophy have much larger asymptotic growth than the classical completed class [2108.00232].

## 6. \(p\)-ary, vectorial, and alternate usages of “completed MM”

Outside the binary Boolean setting, the phrase “completed Maiorana–McFarland class” is used in several adjacent but nonidentical ways. The following summary captures the main strands.

| Strand | Representative form | Relation to completed MM |
|---|---|---|
| \(p\)-ary EA-completed class | \(f(x,y)=x\cdot \pi(y)+g(y)\) | Completed class = all EA-equivalent functions [2507.20715] |
| Weakly regular \(p\)-ary completion principle | MM base plus \(F(\operatorname{Tr}(u_1x),\dots,\operatorname{Tr}(u_\tau x))\) | “Completed/generalized MM” means trace-augmentation of an MM base [1508.05672] |
| Vectorial \(p\)-ary secondary construction | \(F(x,y)=xT(y)+g(y)\) | MM supplies the core \(G\) satisfying \((P_U)\) [2211.11516] |
| Vectorial binary maximal bent components | MM-derived components with derivative test | Some resulting components are outside the complete MM class [2301.02843] |

In the ternary setting, one paper defines completion exactly as EA-closure: a class is complete if it is a union of EA-equivalence classes, and the completed class of \(\mathcal M\) is the smallest complete class containing it. It then proves a derivative criterion: if there exists an \(n/2\)-dimensional subspace \(V\) such that
\[
D_{c,d}f(x)=0\quad \forall c,d\in V
\]
and every nonzero first derivative \(D_cf(x)\) with \(c\in V\) is balanced, then \(f\) is a regular bent function in the completed MM class. This criterion is used to construct ternary degree-4 binomial and trinomial families inside the completed MM class [2507.20715].

A different use of “completed/generalized MM” occurs in the \(p\)-ary weakly regular bent literature. Starting from the MM base
\[
g(x,y)=\operatorname{Tr}_1^k(x\pi(y))+\operatorname{Tr}_1^k(by),
\]
with \(\pi\) a linearized permutation polynomial, the paper adds an arbitrary reduced polynomial in trace coordinates:
\[
f(x,y)=\operatorname{Tr}_1^k(x\pi(y))+\operatorname{Tr}_1^k(by)
+F\Big(\operatorname{Tr}_1^k(u_1^{(1)}x+u_1^{(2)}y),\dots,\operatorname{Tr}_1^k(u_\tau^{(1)}x+u_\tau^{(2)}y)\Big).
\]
If the trace-orthogonality condition
\[
\operatorname{Tr}_1^k\!\left( u_i^{(2)}\pi^{-1}(u_j^{(1)})+u_j^{(2)}\pi^{-1}(u_i^{(1)}) \right)=0,\qquad 1\le i\le j\le \tau,
\]
holds, then \(f\) remains weakly regular bent. Here “completion” refers to a secondary trace-polynomial augmentation of an MM bent function, not to an EA-closure [1508.05672].

In vectorial \(p\)-ary work, the classical MM family
\[
F(x,y)=xT(y)+g(y),\qquad x,y\in\mathbb F_{p^m},
\]
serves as the base object in a \((P_U)\)-driven secondary construction. If the component duals satisfy
\[
G_\lambda^\ast\!\left(x+\sum_{i=1}^t u_iw_i\right)=G_\lambda^\ast(x)+\sum_{i=1}^t w_i g_i(x),
\]
then
\[
F=G+H
\]
with trace-coordinate perturbation \(H\) is again vectorial \(p\)-ary weakly regular bent; this yields new infinite families from the \(p\)-ary MM class without defining a separate completed class [2211.11516].

Finally, in the vectorial binary setting of maximal bent components, the phrase “complete Maiorana–McFarland class” is tied to a second-order derivative condition. Starting from MM bent components
\[
G_{a,b}(y,z)=\operatorname{Tr}_{2^m/2}(ay\phi(z)+bz),
\]
a secondary construction produces new vectorial functions with \(2^n-2^m\) bent components. For the explicit family
\[
F(y,z) = (y\phi(z),z) + (u_{1,1}z,0)\operatorname{Tr}_{2^m/2}(u_{2,1}y+u_{2,2}z),
\]
the paper computes
\[
D_{(w_1,0)}D_{(w_2,0)}F_{a,b}(y,z)
\]
and shows that the complete-MM derivative criterion fails, so the constructed components are outside the complete Maiorana–McFarland class [2301.02843].

Taken together, these strands show that “completed Maiorana–McFarland class” has a stable Boolean-EA meaning, but its extensions split into at least three non-equivalent ideas: EA-completion, trace-coordinate completion of a fixed MM core, and derivative-based tests for inclusion or exclusion in vectorial and \(p\)-ary settings.

Source: https://www.emergentmind.com/topics/completed-maiorana-mcfarland-class