Rank-Two Frobenius-Linearized Normal Forms and Orthoderivative Dual Coordinates in Quadratic APN Maps

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Rank-Two Frobenius-Linearized Normal Forms and Orthoderivative Dual Coordinates in Quadratic APN Maps Jingchuan Ma, Yanhua Liu, and Qiaoyun Huang

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arXiv:2608.11939v1 [cs.CR] 12 Aug 2026

Abstract We classify binary-linear two-term Frobenius-linearized operators L(Y ) = AY σ + BY on K 3 , where K is a finite extension of F2 and σ is a fixed nontrivial Frobenius automorphism of K with fixed field F2 . Under a coefficient-rank and binary-kernel condition, if A and B both have K-rank two and L has a one-dimensional kernel over F2 , then invertible K-linear input and output changes reduce L, for this fixed σ, to the single canonical model (α, β, γ) 7→ (ασ + α, β σ , γ). The proof constructs the coordinate frames from the two coefficient-kernel directions and the binary kernel. In these coordinates, the first dual output row is exactly the unique nonzero trace-adjoint normal, with an exact K-valued normalization. For pure σ-quadratic almost perfect nonlinear maps, this identifies the orthoderivative by πF (X)T F (X) = 1; in odd extension degree it also yields permutation behavior and a bijection from the projective plane to its dual. The triprojective construction of Göloğlu and Kölsch and the cubic norm-twist construction of Li, Zhou, Li, and Qu provide two realizations arising from different algebraic constructions. The triprojective case further admits a determinant factorization and a complete dual frame, whereas the norm-twist realization shows that the pure-map consequences do not follow from the operator theorem alone. A natural Gold representation has coefficient-rank pair (3, 3), delimiting the rank-two subclass. The normal form also supplies exact extension-field labels for known component-radical and Walsh-support relations. Index Terms APN functions, finite fields, orthoderivatives, linearized operators, normal forms

I. I NTRODUCTION UADRATIC almost perfect nonlinear (APN) maps carry two geometries that are usually described at different levels. Binary polarization gives, for each nonzero direction X, a binary-linear map DX F with kernel F2 X and image a binary trace hyperplane. Its unique nonzero trace normal is the orthoderivative πF (X), an invariant that also controls radicals of Boolean components and their Walsh support [1], [2]. Many explicit APN constructions, however, are written on a lowerdimensional vector space over an extension field K. In such presentations the same derivative may split into a σ-semilinear term and a K-linear term. The binary trace-hyperplane description alone does not determine the K-kernel geometry of these two coefficient matrices, nor does it explain when that extension-field geometry forces a canonical normal form and an exact K-valued dual coordinate. This paper resolves that interaction for a three-coordinate rank-defect pattern arising naturally in the APN constructions studied below. Let K = F2m , fix a nontrivial Frobenius automorphism σ with fixed field F2 , and consider

Q

L(Y ) = AY σ + BY,

Y ∈ K 3.

(1)

dimF2 ker L = 1.

(2)

Assume rankK A = rankK B = 2,

The central result is a normal-form classification: every operator satisfying (2) is, up to invertible K-linear changes of input and output coordinates, equivalent to Nσ (α, β, γ) = (ασ + α, β σ , γ). (3) Conversely, every left-right K-linear transform of Nσ has coefficient-rank pair (2, 2) and binary nullity one. Thus the rank condition is not merely a convenient way to construct moving frames: together with the binary kernel condition it characterizes a single left-right equivalence class of two-term Frobenius-linearized operators for this fixed σ. Jingchuan Ma is with the Department of Computer Engineering, Fuzhou University Zhicheng College (e-mail: kiciot@qq.com; ORCID: 0009-0001-3703471X). Yanhua Liu is with Fuzhou University Zhicheng College (e-mail: Lyhwa@fzu.edu.cn; ORCID: 0000-0002-6076-9968). Qiaoyun Huang is with Fuzhou University Zhicheng College (e-mail: huangqy@fdzcxy.edu.cn; ORCID: 0009-0008-2122-7307). Qiaoyun Huang is the corresponding author. This research received no external funding.

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The proof exposes the mechanism behind the classification. If X generates the binary kernel, R generates kerK B, and S σ generates kerK A, then X, R, S are forced to be a K-basis. Their images produce a second K-basis, giving matrices E, J ∈ GL(3, K) such that J −1 ◦ L ◦ E = Nσ . (4) The first row of J −1 is exactly the unique nonzero trace-adjoint normal to the binary image of L, and its pairing with the first output-frame vector is one in K. This is the bridge between the two geometries: the binary image remains the trace hyperplane {Z : Tr(nT Z) = 0}, whereas the same vector n defines the K-linear covector Z 7→ nT Z and hence a point [n] of the dual K-projective plane. These objects are related but should not be conflated: the derivative image is a binary trace hyperplane, not a K-projective line. For a pure σ-quadratic map F : K 3 → K 3 , polarization has the form (1) in every direction and the two coefficient halves both evaluate to F (X) at the derivative direction. Therefore, if F is APN and every nonzero derivative has coefficient-rank pair (2, 2), the normal form identifies the orthoderivative by the exact field identity πF (X)T F (X) = 1

in K.

(5)

When m is odd, its trace is one. This places F (X) outside the binary derivative image, yields a direct permutation proof, and combines with pure semiquadratic homogeneity and the standard odd-dimensional orthoderivative bijection to give [X] 7−→ [πF (X)]

(6)

as a bijection from PG(2, K) to its dual plane. Here πF (X) is first the binary trace normal in the derivative-image equation; its K × -class simultaneously defines the projective dual coordinate. Two APN constructions show that the operator class has content beyond a single family or a reverse-engineered definition. The triprojective family of Göloğlu and Kölsch is pure σ-quadratic and realizes the full pure specialization [3]. Explicit adjugate factorizations verify the rank-(2, 2) condition, and additional family-specific contractions give det JX = (det EX )σ+1

(7)

−1 and all three rows of JX . By contrast, the Li–Zhou–Li–Qu cubic norm-twist construction arises from a different extension-field

mechanism [4]. Its derivative again has coefficient-rank pair (2, 2) and hence the same abstract normal form, but its first outputframe vector is b(HX, X) rather than F (X). Consequently the theorem yields πF (X)T b(HX, X) = 1, not automatically (5). The second realization therefore separates the general operator theorem from the stronger pure-map corollary instead of merely duplicating it. Admissible parameters for the relevant norm-twist subfamily exist in every stated extension degree by [5]. The coefficient-rank hypothesis is also a genuine structural separator, not an automatic consequence of the APN property. In the natural cubic-extension presentation of a Gold map, both derivative coefficient operators are invertible, giving rank pair (3, 3) rather than (2, 2). Thus the two positive realizations and the Gold boundary place the normal-form theorem inside a proper representation-dependent subclass of quadratic APN constructions. On the Fourier side, no new abstract support mechanism is claimed. The classical derivative-incidence theory already determines when a Walsh coefficient can be nonzero and gives the known almost-bent amplitude in odd dimension [1, Sec. 4.2 and Th. 13(v)]. What the normal form contributes is the exact extension-field label of the relevant binary trace hyperplane: for the pure class the component indexed by πF (X) has radical F2 X, and WF (U, πF (X)) ̸= 0

⇐⇒

Tr(U T X) = 1.

(8)

The contributions are organized in three layers. 1) General theory: we classify all operators AY σ + BY on K 3 with coefficient-rank pair (2, 2) and binary nullity one up to left-right K-linear equivalence, and identify the first dual coordinate as the exactly normalized trace-adjoint normal. 2) APN consequences and realizations: for pure σ-quadratic APN maps the classification gives (5), odd-degree permutation behavior, and an orthoderivative dual-coordinate bijection. The Göloğlu–Kölsch and Li–Zhou–Li–Qu constructions realize the general theorem through two distinct algebraic mechanisms, with the latter marking the boundary of the pure corollary. 3) Sharper specialization and boundaries: the triprojective family admits explicit adjugates, the determinant identity (7), and a complete dual frame; the same exact normal gives the extension-field label for known radical/Walsh relations; and the natural Gold representation supplies a full-rank scope separator. Section II positions the normal-form claim against the orthoderivative, field-reduction, semilinear, and linearized-map literature. Section III fixes notation. Sections IV and V prove the classification and its pure APN consequences. Sections VI and VII establish the two realizations. Section VIII records the Fourier-side consequences and the Gold boundary. Appendix A gives a finite cross-family separation in the first common binary dimension; it is not used to support the normal-form classification.

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II. R ELATED W ORK A. Quadratic APN Maps, Orthoderivatives, and Walsh Support Carlet, Charpin, and Zinoviev connect APN maps, associated binary codes, bent functions, and Walsh data [1]. In their notation, Theorem 13(v) describes the bent dual of the derivative-image incidence function, while Section 4.2 attaches a unique permutation to a quadratic APN permutation. Identifying that permutation with the modern normal to a derivative image gives the abstract nonzero-Walsh criterion used in Section VIII. We therefore claim neither that support mechanism nor the odd-dimensional almost-bent amplitude as new. Recent work on Walsh spectra of quadratic APN functions emphasizes component ranks and the even-dimensional case [6]. Canteaut, Couvreur, and Perrin introduced the orthoderivative as an effective EA-equivalence invariant for quadratic APN maps and developed algorithms for recovering or testing EA equivalence [2]. Couvreur, Canteaut, and Perrin subsequently generalized orthoderivatives and related them to cofactor matrices of binary Jacobians [7]. These works provide binary existence, uniqueness, covariance, and computational tools. Kaleyski develops a broader invariant framework for deciding EA equivalence [8], and Yoshiara proves that EA and CCZ equivalence coincide for quadratic APN functions [9]. The normalform theorem here addresses a different layer: after choosing a three-coordinate K-module structure, the separate rank defects of the two coefficient matrices, together with binary nullity one, force the canonical two-term Frobenius-linearized normal form and fix its trace normal by a K-valued pairing. This distinction also separates two geometric objects that can otherwise be blurred. Orthoderivative theory gives the binary hyperplane imF2 DX F = {Z : Tr(πF (X)T Z) = 0}. The same vector defines the K-linear covector Z 7→ πF (X)T Z, whose K × -class is a point in the dual K-projective plane. The latter interpretation is what is used in the dual-coordinate bijection; the derivative image itself is not asserted to be a K-linear plane. B. Triprojective and Cubic Norm-Twist Constructions Göloğlu and Kölsch construct triprojective APN functions from projectively semiquadratic maps and a root-free twistedpolynomial condition [3]. Their results establish differential uniformity and, in the relevant odd-degree case, permutation behavior. A companion paper relates projective-plane bijections to commutative semifields [10]. The present paper uses this family as one realization of the rank-(2, 2) operator class, while proving the normal form without its coefficients. Conversely, the determinant and full dual-frame identities in Section VI depend on coefficient contractions specific to this family and are not consequences of the abstract classification. Li, Zhou, Li, and Qu introduced the cubic norm-twist APN construction and proved the APN and semilinear-permutation facts used in Section VII [4]. Bartoli, Calderini, Polverino, and Zullo proved infiniteness of its admissible parameter family [5]. The rank-(2, 2) derivative split proved here gives a second route into the same normal-form class, but its first output-frame vector is not generally F (X). This is precisely why the general operator classification and the pure σ-quadratic APN corollary are distinct levels of the theory. The construction is not used to claim an infinite EA/CCZ separation from the triprojective family; the finite computation in Appendix A is stated with its explicit quantifier. C. Field Reduction, Linearized Maps, and Frobenius-Twisted Operators Field reduction and Desarguesian spreads provide the standard language for passing between extension-field subspaces and binary projective geometry [11]. In particular, nondegeneracy of the trace gives the generic identity CoreK {Y : Tr(v T Y ) = 0} = kerK (v T ). This identity recovers the largest K-linear subspace inside a binary trace hyperplane, but it does not produce the two coefficientkernel directions or the left-right normal form (20). The finite-field semilinear literature includes normal forms for globally semilinear transformations [12]. For linearized maps, Berson develops a multivariate polynomial-map framework with a polynomial-matrix representation [13], while Wu and Liu study the algebra of linearized polynomials through Dickson matrices and adjugates [14]. Kernel and root criteria for sums of Frobenius powers are also developed in the linearized/projective-polynomial setting [15]. These works provide general algebraic representations and structural tools for linearized maps. The operator considered here is the binary-linear two-term form AY σ + BY : its first summand is σ-semilinear and its second is K-linear, with separate coefficient-rank defects. The classification depends simultaneously on those two K-rank defects and on the binary nullity of the sum. We are not aware of a result giving the same rank-(2, 2)/binary-nullity-one left-right normal form under these hypotheses.

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III. P RELIMINARIES Let K = Fq , where q = 2m and m > 1. Fix an integer k with 1 ≤ k < m and gcd(k, m) = 1, and write k

m−k

σ : x 7−→ x2 ,

τ : x 7−→ x2

.

Thus σ and τ are inverse automorphisms of K, and the fixed field of σ is F2 . In exponent notation, σ 2 denotes the second k Frobenius iterate and expressions involving σ denote the corresponding integer exponents; for example, xσ+1 = x2 +1 and 2 2k k xσ +σ+1 = x2 +2 +1 . Vectors are columns, xT y is the ordinary K-valued dot product, and Tr = TrK/F2 is the absolute trace. We write rankK when the base field matters explicitly; after its first occurrence in a local argument, an unsubscripted matrix rank is over K. The trace pairing on K 3 is ⟨x, y⟩Tr = Tr(xT y). It is a nondegenerate F2 -bilinear form. If L : K 3 → K 3 is F2 -linear, then L∗ denotes its adjoint for this pairing. Consequently, (im L)⊥Tr = ker L∗ .

(9)

We repeatedly use the following elementary trace fact. Lemma 1: The map h : K → K, h(α) = ασ + α, has kernel F2 and image ker Tr. Proof: The kernel is the fixed field of σ, hence is F2 . Moreover, trace invariance under Frobenius gives Tr(ασ + α) = 0. The image and ker Tr both have binary dimension m − 1, so they are equal. Two-term decomposition lemma. The nontriviality of σ makes the Y σ and Y coefficient matrices intrinsic. Indeed, suppose C, D ∈ M3 (K) satisfy CY σ + DY = 0 for every Y ∈ K 3 . For the jth standard basis vector ej , write cj , dj for the corresponding columns and set Y = tej . Then tσ cj + tdj = 0 for every t ∈ K. Taking t = 1 gives dj = cj in characteristic two, and hence cj (tσ + t) = 0 for every t. Since σ ̸= idK , some t satisfies tσ ̸= t, so cj = dj = 0. This holds for every column, proving C = D = 0. Let F : K 3 → K 3 satisfy F (0) = 0. Its polar derivative is DX F (Y ) = F (X + Y ) + F (X) + F (Y ). The map F is quadratic if every DX F is F2 -linear. It is almost perfect nonlinear (APN) if every affine derivative equation in a nonzero direction has at most two solutions. For quadratic F , this is equivalent to kerF2 DX F = F2 X

for every X ̸= 0.

(10)

We call F pure σ-quadratic if each coordinate is a K-linear combination of monomials xσi xj . By the preceding decomposition lemma, such a map has a unique decomposition DX F (Y ) = AX Y σ + BX Y,

AX , BX ∈ M3 (K),

(11)

where Y σ means coordinatewise application of σ. Lemma 2 (Euler splitting): For a pure σ-quadratic map, AX X σ = BX X = F (X).

(12)

Proof: It is enough to polarize one monomial. The polar of cxσi xj is cXiσ Yj + cYiσ Xj . The first term belongs to BX Y and the second to AX Y σ . Substituting Y = X in either term gives cXiσ Xj . Summing over all coordinate monomials proves both equalities in (12). For V ̸= 0, the component qV (Y ) = Tr(V T F (Y )) is a binary quadratic form with polar form
BV (X, Y ) = Tr V T DX F (Y ) . (13) Its radical is denoted by Rad(BV ). If F is quadratic APN, then (10), (9), and binary rank–nullity give a unique nonzero vector πF (X) such that ker DX F ∗ = F2 πF (X), im DX F = {Z ∈ K 3 : Tr(πF (X)T Z) = 0}. (14) This is the orthoderivative convention of [2], [7]. Finally, our Walsh-transform convention is WF (U, V ) =

X Y ∈K 3

T

(−1)Tr(U Y +V

T

F (Y ))

.

(15)

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IV. A N ORMAL F ORM FOR R ANK -T WO F ROBENIUS -L INEARIZED O PERATORS We now prove the operator classification independently of any APN family. The map AY σ + BY is binary-linear: its first summand is σ-semilinear over K, whereas the full sum is generally not a semilinear transformation over K. Its hypotheses use both underlying fields: the two coefficient matrices have K-rank two, while the full binary-linear operator has binary nullity one. The input and output frames constructed below are the mechanism that produces the normal form. Theorem 3 (Normal-form theorem): Fix a nontrivial Frobenius automorphism σ of K with fixed field F2 . Let L : K 3 −→ K 3 ,

L(Y ) = AY σ + BY,

and assume rankK A = rankK B = 2,

kerF2 L = F2 X

(16)

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for some X ̸= 0. Choose nonzero R, S ∈ K satisfying BR = 0,

AS σ = 0,

and put T = AX σ = BX,

P = ARσ ,

Q = BS.

(17)

Then E = [X R S],

J = [T P Q]

belong to GL(3, K), and for every α, β, γ ∈ K,
L E(α, β, γ)T = J(ασ + α, β σ , γ)T .

(18)

Nσ (α, β, γ)T = (ασ + α, β σ , γ)T ,

(19)

J −1 ◦ L ◦ E = Nσ .

(20)

im L = {tT + pP + rQ : Tr(t) = 0, p, r ∈ K}.

(21)

Equivalently, for one has Moreover, Proof: The equality defining T follows from L(X) = 0 in characteristic two. We first prove that E is nonsingular. If R and S were proportional, then both BR = 0 and ARσ = 0 after rescaling. Hence the full K-line KR would lie in ker L, contradicting | ker L| = 2 < q. Suppose next that X = uR + vS. Then uσ P + vQ = 0. (22) If u = 0 or v = 0, this equality makes KS or KR, respectively, a subspace of ker L. If u, v ̸= 0, then for every t ∈ K, tσ v S ∈ ker L uσ by (22). These are q distinct kernel vectors, again a contradiction. Thus X, R, S are K-independent. Direct substitution now gives (18); at this point we have not assumed that J is nonsingular. If rP + sQ = 0 for some (r, s) ̸= (0, 0), then (rτ )R + sS ∈ ker L. tR +

This is a nonzero vector in ⟨R, S⟩K , whereas the unique nonzero kernel vector X is outside that plane. Therefore P, Q are independent. If T ∈ ⟨P, Q⟩K , then (18) would place im L inside a K-plane, of binary dimension 2m. On the other hand, (16) gives dimF2 im L = 3m − 1 > 2m. Thus T, P, Q are a K-basis, and J is nonsingular. Equation (20) is simply (18) written as an operator identity. Finally, bijectivity of β 7→ β σ and Lemma 1 give exactly (21). The preceding statement yields a classification, not only a coordinate construction. We call L1 and L2 left-right K-linearly equivalent if L2 = J ◦ L1 ◦ E −1 for some E, J ∈ GL(3, K). Corollary 4 (Left-right classification): For the fixed nontrivial automorphism σ above, let L(Y ) = AY σ + BY be a binarylinear operator on K 3 . Then rankK A = rankK B = 2, dimF2 ker L = 1 (23) if and only if there exist E, J ∈ GL(3, K) such that L = J ◦ Nσ ◦ E −1 . Thus all such operators for this fixed σ belong to one left-right K-linear equivalence class, represented by Nσ .

(24)

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Proof: The forward implication is Theorem 3. Conversely, write Nσ (Z) = A0 Z σ + B0 Z,

A0 = diag(1, 1, 0),

B0 = diag(1, 0, 1).

Both matrices have K-rank two and kerF2 Nσ = F2 e1 by Lemma 1. If (24) holds, then A = JA0 (E −1 )σ ,

B = JB0 E −1 ,

so the displayed decomposition has both coefficient ranks equal to two, while kerF2 L = E(F2 e1 ) has binary dimension one. The two-term decomposition lemma in Section III shows that these are the coefficient matrices of L. The first dual output coordinate has a stronger interpretation than a projective label: it identifies the binary trace hyperplane exactly and, at the same time, defines a K-linear covector. Corollary 5 (Exact trace-adjoint coordinate): Under the hypotheses of Theorem 3, let nT = eT1 J −1 . Then im L = {Z ∈ K 3 : Tr(nT Z) = 0},

ker L∗ = F2 n,

(25)

and nT T = 1,

nT P = nT Q = 0.

(26) T

×

Thus n is the unique nonzero binary trace normal, with the exact K-valued normalization n T = 1; independently, its K -class represents the K-linear covector Z 7→ nT Z. Proof: The three identities in (26) are the first-row identities in J −1 J = I. By (21), Tr(nT L(Y )) = 0 for all Y . Hence n ∈ ker L∗ . The binary ranks of L and L∗ are equal, so (16) gives dimF2 ker L∗ = 1. Since n ̸= 0, it is the unique nonzero element. Equation (25) follows. There is also a useful complete dual-frame statement. Let ℓA and ℓB be nonzero generators of the left kernels of A and B, respectively, and define p0 = ℓTB P, q0 = ℓTA Q. Corollary 6 (Complete dual frame): The scalars p0 , q0 are nonzero and  T  n T J −1 = p−1 . 0 ℓB −1 T q0 ℓA

(27)

Proof: Because ℓTB B = 0, the row ℓTB annihilates T = BX and Q = BS; nonsingularity of J forces p0 = ℓTB P ̸= 0. Similarly, ℓTA annihilates T = AX σ and P = ARσ , and therefore q0 ̸= 0. The three displayed rows multiply J to the identity. The classification is stronger than the generic field-reduction identity for a trace hyperplane. The two coefficient-kernel directions construct the input frame, while the binary-nullity condition forces the output frame and hence the canonical model. Family-specific identities relating det E and det J require additional coefficient algebra and do not follow from Theorem 3. V. P URE σ-Q UADRATIC APN C ONSEQUENCES We specialize the operator theorem to the derivative of a pure σ-quadratic APN map. The specialization first yields an exact field-valued normalization. Oddness of the total binary dimension is used only later. Theorem 7 (Exact orthoderivative normalization): Let F : K 3 → K 3 be pure σ-quadratic and APN, and suppose rankK AX = rankK BX = 2

for every X ̸= 0

(28)

in the decomposition (11). Then πF (X)T F (X) = 1 in K

(29)

for every X ̸= 0. Proof: Fix X ̸= 0. APN gives kerF2 DX F = F2 X. Apply Theorem 3 to DX F . By Lemma 2, the first output-frame vector is TX = AX X σ = BX X = F (X). −1 Corollary 5 says that the unique nonzero trace-adjoint normal is the first row of JX and pairs to one with TX . By definition (14), that normal is πF (X), proving (29). In particular, the answer is independent of the choices of the two coefficient-kernel generators. We recall the following standard odd-dimensional property [1, Sec. 4.2] and [2] and include the short counting proof for completeness. It supplies the injectivity needed for projectivization without assuming a correlation or polarity. N Lemma 8 (Standard odd-dimensional orthoderivative bijection): Let G : FN 2 → F2 be quadratic APN with N odd. Then every nonzero component polar form has a one-dimensional radical, and the orthoderivative is a bijection on FN 2 \ {0}.

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Proof: For each X ̸= 0, APN gives a unique nonzero normal V = πG (X) to im DX G. By (13), this is equivalent to X ∈ Rad(BV ). Hence the number of incidence pairs (V, X),

V ̸= 0,

X ∈ Rad(BV ) \ {0},

is exactly 2N − 1 when counted by X. Every alternating bilinear form on an odd-dimensional binary space is singular, so each of the 2N − 1 nonzero components contributes at least one such X. Equality in the count forces every component radical to have exactly one nonzero vector. Thus each radical is an F2 -line, and X 7→ πG (X) is a bijection on the nonzero vectors. Theorem 9 (Odd-degree permutation): Under the hypotheses of Theorem 7, if m is odd, then F is a permutation of K 3 . Proof: For X ̸= 0, (29) and oddness of m give
Tr πF (X)T F (X) = Tr(1) = 1. Equation (14) therefore shows that F (X) ∈ / im DX F . If F (Y + X) = F (Y ), then quadraticity and F (0) = 0 give DX F (Y ) = F (X), a contradiction. Thus no two inputs differing by a nonzero X have the same image. The finite map F is injective and hence bijective. Theorem 10 (Orthoderivative dual-coordinate bijection): Under the hypotheses of Theorem 9, πF (λX) = λ−(σ+1) πF (X)

(λ ∈ K ∗ , X ̸= 0),

(30)

and ΦF : PG(2, K) −→ PG(2, K)∗ ,

[X] 7−→ [πF (X)]

(31)

is a bijection. Proof: Pure σ-quadratic homogeneity gives F (λX) = λσ+1 F (X),

DλX F (λY ) = λσ+1 DX F (Y ).

It follows from the trace pairing that λ−(σ+1) πF (X) is the nonzero adjoint normal to im DλX F . Uniqueness in (14) proves (30), which makes (31) well defined. Because m is odd and gcd(k, m) = 1, gcd(2k + 1, 2m − 1) = 1. (32) Indeed, if an odd integer d divides both numbers, the multiplicative order of 2 modulo d divides both 2k and m, hence is one; thus d = 1. Therefore λ 7→ λ−(σ+1) permutes K ∗ . Suppose [πF (X)] = [πF (Y )]. There is a µ ∈ K ∗ such that πF (Y ) = µπF (X). Choose λ ∈ K ∗ with µ = λ−(σ+1) . By (30), πF (Y ) = πF (λX). Lemma 8, applied in the odd binary dimension 3m, gives Y = λX. Hence [Y ] = [X]. The map is injective, and the two finite projective planes have the same number of points, so it is bijective. The target in (31) is the dual plane for the following reason. Equation (14) first identifies πF (X) as the binary trace normal in imF2 DX F = {Z : Tr(πF (X)T Z) = 0}. Independently, the same vector defines the K-linear covector Z 7→ πF (X)T Z; its K × -class is therefore a point of the dual K-projective plane. The theorem does not assert that ΦF is a correlation or a polarity; such terms impose incidence properties not proved here. VI. T HE T RIPROJECTIVE R EALIZATION We now verify the abstract hypotheses for the family of Göloğlu and Kölsch [3]. Let a, b, c ∈ K with a ̸= 0, and define F = (F1 , F2 , F3 ) : K 3 → K 3 by F1 (x, y, z) = xσ+1 + ay σ z + bxσ y + cxσ z, F2 (x, y, z) = ay σ+1 + z σ x + bz σ y + cxσ y, F3 (x, y, z) = z The admissibility polynomial is

σ+1

(33)

σ

+ x y.

2

p(T ) = aT σ +σ+1 + bT σ+1 + cT + 1.

(34)

We assume that p has no root in K. Under our standing hypothesis gcd(k, m) = 1, the cited construction is APN; when m is odd it is also a permutation [3, Ths. 1.3 and 1.6]. We shall recover the latter conclusion independently from the rank-two frame in Theorem 9. For X = (x, y, z)T , put u = x + by + cz, w = ay σ + bz σ + cxσ . (35) Direct polarization of (33) gives DX F (Y ) = AX Y σ + BX Y , where    σ u az 0 x AX = cy ay x + by  , BX =  z σ y 0 z 0

bxσ w xσ

 ay σ + cxσ . 0 σ z

(36)

8

Define the right-kernel candidates R = (w, z σ , xσ )T , τ

(37)

τ

τ T

S = ((az) , u , (ay) ) ,

(38)

ℓA = (y, z, x + by)T ,

(39)

and the left-kernel candidates σ

σ

σ

σ T

ℓB = (z , x , ay + cx ) .

(40)

Theorem 11 (Explicit rank-two realization): For every X ̸= 0, adj(AX ) = S σ ℓTA ,

(41)

adj(BX ) = RℓTB .

(42)

Consequently, rankK AX = rankK BX = 2, BX R = 0, AX S σ = 0, and Theorem 3 applies to DX F . Proof: Using S σ = (az, u, ay)T , expansion of the nine 2 × 2 cofactors of the matrices in (36) gives   az
adj(AX ) =  u  y z x + by ay and



 w adj(BX ) =  z σ  z σ xσ

xσ

ay σ + cxσ .

These are precisely (41) and (42); all minus signs disappear in characteristic two. Direct multiplication also gives AX S σ = 0 and BX R = 0. None of R, S, ℓA , ℓB vanishes when X ̸= 0. For example, R = 0 first forces x = z = 0 and then ay σ = 0; since a ̸= 0, this forces y = 0. The other three cases follow in the same order from (38)–(40). Thus both adjugates are nonzero, so both matrices have rank at least two. Their displayed nonzero right-kernel vectors show that their determinants vanish. Each rank is therefore exactly two. The root-free hypothesis and the cited APN theorem give kerF2 DX F = F2 X. The three hypotheses of Theorem 3 are now verified. Notice that the adjugate identities themselves used neither APN nor the root-free condition. Set EX = [X R S], ∆X = det EX , PX = AX Rσ , QX = BX S, JX = [F (X) PX QX ]. (43) The first output vector is F (X) by Lemma 2. Theorem 3 proves ∆X ̸= 0 and JX ∈ GL(3, K). We next determine its determinant and all three dual rows. For vectors in K 3 , let r × s denote the usual cross product; in characteristic two, subtraction in its coordinates is addition. We use the identity (r × s)T (u × v) = (rT u)(sT v) + (rT v)(sT u). (44) Theorem 12 (Determinant factorization and full dual frame): For every X ̸= 0, det JX = ∆σ+1 X as an identity of K-valued functions. Moreover,   πF (X)T −1 JX = (∆σX )−1 ℓTB  , T ∆−1 X ℓA

πF (X)T =

(45)

(PX × QX )T . ∆σ+1 X

(46)

Proof: Direct multiplication using (36) gives the family-specific identities T BX ℓA = X × R,

ℓA × ℓB = F (X),

ATX ℓB = X σ × S σ .

(47)

The left-kernel relations in Theorem 11 and (47) yield the six contractions ℓTB ℓTA

F (X) PX 0 ∆σX 0 0

QX 0 ∆X .

(48)

9

For instance, T ℓTA QX = (BX ℓA )T S = (X × R)T S = ∆X ,

while ℓTB PX = (ATX ℓB )T Rσ = (X σ × S σ )T Rσ = ∆σX . The remaining four entries are zero because F (X) = ℓA × ℓB or because the appropriate row is a left-kernel vector. Now apply (44): det JX = F (X)T (PX × QX ) = (ℓA × ℓB )T (PX × QX ) = (ℓTA PX )(ℓTB QX ) + (ℓTA QX )(ℓTB PX ) = ∆σ+1 X . This proves (45). The first row in (46) is the normalized cofactor row and hence pairs to one with F (X) and to zero with PX , QX . The last two rows follow from (48). Thus their product with JX is the identity, completing the proof. The root-free hypothesis also has the twisted-polynomial interpretation recalled in [3, Sec. 2.3, Th. 2.5]: in K[t; σ], where tα = ασ t, the polynomial (34) is root free in K exactly when at3 + bt2 + ct + 1 has no linear right divisor. For the present frames, the following direct equivalent is the one needed here. Proposition 13 (Root-free frame criterion): The polynomial p in (34) is root free in K if and only if ∆X ̸= 0 for every X ̸= 0. Proof: If p is root free, the APN theorem quoted above and Theorem 3 give EX ∈ GL(3, K) for every X ̸= 0. Conversely, if p(s) = 0, then s ̸= 0. For X = (1, sσ+1 , s)T the root equation gives

2

w = asσ +σ + bsσ + c = s−1 . Equations (37) and (43) then give R = s−1 X and ∆X = 0. Combining Theorem 3 with (46) also describes the derivative image exactly. If Y = αX + βR + γS, then DX F (Y ) = JX (ασ + α, β σ , γ)T ,

(49)

im DX F = {tF (X) + pPX + rQX : Tr(t) = 0}.

(50)

and hence The ordinary K-plane ⟨PX , QX ⟩K is therefore the canonical slice im DX F ∩ kerK (πF (X)T ) = ⟨PX , QX ⟩K .

(51)

The full derivative image is a binary trace hyperplane of size q 3 /2; it is not itself the K-plane in (51). Finally, (45) is a family-specific finite-field identity: it uses the Frobenius relations in K and is not asserted as a formal polynomial identity over unrestricted indeterminates. VII. A C UBIC N ORM -T WIST R EALIZATION The normal-form theorem is not confined to pure σ-quadratic maps. We show this using the cubic norm-twist construction of Li, Zhou, Li, and Qu [4]. Let L = F23m and K = F2m . Define m

ρ(z) = z 2 ,

s

τ (z) = z 2 ,

σ = τ |K ,

where gcd(s, m) = 1. Let Mµ denote multiplication by µ ∈ L, and put H = (ρ + Mµ )τ,

Λ = H + I.

(52)

NL/K (µ) ̸= 1

Λ permutes L.

(53)

b(a, w) = aρ(w) + wρ(a),

(54)

∗

Choose v ∈ K and µ ∈ L such that and

Writing Q0 (z) = zρ(z), the family is F (z) = Q0 (Λz) + vQ0 (z).

(55)

10

Theorem 6 of [4] proves that (55) is APN. Bartoli, Calderini, Polverino, and Zullo prove the existence of admissible parameters for every m ≥ 3 in the s = 1 subfamily [5, Th. 4.6 and Cor. 4.7]. Lemma 14: For a ̸= 0, the K-linear map ψa : L → L given by ψa (w) = b(a, w) has kernel aK and rank two over K. Proof: Write w = at. Since ρ fixes K, b(a, at) = aρ(a)(ρ(t) + t). This vanishes exactly when t ∈ Fix(ρ) = K. Thus the kernel is the one-dimensional K-space aK, and rank–nullity in the cubic extension gives rank two. Theorem 15 (Norm-twist rank-two realization): Fix any K-basis of L. For every X ̸= 0, the derivative of (55) has a decomposition DX F (Y ) = AX Y σ + BX Y, rankK AX = rankK BX = 2. (56) Theorem 3 applies with RX = ΛX + vX,

SX = H −1 (ΛX),

TX = b(HX, X),

PX = b(ΛX, H(RX )),

(57) QX = b(RX , SX ).

(58)

In particular, its exact adjoint normal satisfies πF (X)T TX = 1.

(59)

Proof: First, H is invertible. Indeed, if (ρ + Mµ )(z) = 0 for a nonzero z, then ρ(z) = µz. Applying ρ three times gives z = NL/K (µ)z, contrary to (53); τ is itself a permutation. The map H is σ-semilinear over K, so in the fixed basis there is a C ∈ GL(3, K) such that H(Y ) = CY σ . Polarizing (55), expanding Λ = H + I, and using bilinearity of b give DX F (Y ) = b(ΛX, ΛY ) + vb(X, Y ) = b(ΛX, H(Y )) + b(ΛX + vX, Y ).

(60)

Thus AX Y σ = b(ΛX, CY σ ) and BX Y = b(ΛX + vX, Y ). Since Λ permutes L, ΛX ̸= 0. Lemma 7 of [4] states, under the hypotheses above, that H + βI permutes L for every β ∈ K. Taking β = 1 + v gives ΛX + vX = (H + (1 + v)I)X ̸= 0.

(61)

Lemma 14, together with invertibility of C, now proves both rank statements in (56). The APN theorem cited above gives kerF2 DX F = F2 X, so Theorem 3 applies. The kernel description in Lemma 14 gives the choices in (57): BX RX = 0 and σ AX SX = b(ΛX, H(SX )) = b(ΛX, ΛX) = 0.

Finally, symmetry and alternation of b give AX X σ = b(ΛX, HX) = b(HX, X), BX X = b(ΛX + vX, X) = b(HX, X). This proves the formula for TX ; the formulas for PX , QX are direct substitutions into (60). Corollary 5 proves (59). The distinction from the pure specialization is exact. Since Λ = H + I, expansion of the norm form gives F (X) = Q0 (HX) + b(HX, X) + (1 + v)Q0 (X).

(62)

Thus the first frame vector TX = b(HX, X) is not identified with F (X) by an Euler split. Equation (59) therefore does not imply πF (X)T F (X) = 1. We draw no automatic permutation, dual-coordinate bijection or triprojective determinant conclusion for this family. The result is a second natural realization of the operator theorem, not a proof that the two APN constructions are inequivalent under arbitrary binary EA or CCZ maps. In the first common binary dimension 3m = 9, an exhaustive calculation does separate every admissible instance of the two parameterizations by the differential spectrum of its orthoderivative. This finite result and its EA/CCZ justification are stated in Appendix A; no conclusion for m > 3 is claimed.

11

VIII. F OURIER -S IDE C ONSEQUENCES AND S TRUCTURAL B OUNDARIES The normal form fixes the extension-field coordinate attached to each derivative direction. The abstract incidence and support mechanism is classical in equivalent form [1, Sec. 4.2 and Th. 13(v)]; the point of this section is therefore not a new support theorem, but the exact K-valued label supplied by Theorem 7 and, for the triprojective family, the explicit dual frame in Theorem 12. Theorem 16 (Radical incidence and pointwise Walsh support): Let F satisfy the hypotheses of Theorem 7, and assume m is odd. Then {(V, X) : V ̸= 0, X ∈ Rad(BV ) \ {0}} = {(πF (X), X) : X ̸= 0}. (63) For every X ̸= 0 and U ∈ K 3 , WF (U, πF (X)) ̸= 0

⇐⇒

Tr(U T X) = 1 .

(64)

Whenever it is nonzero, |WF (U, πF (X))| = 2(3m+1)/2 .

(65)

Proof: Equation (13) shows that X ∈ Rad(BV ) exactly when V ∈ ker DX F ∗ . For X ̸= 0, the latter kernel has unique nonzero element πF (X), proving (63). Lemma 8 shows in addition that the radical of the component indexed by πF (X) is exactly F2 X. Put QX (Y ) = Tr(πF (X)T F (Y )). Translating the Walsh sum by X and using X ∈ Rad(BπF (X) ) gives T

WF (U, πF (X)) = (−1)Tr(U X)+QX (X) WF (U, πF (X)). It follows that the coefficient vanishes if Tr(U T X) + QX (X) = 1. Conversely, choose a binary complement W to F2 X. The sum factors into the sum over the radical and the Walsh transform of the induced nondegenerate quadratic form on the even-dimensional space W . When Tr(U T X) + QX (X) = 0, the radical factor is two and the nondegenerate quadratic sum is nonzero. This proves that nonvanishing is equivalent to Tr(U T X) = QX (X). By Theorem 7 and oddness of m, QX (X) = Tr(πF (X)T F (X)) = Tr(1) = 1, which yields (64). The induced nondegenerate form has binary dimension 3m−1; the standard quadratic Gauss-sum magnitude is 2(3m−1)/2 . Including the radical factor two proves (65). The magnitude (65) is the known almost-bent amplitude of odd-dimensional quadratic APN components [1], [6]; it is not claimed as a new spectrum theorem. Likewise, (64) is a native K 3 realization of the known support relation, not a new coding-theory consequence. We finish by showing that the coefficient-rank assumption is a genuine restriction. Let L = F23m over K = F2m , and let t t satisfy gcd(t, 3m) = 1. Put σ = Frobt2 K , that is, σ(x) = x2 for x ∈ K. In a fixed K-basis of L, the Frobenius map t Y 7→ Y 2 is σ-semilinear, so t Y 2 = CY σ for some C ∈ GL(3, K). (66) t

Proposition 17 (Gold full-rank boundary): For the Gold map G(z) = z 2 +1 in the natural cubic-extension model above, the derivative split DX G(Y ) = MX CY σ + MX 2t Y (67) has rankK (MX C) = rankK (MX 2t ) = 3 for every X ̸= 0, where Mz is multiplication by z on L. Consequently, Theorem 3 does not apply in this representation. t t Proof: Polarization gives DX G(Y ) = X 2 Y + XY 2 , which is (67) by (66). Multiplication by a nonzero field element is an invertible K-linear map, and C is invertible. Both displayed coefficient operators are therefore nonsingular. The proposition is representation-specific: it delimits the natural L/K decomposition of the Gold map. It does not assert that coefficient ranks are invariant under arbitrary binary equivalence or under every possible extension-field presentation. IX. C ONCLUSION For a fixed nontrivial Frobenius automorphism σ with fixed field F2 , we classified binary-linear two-term Frobenius-linearized operators L(Y ) = AY σ + BY on K 3 whose two coefficient matrices have K-rank two and whose full binary kernel has dimension one. Up to invertible K-linear input and output changes, every such operator is the canonical model (α, β, γ) 7−→ (ασ + α, β σ , γ). The coefficient-kernel directions provide the frames that prove the classification, while the first row of the inverse output frame is exactly the unique nonzero binary trace normal and is normalized by a K-valued pairing.

12

For pure σ-quadratic APN derivatives, the first output-frame vector is F (X). The operator normal form therefore yields πF (X)T F (X) = 1, from which the odd-degree permutation result and the bijection [X] 7→ [πF (X)] to the dual projective plane follow. The Göloğlu–Kölsch triprojective family realizes this pure specialization and admits additional determinant and full-dual-frame identities. The Li–Zhou–Li–Qu norm-twist family realizes the same abstract operator class through a different first output vector, showing that those pure conclusions are not consequences of rank-(2, 2) alone. The natural cubic-extension representation of Gold maps has coefficient-rank pair (3, 3), so the normal form describes a proper structural subclass rather than APN derivatives in general. Within the pure subclass, the exact normal also supplies an extension-field coordinate for the known binary component-radical and Walsh-support relations. A remaining structural question is which further APN constructions admit rank-deficient coefficient splittings that fall into this or related linearized normal-form classes. C ONFLICT OF I NTEREST None of the authors have a conflict of interest to disclose. A PPENDIX A F INITE n = 9 C ROSS -FAMILY S EPARATION This appendix records a finite machine-assisted result; it is not used in the proof of any infinite theorem. Extend every orthoderivative by πF (0) = 0 and define δπF (a, b) = |{x : πF (x + a) + πF (x) = b}|. The orthoderivative differential spectrum is the multiset of these integers over all a ̸= 0 and all b. If quadratic APN maps F and G are EA-equivalent, polarization of the EA relation gives BG (x, y) = ABF (Bx, By) where A and B are the linear parts of the output and input affine permutations, respectively. Taking the unique derivative-image normals yields πG (x) = A−T πF (Bx). (68) Therefore their orthoderivatives are linearly equivalent and have the same differential spectrum. This invariant viewpoint is consistent with the orthoderivative-based EA methods of [2], [8]. Yoshiara proves that two quadratic APN functions are CCZ-equivalent if and only if they are EA-equivalent [9, Th. 1]. Proposition 18 (Exhaustive separation at n = 9): At m = 3, every admissible triprojective parameter instance over F38 is binary EA-inequivalent, and hence CCZ-inequivalent, to every admissible Li–Zhou–Li–Qu norm-twist parameter instance over F512 . Proof: Exact polynomial-basis arithmetic exhausts both parameter spaces. On the triprojective side, both k = 1, 2, every a ∈ F∗8 , and every b, c ∈ F8 are tested against (34), leaving 292 instances. On the norm-twist side, s ∈ {1, 2, 4, 5, 7, 8}, all µ ∈ F512 satisfying the norm condition and the permutation condition on Λ, and all v ∈ F∗8 are exhausted, leaving 5292 instances. For each truth table, exact binary Gaussian elimination recovers the unique normal to every nonzero derivative image; the implementation also checks binary derivative rank eight. It then counts all 511 · 512 values δπF (a, b). The 292 target instances give two distinct spectra, each with multiplicity 146. The 5292 norm instances give eight spectra, four with multiplicity 189 and four with multiplicity 1134. The two sets of spectra are disjoint. The invariance proved in (68) gives EA separation for every cross pair, and Yoshiara’s theorem gives CCZ separation. The computation uses the irreducible moduli 0xb for F8 and 0x211 for F512 . Independent cross-checks repeat the full target enumeration with modulus 0xd, repeat the complete s = 1 norm slice with modulus 0x203, verify normal annihilation and APN kernels directly for a fixed cross pair, and verify invariance under a nontrivial deterministic EL copy. The exact enumerator and regression tests are provided in the ancillary files accompanying this preprint. These facts establish Proposition 18 only for m = 3; no parameter-uniform EA or CCZ separation is claimed.

13

R EFERENCES [1] C. Carlet, P. Charpin, and V. Zinoviev, “Codes, bent functions and permutations suitable for DES-like cryptosystems,” Designs, Codes and Cryptography, vol. 15, no. 2, pp. 125–156, 1998. [2] A. Canteaut, A. Couvreur, and L. Perrin, “Recovering or testing extended-affine equivalence,” IEEE Transactions on Information Theory, vol. 68, no. 9, pp. 6187–6206, 2022. [3] F. Göloğlu and L. Kölsch, “Triprojective almost perfect nonlinear permutations and functions,” 2026, arXiv:2605.17545, version 1. [4] K. Li, Y. Zhou, C. Li, and L. Qu, “Two new families of quadratic APN functions,” IEEE Transactions on Information Theory, vol. 68, no. 7, pp. 4761–4769, 2022. [5] D. Bartoli, M. Calderini, O. Polverino, and F. Zullo, “On the infiniteness of a family of APN functions,” Journal of Algebra, vol. 598, pp. 68–84, 2022. [6] S. H. Bénéteau, N. Goluboff, L. Kölsch, and D. Vaghasiya, “On the Walsh spectra of quadratic APN functions,” IEEE Transactions on Information Theory, vol. 72, no. 7, pp. 5207–5216, 2026. [7] A. Couvreur, A. Canteaut, and L. Perrin, “On the properties of the Ortho-Derivatives of quadratic functions,” in WCC 2024—The Thirteenth International Workshop on Coding and Cryptography, Perugia, Italy, Jun. 2024. [8] N. Kaleyski, “Deciding EA-equivalence via invariants,” Cryptography and Communications, vol. 14, no. 2, pp. 271–290, 2022. [9] S. Yoshiara, “Equivalences of quadratic APN functions,” Journal of Algebraic Combinatorics, vol. 35, no. 3, pp. 461–475, 2012. [10] F. Göloğlu and L. Kölsch, “Commutative semifields from bijections of the Desarguesian plane,” Journal of the London Mathematical Society, vol. 114, no. 1, p. e70635, 2026. [11] G. V. de Voorde, “Desarguesian spreads and field reduction for elements of the semilinear group,” Linear Algebra and its Applications, vol. 507, pp. 96–120, 2016. [12] U. Dempwolff, J. C. Fisher, and A. Herman, “Semilinear transformations over finite fields are Frobenius maps,” Glasgow Mathematical Journal, vol. 42, no. 2, pp. 289–295, 2000. [13] J. Berson, “Linearized polynomial maps over finite fields,” Journal of Algebra, vol. 399, pp. 389–406, 2014. [14] B. Wu and Z. Liu, “Linearized polynomials over finite fields revisited,” Finite Fields and Their Applications, vol. 22, pp. 79–100, 2013. [15] G. McGuire and J. Sheekey, “A characterization of the number of roots of linearized and projective polynomials in the field of coefficients,” Finite Fields and Their Applications, vol. 57, pp. 68–91, 2019.

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