General constructions of normal bent partitions related to vectorial dual-bent functions

1. Introduction

• Bent functions have maximum Hamming distance from affine functions and constant absolute-square Walsh coefficients, with links to combinatorics, coding, sequences, and cryptography.The introduction also notes that complete classification remains elusive, particularly for p-ary bent functions.Source: §1 · p. 1
• Bent partitions were introduced because bentness is a property of the partition, not only of an individual function..Source: §1 · p. 2
• A bent partition is one for which functions constant on one cell and meeting the specified distribution across the other cells are bent.The definition requires each Dj to contain exactly K/p elements of the partition other than A0.Source: §1 · p. 2
• For depth K>4, all bent partitions known so far are Type I, with A0 an m-dimensional subspace; these are normal bent partitions.The introduction defines normal bent partitions as partitions whose distinguished cell is an m-dimensional subspace.Source: §1 · p. 2
• The paper uses permutation polynomials and normal m-partitions to construct normal bent partitions, with particular choices covering previously known examples.The construction combines elements of m-partitions into partitions of Fq2; the sets need only be disjoint and cover the whole space.Source: §1 · p. 2

2. Normal Bent Partitions of Type I (Construction 1)

• The construction forms a normal m-partition from U and the Uj sets, then combines its elements into a normal partition.The resulting A1(γ) sets have size p^(m−k)(q−1).Source: §2 · p. 3
• When B is linear, f1 is bent exactly for the stated trace-form B, and the resulting function is regular with components in the completed original Maiorana–McFarland class.With trace-form B, each component function is bent.Source: §2 · pp. 4, 6
• If π1^-1(x) reduced modulo x^pk−x is linearized, S1 is a normal bent partition and all bent functions it generates are regular.For q>3, every bent function constant on S1 also has the prescribed distribution across its cells.Source: §2 · pp. 4, 7
• The construction also applies for p=2 by changing β and defining the Uj bijections accordingly.The proof is stated to follow the same argument as Theorem 2.1.Source: §2 · p. 7

Construction 2

• The construction combines the sets Vⱼ into a normal partition of Fq², with |A₂(γ)| = pᵐ⁻ᵏ(q − 1) for every γ ∈ Fpᵏ.The Vⱼ are disjoint apart from zero, and the construction first forms a normal m-partition.Source: §3 · p. 8
• For a + a^q = 0, Theorem 3.1 gives Wf₂(a) = qζc and f₂*(a) = c; for a ∈ Fq*, it gives Wf₂(a) = q and f₂*(a) = 0.These coefficient statements apply to any f₂ given by (3.3).Source: §3 · p. 8
• If B is linear on Fpᵏ, f₂ is bent exactly when B(γ) = Trₖ(t′γ) for some t′, and then f₂ is regular.The theorem also identifies its dual in this case as belonging to the completed original Maiorana–McFarland class with associated subspace V.Source: §3 · p. 8
• Theorem 3.1 also gives a bentness criterion for the case where π₂⁻¹(x) modulo x^pᵏ−x is linearized, and the result extends to p = 2.The condition is stated for π₂⁻¹ reduced modulo x^pᵏ−x; the binary extension is noted in Remark 3.3.Source: §3 · pp. 8, 12
• When π₂⁻¹(x) modulo x^pᵏ−x is linearized, S₂ is a normal Type I bent partition, and all generated bent functions are regular.For q > 3, every p-ary bent function constant on S₂ has exactly pᵏ−1 elements of S₂ \ {U} in each Dⱼ.Source: §3 · pp. 8, 12

Construction 1 examples

• Construction 1 has bivariate sets U = (0, y) and Uⱼ = (x, jπ₁(x)); with k = m and π₁ the identity, its partition is the Desarguesian spread.The section presents this as a special case of the bivariate representation through ψ.Source: §4 · pp. 12–13
• For π₁(x) = xᵉ with gcd(e, q − 1) = 1 and e ≡ pˡ (mod pᵏ − 1), S₁ is called a generalized Desarguesian spread.The paper notes that its elements need not be subspaces, so it reserves “spread” for partitions whose elements have subspace structure.Source: §4 · p. 13
• When k = m and π₁ is linear, S₁ is a spread and its bent functions belong to the partial spread class.With π₁ the identity, S₁ reduces to the Desarguesian spread.Source: §4 · p. 13

Construction 2 examples

• Construction 2 has bivariate sets V = (x, 0) and Vⱼ = (jπ₂(y), y); with k = m and π₂ the identity, S₂ is the Desarguesian spread.The corresponding function reduces to the stated special-case form.Source: §4 · pp. 13–14
• For π₂(x) = xᵈ with gcd(d, q − 1) = 1 and d ≡ pˡ (mod pᵏ − 1), S₂ is called a generalized Desarguesian spread.Bent functions from generalized Desarguesian spreads have been called generalized PSap bent functions.Source: §4 · p. 14

• Each component of F* is the dual of the corresponding component F_-t, whereas the varying G◦L_t cannot be called a vectorial dual of F.Source: §5 · pp. 15–16
• Proposition 5.4 establishes that F is vectorial dual-bent and satisfies Condition A.Source: §5 · p. 16
• For nonzero t1 + t2, the sum of the duals of F_t1 and F_t2 equals the dual of F_t1+t2.Source: §5 · p. 16
• Joining U with some A(γ) yields a bent partition satisfying Condition C, and the cited result also gives this property for the preimage partition defined by F.Source: §5 · p. 16
• With appropriate π1, the normal bent partition S1 satisfies Condition C.Source: §5 · p. 16
• With an appropriate π2, the normal bent partition S2 satisfies Condition C; the same holds for the modified partitions when p = 2.Source: §5 · p. 17