Constructions and Characterizations of $s$-Plateaued Partitions

Abstract: Bent partitions play a significant role in constructing bent functions and have rich connections with coding theory and combinatorics. In this paper, we introduce $s$-plateaued partitions, which generalize the bent partitions. Let $Γ={A_{i}, 1 \leq i \leq K}$ be a partition of $V_{n}<sup>{(p)}$, where $V_{n}<sup>{(p)}$ is an $n$-dimensional vector space over the prime field $\mathbb{F}<em>{p}$ and $p \mid K$. Then $Γ$ is called an $s$-plateaued partition of $V</em>{n}<sup>{(p)}$ of depth $K$ if each $p$-ary function $$f: V_{n}^{(p)} \rightarrow \mathbb{F}<em>{p}$$ for which every $j \in \mathbb{F}</em>{p}$ has exactly $\frac{K}{p}$ of sets $A_{i}$ in $Γ$ in its preimage set, is a $p$-ary $s$-plateaued function. By using an $s$-plateaued partition, a large number of $p$-ary $s$-plateaued functions, vectorial $s$-plateaued functions and generalized $s$-plateaued functions can be constructed. In particular, $0$-plateaued partitions are just bent partitions. In general, $s$-plateaued partitions are much more complicated than bent partitions. We analyze the possible cardinality of $A_{i}$ of an $s$-plateaued partition. We give some explicit constructions of $s$-plateaued partitions for which any generated $p$-ary $s$-plateaued function has no nonzero linear structure. We give a characterization of an $s$-plateaued partition $Γ={A_{i}, 1 \leq i \leq K}$, where $p$ is odd, $K \geq 5$ and $-A_{i}=A_{i}, 1 \leq i \leq K$. Based on which, we show that if $p \geq 5$, then the preimage set partition of a $p$-ary $s$-plateaued function $$f: V_{n}^{(p)} \rightarrow \mathbb{F}<em>{p}$$ with $f(x)=f(-x)$ is an $s$-plateaued partition if and only if $f$ is of $(p-1)$-form, where $n+s$ is even.When $s=0$, we partially address an open problem on whether a bent partition $Γ$ of $V</em>{n}<sup>{(p)}$ of depth $p<sup>{\frac{n}{2}}$ must be obtained from spreads.

• The paper establishes a novel definition of s-plateaued partitions that generalize bent partitions by proving their equivalence with the systematic generation of vectorial and generalized s-plateaued functions.
• It details explicit constructions of s-plateaued partitions using linear, affine, and trace mapping methods that yield functions with no nonzero linear structures, enhancing cryptographic resistance.
• The work provides complete cardinality characterizations and symmetry conditions for s-plateaued partitions, partially resolving open problems related to maximal bent partition constructions.

Constructions and Characterizations of $s$-Plateaued Partitions

Motivation and Problem Setting

This paper extends the structural foundations of bent partitions by introducing the concept of $s$-plateaued partitions in the context of finite vector spaces $V_n^{(p)}$ over the prime field $\mathbb{F}_p$. An $s$-plateaued partition is a partition $\Gamma = \{A_i: 1 \leq i \leq K\}$, where $p \mid K$, with the property that every $p$-ary function $f : V_n^{(p)} \to \mathbb{F}_p$ that maps each $j \in \mathbb{F}_p$ to exactly $s$0 index sets $s$1 is an $s$2-plateaued function. This notion generalizes bent partitions (the case $s$3), a foundational tool in the construction of cryptographically robust functions and in combinatorial designs.

$s$4-Plateaued functions are pivotal in both the cryptographic and combinatorial landscapes due to their high nonlinearity, diversified spectrum distribution (quantified by the Walsh transform), and connections to linear codes, difference sets, association schemes, and related algebraic structures. The characterization and explicit construction of $s$5-plateaued partitions are notably more intricate in the $s$6 regime, in contrast to the bent ($s$7) case, due to the broader class of possible value distributions and structural properties.

Main Contributions

Theoretical Foundations

The authors formally define $s$8-plateaued partitions and establish their equivalence with the systematic generation of $s$9-ary, vectorial, and generalized $V_n^{(p)}$0-plateaued functions. Specifically, they prove that, for $V_n^{(p)}$1 with $V_n^{(p)}$2, the following are equivalent:

• $V_n^{(p)}$3 is an $V_n^{(p)}$4-plateaued partition,
• Any vectorial map with fibers partitioned by $V_n^{(p)}$5 is a vectorial $V_n^{(p)}$6-plateaued function,
• Any generalized $V_n^{(p)}$7-ary function as above is a generalized $V_n^{(p)}$8-plateaued function.

The paper provides a comprehensive characterization of the possible cardinalities of sets $V_n^{(p)}$9 arising in an $\mathbb{F}_p$0-plateaued partition, and details how these possibilities diverge sharply from the bent case, especially with respect to the balanced/unbalanced nature of the induced functions and their impact on code and design theory.

Explicit Constructions

The authors present several explicit constructions for $\mathbb{F}_p$1-plateaued partitions, emphasizing the nontrivial regime where any resulting $\mathbb{F}_p$2-ary $\mathbb{F}_p$3-plateaued function has no nonzero linear structure—critical for cryptographic resistance to linear attacks. The constructions employ:

• Linear and affine mappings governed by presemifield theory,
• Structure-preserving permutations and trace representations,
• Properties of vectorial plateaued functions with isomorphic Walsh supports for all components, and constraints ensuring duals and support structures are aligned,
• Evidence that the construction methodology produces partitions for which the Walsh supports contain a linear basis, directly implying the absence of nonzero linear structures by spectral analysis.

Characterizations and Structural Theorems

The authors give a detailed structure theorem for $\mathbb{F}_p$4-plateaued partitions with the symmetry property $\mathbb{F}_p$5, for $\mathbb{F}_p$6 odd and $\mathbb{F}_p$7. They show that, in this case:

• $\mathbb{F}_p$8 must be a power of $\mathbb{F}_p$9,
• The partition indices can be identified with $s$0, and the corresponding indicator map $s$1 is a vectorial $s$2-plateaued function,
• Additional regularity and support-alignment conditions on $s$3 are both necessary and sufficient for the $s$4-plateaued property; specifically, all Walsh supports match, sign vectors are component-invariant, and the duals of all components are linearly parameterized by a single vectorial function $s$5.

Implications and Partial Resolution of Open Problems

For the special case $s$6, the traditional bent partition scenario, they prove that, under the above symmetry and nontriviality assumptions, any bent partition at maximal depth $s$7 arises from spreads, thereby partially resolving a long-standing open question about the exclusivity of the spread construction for maximal bent partition depth.

Moreover, it is shown that, when $s$8, the partition induced by the preimage structure of any $s$9-ary $\Gamma = \{A_i: 1 \leq i \leq K\}$0-plateaued function $\Gamma = \{A_i: 1 \leq i \leq K\}$1 with $\Gamma = \{A_i: 1 \leq i \leq K\}$2 is an $\Gamma = \{A_i: 1 \leq i \leq K\}$3-plateaued partition if and only if $\Gamma = \{A_i: 1 \leq i \leq K\}$4 is of $\Gamma = \{A_i: 1 \leq i \leq K\}$5-form and $\Gamma = \{A_i: 1 \leq i \leq K\}$6 is even. This connects the algebraic invariance properties of $\Gamma = \{A_i: 1 \leq i \leq K\}$7 directly with the global symmetry constraints of the partition.

Key Results and Numerical Strength

• The paper gives complete cardinality distributions for the sets $\Gamma = \{A_i: 1 \leq i \leq K\}$8 in both the balanced and unbalanced cases, distinguishing, for example, how the possible values depend on $\Gamma = \{A_i: 1 \leq i \leq K\}$9, $p \mid K$0, $p \mid K$1, and the parity of $p \mid K$2.
• The explicit constructions yield $p \mid K$3-plateaued partitions (with large depth and no nonzero linear structure) whose induced functions populate the full spectrum of plateaued maps permitted by the theory.
• The characterization theorems provide necessary and sufficient algebraic criteria, unifying the spectral and combinatorial perspectives.

Practical and Theoretical Implications

The findings open new pathways for constructing cryptographically strong plateaued and bent functions, with direct relevance to symmetric-key design (S-boxes, masking, code design) as well as combinatorial construction of association schemes, partial geometric difference sets, and LP-packings. The highly structured and parameterized approach to partition construction facilitates algorithms for synthesizing functions with precisely tailored spectral and combinatorial properties.

The structural results on the invariance of Walsh supports and the absence of nonzero linear structure have broad implications for the security analysis of cryptographic primitives and for the synthesis of error-correcting codes with predetermined weight spectra.

Open Directions

• The classification and construction of balanced $p \mid K$4-plateaued partitions of higher depth (beyond those in this work, which are predominantly unbalanced) remains open.
• The question of whether invariance of Walsh support across all induced $p \mid K$5-plateaued functions extends from the symmetric case studied to more general $p \mid K$6-plateaued partitions is unsettled.
• Further connections with coding theory and combinatorics for $p \mid K$7 partitions, in analogy to the well-explored bent case, merit systematic investigation.

Conclusion

This work extends the theory of cryptographically significant partitions of finite vector spaces by generalizing bent partitions to the broader and significantly more complex class of $p \mid K$8-plateaued partitions. These theoretical developments are underpinned by explicit algebraic constructions and sharp structural characterizations, advancing both the algebraic understanding and practical methodology for synthesizing functions and designs with targeted spectral and combinatorial properties. The results create new directions for the intersection of spectral graph theory, finite geometry, and cryptography.

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Reference: "Constructions and Characterizations of $p \mid K$9-Plateaued Partitions" (2606.27776)