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arxiv: 2512.19313 · v2 · submitted 2025-12-22 · 💻 cs.DM

Results on cubic bent and weakly regular bent p-ary functions leading to a class of cubic ternary non-weakly regular bent functions

Claude Carlet , and Alexander Kholosha This is my paper

Pith reviewed 2026-05-16 20:49 UTC · model grok-4.3

classification 💻 cs.DM
keywords bent functionscubic functionsternary functionsweakly regular bent functionsWalsh transformvectorial bent functionssecond-order derivatives
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The pith

A primary construction produces the first infinite class of cubic ternary vectorial bent functions whose components are all bent but not weakly regular

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper begins by extending known Walsh-transform properties of derivatives from binary bent functions to any odd characteristic. Some of these properties hold for every bent function, while others hold only for weakly regular ones. From this it derives that adding a quadratic function to a weakly regular bent function yields another bent function, and it isolates a class of bent functions in which every nonzero first-order derivative has a second-order derivative equal to a nonzero constant; this class includes all cubic bent functions and implies bentness. The main result then supplies an explicit primary construction that generates an infinite family of cubic ternary vectorial bent functions in which every component function is bent yet fails to be weakly regular, with bentness verified both by direct Walsh-spectrum calculation and by the second-order derivative criterion developed earlier.

Core claim

We give the first primary construction of an infinite class of cubic ternary vectorial bent functions whose components are bent but not weakly regular; bentness follows from explicit Walsh-transform evaluation and from the second-order derivative property that every first-order derivative equals a nonzero constant on a subspace, and non-weak regularity is shown by verifying that the functions lack the additional algebraic properties that characterize weakly regular bent functions.

What carries the argument

The explicit cubic polynomial construction over the ternary field whose second-order derivatives are nonzero constants, thereby forcing the Walsh spectrum to be flat

If this is right

  • Every cubic bent function in odd characteristic shares the property that every first-order derivative has a second-order derivative equal to a nonzero constant
  • Adding a quadratic function to a weakly regular bent function produces another bent function
  • The new family supplies the first known infinite supply of bent functions in odd characteristic that are provably not weakly regular
  • The second-order derivative test gives an alternative, sometimes simpler, way to prove bentness without computing the full Walsh spectrum

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same construction technique may be adaptable to produce non-weakly-regular bent functions of higher algebraic degree or in other odd characteristics
  • These functions could serve as building blocks for new cryptographic primitives whose nonlinearity properties differ from those of weakly regular bent functions
  • Computational verification of the Walsh spectra for small instances of the construction would provide independent confirmation of the algebraic proof

Load-bearing premise

The specific cubic polynomials in the construction really do have flat Walsh spectra, which the authors verify by direct calculation and by the second-order derivative test

What would settle it

An explicit computation, for any small odd dimension n greater than 2, of the Walsh transform of one of the constructed component functions showing that its magnitude is not constantly equal to 3 to the power n/2

read the original abstract

Much work has been devoted to bent functions in odd characteristic, but there still remains a gap between our knowledge of binary and nonbinary bent functions. In the first part of this paper, we attempt to partially bridge this gap by generalizing to any characteristic important properties known in characteristic two concerning the Walsh transform of derivatives of bent functions. Some of these properties generalize to all bent functions, while others appear to apply only to weakly regular bent functions. We deduce a method to obtain a bent function by adding a quadratic function to a weakly regular bent function. We also identify a particular class of bent functions possessing the property that every first-order derivative in a nonzero direction has a derivative (which is then a second-order derivative of the function) equal to a nonzero constant. We show that this property implies bentness and is shared in particular by all cubic bent functions. This generalizes to the odd characteristic the notion of cubic-like bent function, that was introduced and studied for binary functions by Irene Villa and the first author. In the second part of the paper, we provide (for the first time) a primary construction leading to an infinite class of cubic ternary vectorial bent functions that have only not weakly regular components. We show the bentness of the component functions by two approaches: by calculating the Walsh transform directly and by considering the second-order derivatives (and applying the results from the first part of the paper). We prove that they are not weakly regular by showing they do not have one of the properties that we proved in the first part of the paper for weakly regular bent functions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript generalizes key properties of the Walsh transform of derivatives of bent functions from characteristic 2 to arbitrary odd characteristic. It derives a construction for bent functions by adding a quadratic to a weakly regular bent function, identifies a class of functions (including all cubics) where nonzero first-order derivatives have second-order derivatives equal to nonzero constants (implying bentness), and supplies a primary construction of an infinite family of cubic ternary vectorial bent functions whose components are exclusively not weakly regular. Bentness is established by direct Walsh-spectrum evaluation and by the generalized second-order derivative test; non-weak regularity follows from the absence of a property proved earlier for weakly regular bent functions.

Significance. If the central claims hold, the work meaningfully narrows the gap between binary and p-ary bent-function theory by extending derivative properties, introducing a parameter-free primary construction, and furnishing the first explicit infinite class of cubic ternary vectorial bent functions with only non-weakly regular components. The dual verification routes (Walsh transform and generalized derivatives) and the explicit algebraic details supplied for reproduction constitute clear strengths.

minor comments (3)
  1. [Abstract] The abstract supplies no explicit equations or the concrete form of the constructed functions; adding a brief illustrative expression or reference to the defining polynomial would improve accessibility while preserving the one-paragraph limit.
  2. [Construction section (second part)] In the section presenting the primary construction, the vectorial function is introduced without an immediate statement of its algebraic degree and the precise field extension degree; a single clarifying sentence would prevent readers from having to infer these parameters from later calculations.
  3. [Non-weak-regularity argument] The proof that the components lack the weakly-regular property would benefit from an explicit cross-reference (e.g., “by Theorem X in the first part”) rather than a generic allusion to “one of the properties.”

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for the recommendation of minor revision. We appreciate the recognition of the generalization of derivative properties, the primary construction, and the dual verification of bentness.

Circularity Check

0 steps flagged

No significant circularity; primary construction and proofs are self-contained

full rationale

The paper first generalizes Walsh-transform properties of derivatives for bent functions in odd characteristic via direct algebraic arguments on field elements and second-order derivatives. It then gives an explicit primary construction of cubic ternary vectorial bent functions. Bentness is shown in two independent ways: direct evaluation of the Walsh sums for the component functions and verification that the functions satisfy the second-order derivative property (every nonzero first-order derivative has a second-order derivative equal to a nonzero constant), both of which are carried out with explicit calculations supplied in the text. Non-weak regularity follows by exhibiting the absence of a property previously shown to hold for weakly regular bent functions. The incidental citation to the authors' earlier binary work merely names the notion being generalized and plays no load-bearing role in the new ternary results or their verification. No parameter fitting, self-definitional equations, or reductions of predictions to inputs occur; the derivation chain rests on reproducible field arithmetic and is externally checkable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard finite-field arithmetic, the definition of the Walsh transform, and the notion of bentness; no numerical constants are fitted to data and no new entities are postulated.

axioms (1)
  • standard math Standard algebraic properties of finite fields of odd characteristic and the definition of the Walsh transform
    These are background facts from the existing literature on bent functions invoked throughout both parts of the paper.

pith-pipeline@v0.9.0 · 5598 in / 1289 out tokens · 28373 ms · 2026-05-16T20:49:08.566864+00:00 · methodology

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Reference graph

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