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arxiv: 2604.08311 · v1 · submitted 2026-04-09 · 💻 cs.IT · math.IT

On quadratic binomial vectorial functions with maximal bent components

Xianhong Xie , Yi Ouyang , Shenxing Zhang This is my paper

Pith reviewed 2026-05-10 17:16 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords quadratic binomial vectorial functionsbent componentsaffine equivalencefinite fields of even characteristicnonlinearity boundsdifferential uniformityimage set cardinalityF_{2^n}
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The pith

A quadratic binomial vectorial function over F_{2^n} with the maximum number of bent components is affine equivalent to one of two explicit forms when n is even, the exponents have 2-adic weight at most 2, the minimal field dimension ell(n)

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

The paper classifies all binomial vectorial functions F(x) = x^{d1} + x^{d2} over the finite field F_{2^n} with n even that achieve the largest possible count of bent component functions. When the 2-adic weights of the exponents are at most two and the gcd of the exponents with 2^m minus one is greater than one, the authors show that F must be affine equivalent to either x raised to 2^m plus one or to x to the power 2^i times (x plus x to the 2^m), provided the minimal dimension ell(n) of the F2-span generated by a primitive element exceeds m. This classification then supplies explicit lower bounds on the nonlinearity of F and upper bounds on its differential uniformity expressed directly in terms of the number of distinct values F attains. Readers in cryptography would care because bent components are the Boolean functions that resist linear attacks most effectively, and knowing every maximal case narrows the search for secure S-boxes and clarifies the structural limits of quadratic constructions.

Core claim

We prove that F(x) = x^{d1} + x^{d2} over F_{2^n} with n = 2m, wt_2(d1) <= 2, wt_2(d2) <= 2, and exactly 2^n - 2^m bent components is affine equivalent to x^{2^m + 1} or x^{2^i}(x + x^{2^m}) whenever ell(n) > m and gcd(d1, d2, 2^m - 1) > 1. Under the same hypotheses we also derive two explicit bounds, one on nonlinearity and one on differential uniformity, that depend only on the cardinality of the image set of F.

What carries the argument

The quantity ell(n) defined as the smallest F_2-dimension of the cyclic span F_2[sigma]gamma over all primitive elements gamma, which, together with the gcd condition on the exponents, forces the binomial to reduce to one of the two listed monomial or binomial shapes via affine equivalence.

If this is right

  • All such maximal-binomial functions obey concrete lower bounds on nonlinearity that improve with larger image cardinality.
  • Their differential uniformity is likewise bounded above in terms of the same image cardinality.
  • Any cryptographic construction relying on quadratic binomials with maximal bent components can be restricted to checking only the two affine classes.
  • The image-size method used for the bounds becomes a general tool for estimating crypto metrics once the function is known to lie in these families.

Where Pith is reading between the lines

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

  • Removing or relaxing the ell(n) > m hypothesis may reveal additional inequivalent forms that can be searched exhaustively for small even n.
  • The two classified families may admit further closed-form expressions for their full Walsh spectra or for the exact size of their image sets.
  • These results suggest a route to classifying higher-degree binomials with many bent components by first reducing degree via the same dimension condition.
  • In coding-theory terms the image-cardinality bounds translate directly into minimum-distance estimates for the associated linear codes.

Load-bearing premise

The assumption that ell(n) exceeds m is required to guarantee that only the two listed forms appear, but the paper does not prove the condition is necessary for the classification to hold.

What would settle it

An explicit binomial F(x) = x^{d1} + x^{d2} with wt_2(di) <= 2, gcd(d1,d2,2^m-1) > 1, exactly 2^n - 2^m bent components, yet ell(n) <= m and F not affine equivalent to either listed form, or the same for a field where ell(n) > m but an extra inequivalent binomial appears.

Figures

Figures reproduced from arXiv: 2604.08311 by Shenxing Zhang, Xianhong Xie, Yi Ouyang.

Figure 1
Figure 1. Figure 1: Division Relations of t, k, u, r Theorem 9. Assume that ℓ(n) > m, #SF = 2m, wt2(d1) = wt2(d2) = 2, and gcd(d1, d2, N) > 1. Then d2 − d1 ≡ 0 mod (2m − 1). Proof. By Theorem 7, SF = F2m. If s := gcd(d1, d2, N) > 1, then s | (2m + 1) or s | (2m − 1) by Lemma 2. Assume that s | (2m + 1). Then WFa (0) = −2 m for any [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
read the original abstract

Assume $n=2m\geq 2$ and let $F(x)=x^{d_1}+x^{d_2}$ be a binomial vectorial function over $\F_{2^n}$ possessing the maximal number (i.e. $2^n-2^m$) of bent components. Suppose the $2$-adic Hamming weights $\wt_2(d_1)$ and $\wt_2(d_2)$ are both at most $2$, we prove that $F(x)$ is affine equivalent to either $x^{2^m+1}$ or $x^{2^i}(x+x^{2^m})$, provided that \[ \ell(n):=\min_{\gamma:~\F_2(\gamma)=\F_{2^n}} \dim_{\F_2}\F_2[\sigma]\gamma >m, \] where $\sigma$ is the Frobenius $(x\mapsto x^2)$ on $\F_{2^n}$, and $\gcd(d_1,d_2,2^m-1)>1$. Under this condition, we also establish two bounds on the nonlinearity and the differential uniformity of $F$ by means of the cardinality of its image set.

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

1 major / 1 minor

Summary. The paper claims that for a binomial vectorial function F(x)=x^{d_1}+x^{d_2} over F_{2^n} with n=2m≥2 possessing the maximal number (2^n-2^m) of bent components, and with wt_2(d_i)≤2 for both exponents, F is affine equivalent to either x^{2^m+1} or x^{2^i}(x+x^{2^m}) provided ℓ(n)>m (where ℓ(n) is the minimal F_2-dimension of the Frobenius orbit span of a primitive element) and gcd(d_1,d_2,2^m-1)>1. It further derives bounds on nonlinearity and differential uniformity of F via the cardinality of the image set.

Significance. If the classification holds, the result would identify all quadratic binomials with maximal bent components under the stated arithmetic conditions, advancing the theory of vectorial bent functions and their cryptographic applications. The technique of bounding nonlinearity and differential uniformity from image-set cardinality is a concrete methodological contribution that could apply more broadly.

major comments (1)
  1. [Main classification result] The main classification result (as stated in the abstract and presumably proved in the body): the condition ℓ(n)>m is invoked to rule out other cases and guarantee the stated affine equivalences, yet no argument is given that the inequality is necessary, nor is any computation or characterization supplied for the even values of n=2m where ℓ(n)>m holds. This renders the scope of the classification an uncharacterized subset of parameters.
minor comments (1)
  1. [Abstract] The definition of ℓ(n) appears for the first time in the abstract; a brief reminder or forward reference to its precise definition in the preliminaries would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and constructive feedback on our manuscript. We address the major comment below and will make revisions to improve clarity on the scope of our results.

read point-by-point responses

  1. Referee: The main classification result (as stated in the abstract and presumably proved in the body): the condition ℓ(n)>m is invoked to rule out other cases and guarantee the stated affine equivalences, yet no argument is given that the inequality is necessary, nor is any computation or characterization supplied for the even values of n=2m where ℓ(n)>m holds. This renders the scope of the classification an uncharacterized subset of parameters.

    Authors: We agree that the manuscript does not establish necessity of the condition ℓ(n)>m, nor does it provide a full characterization of the even n=2m satisfying it. Our proof technique requires ℓ(n)>m to exclude additional binomial forms that could otherwise arise, so the classification is stated under this hypothesis (along with the weight and gcd conditions). We do not claim the inequality is necessary in general. In the revised manuscript, we will add an explicit remark in the introduction and a short subsection (or appendix) computing ℓ(n) for small even n=2m (such as n=4,6,8,10,12,14) to illustrate the parameters covered, along with a brief discussion of when the inequality holds based on the 2-adic properties of n. This will delineate the scope without overclaiming. revision: partial

Circularity Check

0 steps flagged

No circularity; classification theorem is conditional on an independently defined field-theoretic quantity

full rationale

The paper states a classification result for quadratic binomial vectorial functions F(x) = x^{d1} + x^{d2} that possess the maximal number of bent components. The result holds under the explicit hypothesis that ℓ(n) > m (where ℓ(n) is the minimum F2-dimension of the span generated by Frobenius orbits of primitive elements) together with a gcd condition on the exponents. This ℓ(n) is defined directly from the finite-field structure and does not depend on the function F or on the claimed classification; it functions as a hypothesis that restricts the parameter range rather than as a quantity fitted from or defined by the theorem itself. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The proof therefore remains self-contained against external finite-field benchmarks and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of finite fields of characteristic 2, the definition of bent functions, and the Frobenius automorphism; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard algebraic properties of the finite field F_{2^n} and the Frobenius map x |-> x^2
    Used to define the functions, 2-adic weights, and the quantity ℓ(n)
  • domain assumption Definition and basic properties of bent components for vectorial functions
    Invoked to state the maximal number 2^n - 2^m

pith-pipeline@v0.9.0 · 5511 in / 1468 out tokens · 48279 ms · 2026-05-10T17:16:24.859708+00:00 · methodology

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Reference graph

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