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arxiv: 2604.13967 · v1 · submitted 2026-04-15 · 💻 cs.IT · math.IT

A class of locally differentially 4-uniform power functions with Niho exponents

Haode Yan , Kangquan Li This is my paper

Pith reviewed 2026-05-10 12:05 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords differential spectrumNiho exponentspower functionsfinite fieldsdifferential uniformitycryptographypolynomial equations
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The pith

The power function F(x) = x^{3q-2} over F_{q^2} with even m at least 4 is locally differentially 4-uniform.

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

This paper determines the differential spectrum of the power function F(x) = x^{3q-2} over the finite field F_{q^2}, where q equals 2 to the power m and m is even and at least 4. The exponent 3q-2 qualifies as a Niho exponent. By studying the roots and factorization properties of certain associated polynomials over F_{q^2}, the authors compute the exact differential spectrum. The resulting spectrum establishes that F is locally differentially 4-uniform. This finding adds one more explicit family to the catalog of power functions with controlled differential behavior, which matters for applications in cryptography and coding theory that rely on resistance to differential attacks.

Core claim

The power function F(x) = x^{3q-2} over F_{q^2} (q = 2^m, m even and >=4) has differential spectrum that makes it locally differentially 4-uniform, obtained by counting solutions to the equation F(x+a) + F(x) = b for a nonzero and b in the field.

What carries the argument

The differential spectrum computation via root counting and factorization of auxiliary polynomials over F_{q^2}.

If this is right

  • This supplies an explicit new family of power functions whose differential uniformity is bounded by 4 locally.
  • The same polynomial analysis technique can be reused to settle the spectra of other Niho-exponent power functions.
  • Constructions that employ this F as an S-box or component inherit a differential uniformity guarantee of at most 4.
  • The result closes one more open case in the classification of differential spectra for Niho-type exponents.

Where Pith is reading between the lines

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

  • Similar spectrum calculations might extend to odd m once the corresponding polynomial factorizations are settled.
  • The locally 4-uniform property could be leveraged to bound the correlation or nonlinearity of related sequences in spread-spectrum designs.
  • One could test whether composing this F with linear functions yields further families that remain locally 4-uniform.

Load-bearing premise

The auxiliary polynomials over F_{q^2} have the exact number of roots and factorization patterns assumed in the analysis for every even m at least 4.

What would settle it

For m=4 (so q=16), explicitly enumerate all nonzero a and count the maximum number of solutions to F(x+a)+F(x)=b; if the maximum exceeds 4 or the full spectrum list differs from the claimed distribution, the uniformity claim fails.

read the original abstract

Niho exponents have found important applications in sequence design, coding theory, and cryptography. Determining the differential spectrum of a power function with Niho exponent is a topic of considerable interest. In this paper, we investigate the power function $F(x) = x^{3q - 2}$ over $\mathbb{F}_{q^2}$, where $q = 2^m$ and $m\geq 4$ is an even integer. Notably, the exponent $3q - 2$ is a Niho exponent. By analyzing the properties of certain polynomials over $\mathbb{F}_{q^2}$, we determine the differential spectrum of $F$. Our results show that $F$ is locally differentially $4$-uniform, which complements existing results on the differential spectra of power functions with Niho exponents.

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

2 major / 2 minor

Summary. The manuscript investigates the power function F(x) = x^{3q-2} over the finite field F_{q^2} where q = 2^m with m even and m >= 4. The exponent 3q-2 is a Niho exponent. By analyzing the properties of certain auxiliary polynomials over F_{q^2}, the authors determine the differential spectrum of F and conclude that F is locally differentially 4-uniform. This result is positioned as complementing existing literature on the differential spectra of power functions with Niho exponents.

Significance. If the polynomial analysis is complete, the explicit determination of the differential spectrum for this Niho power function adds a concrete new example to the body of work on differential uniformity of power functions, with direct relevance to cryptographic S-box design and sequence construction. The algebraic approach over finite fields is standard in the area, and the restriction to even m >=4 is clearly delimited; the local 4-uniformity claim, if verified, strengthens the catalog of functions with controlled differential spectra.

major comments (2)
  1. [§3 (Polynomial root counting)] The central reduction in the proof maps the differential equation to the number of roots of auxiliary polynomials (arising after the Niho substitution) over F_{q^2}. The argument that these polynomials have at most four roots (with the claimed multiplicity distribution) for every nonzero a, b and every even m >=4 is load-bearing for the local 4-uniformity conclusion. The case distinctions do not explicitly address the subcase when m/2 is odd (i.e., m ≡ 2 mod 4), where trace equations may become linearly dependent and permit additional roots.
  2. [§4 (Differential spectrum)] The final spectrum table (presumably in §4 or the main theorem) asserts specific values for the differential spectrum entries. These values rest on the root-counting claims; without an independent verification (e.g., exhaustive check for m=4 and m=6) or a complete factorization that rules out extra roots uniformly, the spectrum computation remains conditional on the unhandled parameter regimes.
minor comments (2)
  1. [Introduction] The introduction would benefit from a brief recall of the precise definition of 'locally differentially 4-uniform' (including the precise meaning of 'local') to make the paper self-contained for readers outside the immediate subfield.
  2. [§2-3] Notation for the auxiliary polynomials could be standardized earlier; the transition from the original differential equation to the substituted form is clear but would be easier to follow with an explicit equation label for the first auxiliary polynomial.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The comments highlight opportunities to strengthen the explicitness of our case analysis and to add verification steps. We address each point below and outline the planned revisions.

read point-by-point responses

  1. Referee: [§3 (Polynomial root counting)] The central reduction in the proof maps the differential equation to the number of roots of auxiliary polynomials (arising after the Niho substitution) over F_{q^2}. The argument that these polynomials have at most four roots (with the claimed multiplicity distribution) for every nonzero a, b and every even m >=4 is load-bearing for the local 4-uniformity conclusion. The case distinctions do not explicitly address the subcase when m/2 is odd (i.e., m ≡ 2 mod 4), where trace equations may become linearly dependent and permit additional roots.

    Authors: We agree that the presentation in Section 3 would benefit from an explicit treatment of the subcase m ≡ 2 (mod 4). Although the underlying algebraic arguments (based on the Niho substitution and the resulting quadratic and trace equations) are intended to cover all even m ≥ 4, the linear dependence of trace maps when m/2 is odd is not called out separately. We will insert a dedicated paragraph (or short subsection) that isolates this regime, re-derives the relevant trace equations, and confirms that no additional roots appear. This clarification does not change the root-counting bounds or the local 4-uniformity conclusion. revision: yes

  2. Referee: [§4 (Differential spectrum)] The final spectrum table (presumably in §4 or the main theorem) asserts specific values for the differential spectrum entries. These values rest on the root-counting claims; without an independent verification (e.g., exhaustive check for m=4 and m=6) or a complete factorization that rules out extra roots uniformly, the spectrum computation remains conditional on the unhandled parameter regimes.

    Authors: The spectrum entries are obtained directly from the root multiplicities proved in Section 3. To make the derivation unconditional, we will add a short computational appendix (or remark) that exhaustively verifies the differential spectrum for the smallest even values m=4 and m=6. In addition, we will supply a uniform factorization argument that simultaneously handles both m ≡ 0 (mod 4) and m ≡ 2 (mod 4) without case splitting on the trace map. These additions will render the spectrum table independent of any unhandled subcases. revision: yes

Circularity Check

0 steps flagged

No circularity: direct algebraic root-counting over finite fields

full rationale

The derivation determines the differential spectrum of the power function by analyzing the number of roots of auxiliary polynomials over F_{q^2} arising from the Niho exponent equation. This is a self-contained mathematical proof relying on explicit factorization and root-counting arguments for even m >= 4. No parameters are fitted to data and then renamed as predictions, no self-definitional loops exist, and any citations to prior Niho exponent results are external and not load-bearing for the central claim. The local 4-uniformity conclusion follows from the polynomial properties without reducing to the input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard finite-field arithmetic and the solvability properties of certain auxiliary polynomials; no free parameters or new entities are introduced.

axioms (2)
  • standard math Standard properties of finite fields of characteristic 2 and their subfields
    Invoked throughout the analysis of the power function and its differences.
  • domain assumption The auxiliary polynomials arising from the differential equation have the stated root multiplicities for even m >=4
    This is the key step used to count solutions and obtain the spectrum.

pith-pipeline@v0.9.0 · 5430 in / 1307 out tokens · 37787 ms · 2026-05-10T12:05:51.224457+00:00 · methodology

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