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arxiv: 2601.02919 · v2 · submitted 2026-01-06 · 🧮 math.NT

Inverses of six classes of permutation polynomials of the form x+γoperatorname{Tr}_q^(q²)(h(x)) over finite fields of even characteristic

Rajesh P. Singh , Dinesh Kumar , Jitendra Prakash This is my paper

Pith reviewed 2026-05-16 17:20 UTC · model grok-4.3

classification 🧮 math.NT
keywords permutation polynomialsfinite fieldscompositional inversetrace functioneven characteristicquadratic extension
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The pith

Compositional inverses are explicitly determined for six classes of permutation polynomials of the form x + γ Tr(h(x)) over finite fields of even characteristic.

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

The paper determines the compositional inverses for six families of permutation polynomials over finite fields F_{q²} where q is a power of two. Each family has the form x plus a scalar times the trace from the quadratic extension of a function h of x. Explicit inverses confirm the maps are bijective and allow direct computation of preimages under these polynomials. The derivations use algebraic properties of the trace map in characteristic two to produce closed-form expressions for the inverses of each class.

Core claim

For each of the six classes, the compositional inverse is given by an explicit polynomial expression, also built from the trace function together with suitably chosen auxiliary functions or coefficients that ensure the composition with the original polynomial recovers the identity on the entire field.

What carries the argument

The absolute trace Tr_q^{q²} from the quadratic extension F_{q²} to the subfield F_q, which linearizes the correction term and permits solving for the inverse by rearranging the defining equation.

Load-bearing premise

The six classes are permutation polynomials on the full field and the algebraic steps used to obtain the inverses apply without exception to every element.

What would settle it

For any of the six classes and a small even q, an explicit element x in F_{q²} such that applying the claimed inverse after the original polynomial fails to return x.

read the original abstract

Recently, Jiang et al. \cite{JIANG2025102522} obtained several classes of Permutation Polynomial of the form $x+\gamma\operatorname{Tr}_q^{q^2}(h(x))$ over finite fields $\mathbb{F}_{q^2},q=2^n$. In this paper, we find the compositional inverse of six classes of permutation polynomials of this form.

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 derives explicit compositional inverses for six classes of permutation polynomials over finite fields F_{q^2} (q=2^n) of the form x + γ Tr_q^{q^2}(h(x)), building directly on the six classes constructed by Jiang et al. The inverses are obtained via algebraic manipulation of the defining equation y = x + γ Tr(h(x)), using linearity of the trace and standard field identities to solve for x in terms of y.

Significance. Explicit inverses for these trace-based permutation polynomials are useful in cryptographic constructions (e.g., S-boxes) and coding theory, where both the forward map and its inverse must be efficiently evaluable. The approach follows the standard pattern of solving the trace equation and yields closed-form expressions without introducing new parameters, strengthening the practical value of the Jiang et al. families.

minor comments (3)
  1. [§2] §2: The six classes should be restated with their exact h(x) polynomials (including any coefficient restrictions) so that the inverse formulas can be checked against the original definitions without consulting the cited paper.
  2. [§3–§8] §3–§8: Each inverse derivation ends with a composition check; adding a short remark that the verification holds identically (rather than only for generic elements) would make the proofs self-contained.
  3. [References] References: Ensure the Jiang et al. citation includes the full bibliographic details (journal, volume, year) rather than only the arXiv number.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. We are pleased that the utility of the explicit inverses for cryptographic and coding applications is recognized.

Circularity Check

0 steps flagged

No significant circularity; derivation is direct algebraic inversion

full rationale

The paper takes the six permutation polynomial classes as given from the external citation Jiang et al. (distinct authors) and derives their compositional inverses via explicit algebraic manipulation of the equation y = x + γ Tr(h(x)) using trace linearity and finite-field identities. No parameter is fitted to data, no result is renamed as a prediction, and no load-bearing step reduces to a self-citation or self-definition. The derivation chain is therefore self-contained against the stated field equations and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard properties of finite fields of characteristic two and the trace function, plus the assumption that the six classes are permutations as previously established.

axioms (2)
  • standard math The trace function Tr_q^{q²} satisfies the standard linearity and surjectivity properties over F_{q²}/F_q in characteristic 2.
    Invoked implicitly when manipulating the polynomial form.
  • domain assumption The six classes are permutation polynomials as constructed in Jiang et al.
    The paper builds directly on those constructions without re-proving the permutation property.

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Works this paper leans on

23 extracted references · 23 canonical work pages

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