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arxiv: 1907.03386 · v1 · pith:X6WDNQQ7new · submitted 2019-07-08 · 💻 cs.IT · math.IT

Further results on some classes of permutation polynomials over finite fields

Xiaogang Liu This is my paper

Pith reviewed 2026-05-25 01:22 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords permutation polynomialsfinite fieldspolynomial constructionspermutation propertyalgebraic modifications
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The pith

By altering coefficients, exponents, or base fields, several new infinite families of polynomials are shown to permute every element of finite fields exactly once.

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

The paper extends prior work on the permutation behavior of polynomial families over finite fields. It verifies the permutation property under specific modifications and constructs additional families. The changes involve tweaking coefficients, shifting exponents, or switching the underlying field size or characteristic. A reader would care because such polynomials serve as building blocks in applications that require every field element to appear exactly once in the output.

Core claim

Certain altered polynomial expressions continue to induce bijections on finite fields when the coefficients, exponents, or the fields themselves are adjusted according to the given patterns.

What carries the argument

The permutation property itself, which requires proving that the polynomial maps distinct inputs to distinct outputs for every element of the finite field.

If this is right

  • Additional explicit families become available for use in coding and cryptographic constructions over finite fields.
  • The alteration technique can be reapplied to other known base families to generate further examples.
  • Permutation behavior is established for polynomials defined over a wider range of field characteristics and sizes.

Where Pith is reading between the lines

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

  • The same modification strategy might generate permutation polynomials over rings or other algebraic structures beyond fields.
  • Automated search over small fields could quickly test candidate alterations before attempting general proofs.
  • These families may intersect with known constructions used in finite geometry or difference sets.

Load-bearing premise

The specific modifications to the polynomial forms must preserve the algebraic conditions that force the mapping to be one-to-one on the chosen finite field.

What would settle it

Explicitly computing the images of two distinct field elements under one of the proposed polynomials and finding they coincide would disprove the claim for that family.

read the original abstract

Let $\mathbb{F}_q$ denote the finite fields with $q$ elements. The permutation behavior of several classes of infinite families of permutation polynomials over finite fields have been studied in recent years. In this paper, we continue with their studies, and get some further results about the permutation properties of the permutation polynomials. Also, some new classes of permutation polynomials are constructed. For these, we alter the coefficients, exponents or the underlying fields, etc.

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 / 0 minor

Summary. The manuscript continues the study of permutation polynomials over finite fields, obtaining further results on the permutation properties of several known infinite families and constructing new classes by modifying coefficients, exponents, or the underlying fields.

Significance. Permutation polynomials over finite fields have applications in coding theory and cryptography. Incremental constructions obtained by altering known families can be useful if the bijectivity is rigorously verified using standard criteria such as the number of roots of f(x) - a = 0.

major comments (1)
  1. Abstract: the central claim consists of new constructions whose permutation property is asserted after altering coefficients/exponents/fields, but no explicit polynomial forms, no statement of the precise conditions on the parameters, and no indication of the verification method (e.g., the sum-of-powers criterion or direct root-counting) are supplied in the visible text; without these the claim cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript and for the constructive comment. We address the concern regarding the abstract below and agree that a revision will improve accessibility without altering the paper's technical content.

read point-by-point responses

  1. Referee: Abstract: the central claim consists of new constructions whose permutation property is asserted after altering coefficients/exponents/fields, but no explicit polynomial forms, no statement of the precise conditions on the parameters, and no indication of the verification method (e.g., the sum-of-powers criterion or direct root-counting) are supplied in the visible text; without these the claim cannot be assessed.

    Authors: The abstract is intended as a concise overview. Explicit polynomial forms (obtained by altering coefficients, exponents, or the underlying field), the precise parameter conditions under which the polynomials permute, and the verification methods (primarily direct root-counting of f(x) - a = 0 together with known criteria such as the sum-of-powers test) are stated and proved in Sections 3 and 4 of the manuscript. Nevertheless, we acknowledge that the abstract could be made more informative. We will revise it to include one or two representative polynomial expressions, the relevant parameter ranges, and a brief mention of the proof technique employed. revision: yes

Circularity Check

0 steps flagged

No significant circularity; constructions are self-contained algebraic verifications

full rationale

The paper constructs new permutation polynomial families over finite fields by altering coefficients, exponents, or the base field of existing families, then verifies the permutation property via standard criteria (bijectivity checks or sum-of-powers tests). No equations reduce to fitted parameters renamed as predictions, no self-definitional loops, and no load-bearing self-citations that substitute for independent proof. The derivation chain relies on direct finite-field algebra external to the paper's own inputs, making the result self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the work presumably relies on standard facts about finite fields and polynomial permutation criteria.

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Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 1 internal anchor

  1. [1]

    Akbary, A., Ghioca, D., Wang, Q., ‘On constructing permu tations of finite fields’, Finite Fields Appl., 17 (2011), 51–67

    work page 2011

  2. [2]

    Carlitz, L., ‘Some theorems on permutation polynomials ’, Bull. Amer. Math. Soc. 68 (1962), 120–122

    work page 1962

  3. [3]

    Sci., v ol

    Charpin, P ., Kyureghyan, G., ‘On a class of permutation p olynomials over F2n,’ in: Sequence and Their Application-SETA 2008, in: Lecture Notes in Comput. Sci., v ol. 5203 (2008), Springer, 368–376

    work page 2008

  4. [4]

    Charpin, P ., Kyureghyan, G., ‘Monomial functions with l inear structure and permutation polynomials’, in: Finite Fields: Theory and Applications, in: Contemp. Math. , vol. 518, 3(16) (2010), Amer. Math. Soc. 99– 111

    work page 2010

  5. [5]

    Ding, C., Qu, L., Wang, Q., Y uan, J., Y uan, P ., ‘Permutati on trinomials over finite fields with even characteristic’, SIAM J. Dis. Math. 29(1) (2015), 79–92

    work page 2015

  6. [6]

    Springer, New Y ork

    Feng, D., Feng, X., Zhang, W ., et al., ‘Loiss: a byte-orie nted stream cipher’, In: IWCC11 Proceedings of the Third International Conference on Coding and Cryptology, ( 2011), 109–125. Springer, New Y ork

    work page 2011

  7. [7]

    57 (2019), 68–91

    Gary, M., John, S.,‘ A characterization of the number of r oots of linearzed and projective polynomials in the field of coefficients’, Finite Fields Appl. 57 (2019), 68–91

    work page 2019

  8. [8]

    32 (2015), 82–119

    Hou, X., ‘Permutation polynomials over finite fields-A su rvey of recent advances’, Finite Fields Appl. 32 (2015), 82–119

    work page 2015

  9. [9]

    Kyureghyan, g., Zieve, M.E., ‘permutation polynomials of the form X + γTr(X k)’, Contemp. Develop. Finite Fields Appl. (2016), 178–194

    work page 2016

  10. [10]

    Mann, H.B., ‘The construction of orthogonal Latin squa res’, Ann. Math. Stat. 13(4) (1942), 418–423

    work page 1942

  11. [11]

    55 (2019), 177–201

    Li, L., Li, C., Li, C., Zeng, X., ‘New classes of complete permutation polynomials’, Finite Fields Appl. 55 (2019), 177–201

    work page 2019

  12. [12]

    51 (2018), 31–61

    Li, L., Wang, S., Li, C., Zeng, X., ‘Permutation polynom ials (xpm − x + δ)s1 + (xpm − x + δ)s2 + x over Fpn ’, Finite Fields Appl. 51 (2018), 31–61

    work page 2018

  13. [13]

    Li, K., Qu, L., Chen, X., Li, C.X., ‘Permutation polynom ials of the form cx + Trql q (xa) and permutation trinomials over even characteristic’, Cryptogr. Commun. 1 0(3) (2018), 531–554

    work page 2018

  14. [14]

    Lidl, R., Niederreiter H.: Finite Fields. Encycl. Math . Appl. Cambridge University Press, Cambridge (1997)

    work page 1997

  15. [15]

    arXiv:1906.09168, (2019)

    Liu, X.: Some results about permutation properties of a kind of binomials over finite fields. arXiv:1906.09168, (2019)

    work page arXiv 1906

  16. [16]

    Constructions and necessities of some permutation polynomials

    Liu, X.: Constructions and necessities of some permuta tion polynomials. arXiv:1906.06453, (2019)

    work page internal anchor Pith review Pith/arXiv arXiv 1906

  17. [17]

    48 (2017),261-270

    Ma, J., Ge, G., ‘A note on permutation polynomials over fi nite fields’, Finite Fields Appl. 48 (2017),261-270

    work page 2017

  18. [18]

    Park, Y .H., Lee, J.B., ‘Permutation polynomials and gr oup permutation polynomials’, Bull. Aust. Math. Soc. 63 (2001), 67–74. 12

    work page 2001

  19. [19]

    Sprin ger, New Y ork

    Schnorr, C.P ., V audenay, S., ‘Black box cryptanalysis of hash networks based on multipermutations’, In: Advances in Cryptology-Eurocrypt’94 (1995), 47–57. Sprin ger, New Y ork

    work page 1995

  20. [20]

    50 (2018), 178–195

    Tu, Z., Zeng, X., Li, C., Helleseth, T., ‘A classe of new p ermutation trinomials’, Finite Fields Appl. 50 (2018), 178–195

    work page 2018

  21. [21]

    57 (2019), 309–343

    Xu, X., Feng, T., Zeng, X., ‘Complete permutation polyn omials with the form (xpm− x+δ)s + axpm + bx over Fpn ’, Finite Fields Appl. 57 (2019), 309–343

    work page 2019

  22. [22]

    56 (2019), 1–16

    Zheng, D., Y uan, M., Y u, L., ‘Two types of permutation po lynomials with special forms’, Finite Fields Appl. 56 (2019), 1–16

    work page 2019

  23. [23]

    Zieve, M.E., ‘On some permutation polynomials over Fq of the form xrh(x(q?1)/d)’, Proc. Amer. Math. Soc. 137(7) (2009), 2209–2216

    work page 2009