Some permutation pentanomials over finite fields of even characteristic

Farhana Kousar; Maosheng Xiong

REVIEW 3 major objections 4 minor 1 cited by

Some permutation pentanomials over finite fields of even characteristic

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read One gcd condition decides whether each of three five-term polynomial families permutes the field.

desk verdict A genuinely useful structural observation about permutation pentanomials, but the central theorem statements as printed are broken and need a thorough revision before the paper can be trusted. read the letter →

arxiv 2412.14641 v1 pith:EU6G2WLA submitted 2024-12-19 math.CO cs.DMmath.NT

Farhana Kousar , Maosheng Xiong This is my paper · ORCID

classification math.COcs.DMmath.NT MSC 11T06

keywords permutationpolynomialspentanomialsfinitefieldsofcharacteristic2linearequivalencerationalfunctionsoverramificationgcdconditionspowermaps

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that three general families of five-term polynomials over $\mathbb{F}_{q^2}$, with $q=2^m$, are permutation polynomials exactly when a small set of gcd conditions holds. Each pentanomial is written as $x^{Q_1+Q_2+1}H(x^{q-1})$ with $Q_1=2^i$, $Q_2=2^j$, and the proof shows that the accompanying rational function $g=N/H$ is, up to fractional linear changes, a power map. When $m$ is odd, each family is linearly equivalent to the bivariate map $(u^{Q_1+Q_2+1}, v^{Q_1+Q_2+1})$; when $m$ is even, it is linearly equivalent to $x^{Q_1+Q_2+1+r(q-1)}$ for an explicit correction $r$ depending on the parities and sizes of $Q_1,Q_2$. This gives a uniform reason why 14 of the 17 pentanomial families found by a recent computer search are permutations, and it extends those 14 families to arbitrary parameter values satisfying the conditions.

What carries the argument

The load-bearing object is the rational function $g_\bullet(x)=N_\bullet(x)/H_\bullet(x)$ attached to $f_\bullet(x)=x^{Q_1+Q_2+1}H_\bullet(x^{q-1})$: the permutation criterion reduces $f_\bullet$ to the requirements $\gcd(Q_1+Q_2+1,q-1)=1$, $H_\bullet$ has no roots on the unit circle $\mu_{q+1}$, and $g_\bullet$ permutes $\mu_{q+1}$. The proof shows that $g_\bullet$ obeys two identities involving $Q(x)=x^2+x+1$: $N_\bullet'(x)H_\bullet(x)+N_\bullet(x)H_\bullet'(x)=Q(x)^{Q_1+Q_2}$ and $Q(g_\bullet(x))H_\bullet(x)^2=Q(x)^{Q_1+Q_2+1}$. These identities force the only ramification points of $g_\bullet$ to be the two roots of $Q(x)$, and its only branch points to be the images of those roots, so a classification of rational functions with two totally ramified points gives $g_\bullet=\eta^{-1}\circ x^{Q_1+Q_2+1}\circ\sigma$ for odd $m$, and $g_\bullet=\rho^{-1}\circ x^{Q_1+Q_2+1-2r_\bullet}\circ\sigma$ for even $m$, with $\eta,\sigma,\rho$ fractional linear maps of the appropriate type on $\mu_{q+1}$. Expanding these compositions yields the claimed linear equivalence of $f_\bullet$ to a power map.

What would settle it

Expand equations (23) and (24) symbolically for several values of $i,j$: any failure of either identity for the stated conditions would break Lemma 10 and hence Theorems 2–4. Alternatively, evaluate $f_A,f_B,f_C$ on all elements of $\mathbb{F}_{q^2}$ for small $q=2^m$ and small $i,j$; a single instance satisfying the gcd conditions that is not a bijection would refute the 'if' direction, and a permutation outside the conditions would refute the 'only if' direction.

Watch

Extended reading notes

Core claim

The central discovery is that the permutation behaviour of the three pentanomial classes is controlled by the single integer $Q_1+Q_2+1$ together with a correction term $r_\bullet$ determined by the parities of $i,j$ and the relative sizes of $Q_1,Q_2$. For $q=2^m$: if $m$ is odd, $f_\bullet$ permutes $\mathbb{F}_{q^2}$ if and only if $\gcd(Q_1+Q_2+1,q-1)=1$ plus a parity condition on $i,j$ that depends on the family, and then $f_\bullet$ is linearly equivalent to coordinatewise exponentiation on $\mathbb{F}_q^2$; if $m$ is even, $f_\bullet$ permutes if and only if $\gcd(Q_1+Q_2+1,q-1)=\gcd(Q_1+Q_2+1-2r_\bullet,q+1)=1$, and then $f_\bullet$ is linearly equivalent to the single power map $x^{Q_1+Q_2+1+r_\bullet(q-1)}$. Because linear equivalence preserves the permutation property, these equivalences yield necessary and sufficient conditions, not merely sufficient ones, and the three classes include the 14 starred families from the earlier search as special cases.

Load-bearing premise

The whole reduction to power maps rests on two polynomial identities (equations 23 and 24) that are asserted with the phrase 'here it is easy to check' but not derived; if either identity failed for some pair $(i,j)$, the claimed linear equivalence and the theorems would not follow.

Editorial extensions

If this is right

For each of the three families, testing whether a pentanomial is a permutation is reduced to one or two gcd computations, and the conditions are both necessary and sufficient.
Fourteen of the seventeen pentanomial families from the earlier search are recovered as special cases, so their individual case-by-case proofs can be replaced by the uniform argument.
Every polynomial in these families is linearly equivalent to a monomial map, so any function-theoretic property invariant under linear equivalence is shared with the corresponding power map.
The gcd conditions are satisfied for infinitely many exponent pairs $(i,j)$, so the three classes go well beyond the original finite search range $4\le t<100$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the two asserted identities hold for all $i,j$, the same two-ramification-point strategy may also accommodate the three remaining families from the earlier search, should they fit a similar shape after a suitable correction term.
The argument is specific to characteristic two because it uses $Q(x)=x^2+x+1$ and the structure of $\mu_{q+1}$; an odd-characteristic analogue would need a different quadratic and a different normalizing family of fractional linear maps.
A direct computational test over small $q$ and small $i,j$ could check the 'only if' direction exhaustively, which the earlier paper did not claim.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core observation is good: 14 of the 17 permutation pentanomial families from Zhang et al. are disguised power maps, and the paper turns them into three infinite classes with explicit permutation conditions. That is a real reorganization of a small but active subfield, and it gives people a recipe for spotting linear equivalence to power maps. The paper is also honest that the technique comes from Ding and Zieve [9]; it does not oversell its own originality.

What is actually new: Theorems 2–4, which state if-and-only-if permutation conditions for the three classes f_A, f_B, f_C and identify their power-map equivalents. Theorem 1, the worked example for family 17, is clean and checkable. I spot-checked the factorization in Theorem 1 and the permutation behavior in one family; the arithmetic works.

The soft spots are serious, but they look correctable rather than fatal. The r_A table in (10) is internally inconsistent: for i odd and j even it sets r_A = 0 in the first line and also sets r_A to a positive value in the later lines, with one of those later cases always applying since Q1 and Q2 are powers of 2. The r_C table in (14) omits the case i and j both even, which is exactly the case for the f_C families in Table 1. Table 1 also reproduces conditions for families 6 and 14 that conflict with the necessary gcd(t, q−1) = 1 condition. Because Lemma 8, Lemma 9, and the final theorems all depend on these r values, the even-m part of the main theorem is not well-defined as printed.

On top of that, equations (23) and (24) are load-bearing: the entire ramification argument in Lemma 10 rests on them. They are asserted with “Here it is easy to check” and no derivation. I have not found a counterexample, and the spot-checks I did are consistent, but these identities need to be verified independently, not left to the reader.

If the tables are fixed and (23)–(24) check out, the paper is a useful contribution: the argument via ramification and linear equivalence is coherent, and the credit to [9] is appropriate. As it stands, the printed theorem statements are not reliable.

This paper deserves a serious referee, but the referee should demand a major revision. The authors need to correct the r_A and r_C tables, reconcile Table 1 with their own conditions, and either prove or explicitly verify (23)–(24). Once those are done, I would be comfortable citing it.

Referee Report

3 major / 4 minor

Summary. The paper studies permutation pentanomials over F_{q^2}, q=2^m, of the form f(x)=x^t H(x^{q-1}). Its main results are Theorems 2–4, which give necessary and sufficient conditions for three general families f_A, f_B, f_C to permute F_{q^2}, and which explain 14 of the 17 families previously found by Zhang et al. The proof strategy is to write f_•(x)=x^{Q_1+Q_2+1}H_•(x), introduce the rational function g_•=N_•/H_• on the unit circle, use two asserted polynomial identities (23) and (24) to control ramification, and then deduce that g_• is a composition of degree-one rational maps with a power map. Lemma 11 then lifts this to a linear equivalence of f_• to either a bivariate power map (m odd) or a univariate power map (m even). Theorem 1 gives an explicit self-contained proof for one of the 14 families using linear permutations.

Significance. If the main results are correct, the paper provides a genuinely simpler and more conceptual explanation for a substantial part of the existing classification of permutation pentanomials, and it extends those families to three infinite classes with clean permutation criteria. The reduction to linear equivalence with power maps is a useful observation, and the paper is honest in crediting the technique to [9]. However, the current version is not reliable as written: the r_A and r_C tables in the central theorems are defective, making the even-m classification undefined for some parameter ranges that include the paper's own examples, and the load-bearing identities (23) and (24) are asserted without proof. Because these issues are local and likely fixable, the paper warrants a major revision rather than rejection.

major comments (3)

[§3, Eq. (10), Theorem 2] The table defining r_A is not well-defined. For i≡1, j≡0 (mod 2), the first line assigns r_A=0, while the last two lines assign r_A=Q_1+1 if Q_1<Q_2 and r_A=Q_2 if Q_1≥Q_2. Since Q_1 and Q_2 are powers of 2, one of those inequalities always holds, so r_A is assigned two different values. Moreover, there is no row for i≡j≡1 (mod 2), even though this is precisely the parity pattern of all f_A families appearing in Table 1 (families 2, 9, 11). Lemma 8's claim that gcd(N_A,H_A)=Q(x)^{r_A}, and Lemma 9's use of r_A to determine roots in µ_{q+1}, therefore do not make sense for these parameters. The proof of Case A in Lemma 8 says 'the other cases are similar' but does not resolve this contradiction. The table must be corrected and every parity case checked.
[§3, Eq. (14), Theorem 4] The table for r_C has no entry for i≡j≡0 (mod 2). This is not a cosmetic omission: the three f_C families in Table 1 (families 6, 14, 16) all have both i and j even. For those families, the condition gcd(Q_1+Q_2+1−2r_C, q+1)=1 in Theorem 4(ii) is undefined. The proof of Case C in Lemma 8 itself derives HC(ω)≠0 when i≡j≡0, so r_C=0 in that case, which suggests the intended entry is 0; but the published table omits it, and Lemma 8 does not state the resulting r_C value. This must be fixed before the theorem can be evaluated.
[§3, Eqs. (23) and (24)] The two identities N'_•(x)H_•(x)+N_•(x)H'_•(x)=Q(x)^{Q_1+Q_2} and Q(g_•(x))H_•(x)^2=Q(x)^{Q_1+Q_2+1} are the foundation of the entire ramification argument. They imply Eqs. (31) and (32), which are what make Lemma 10 and hence Lemma 11 possible. They are introduced with the phrase 'Here it is easy to check' and no derivation. These identities are not self-evident from (16)–(21), and a failure for some (i,j) would invalidate the classification of g_• and the linear equivalence of f_•. The authors should provide a full verification, preferably as a separate lemma, for all three families; this is a load-bearing step, not a routine detail that can be left to the reader.

minor comments (4)

[§3, proof of Lemma 8, Case A] The displayed expansion of H_A(x+ω) omits the constant term H_A(ω). This omission is likely responsible for the inconsistency in the r_A table, since for i≡j≡1 the constant term is nonzero and gives r_A=0, while the displayed expansion without the constant term suggests a positive valuation. The expansion should be written in full, including the constant term.
[§4, Lemma 11, Case 1] The sentence 'Case 1: t is odd' should read 'Case 1: m is odd', since t=Q_1+Q_2+1 is always odd for powers of 2. This is a typo but could confuse the reader.
[§4, Proofs of Theorems 2–4] In the even-m case, the text writes P_2(x)=x^{Q_1+Q_2+1+m_•(q−1)}; here m_• is undefined and should be r_• from the relevant theorem. The proof also jumps from the permutation condition of this power map to the two displayed gcds without showing the standard equivalence gcd(E,q^2−1)=1 ⇔ gcd(Q_1+Q_2+1,q−1)=gcd(Q_1+Q_2+1−2r_•,q+1)=1. Adding this one-line derivation would make the proof complete.
[§3, Eq. (22)] The identity N_•(x)=x^{Q_1+Q_2+1}H_•(x^{-1}) is stated as 'easy to check'; it is in fact immediate from (16)–(21), but a one-line verification for each family would remove any doubt and would also make the subsequent use of this identity clearer.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the linear-equivalence proof is self-contained and the [30] families function only as external benchmarks.

full rationale

The derivation chain is not circular. The paper's criterion for f_•(x)=x^{Q1+Q2+1}H_•(x^{q-1}) to permute F_{q^2} is a standard external theorem (gcd(t,q-1)=1, H_• has no roots in µ_{q+1}, and g_• permutes µ_{q+1}), and the paper then proves that g_• is degree-one equivalent to x^{Q1+Q2+1} or x^{Q1+Q2+1-2r_•} using the polynomial identities (22)-(24), Lemma 8 on gcd(N_•,H_•)=Q(x)^{r_•}, and the ramification argument in Lemma 10. The target power-map permutations are not assumed: their permutation criteria (gcd(n,q-1)=1 and gcd(n,q^2-1)=1, split via q-1 and q+1) are standard and checked at the end of the proof. The 14 families taken from [30] are external benchmarks that the general classes f_A, f_B, f_C reproduce; no parameter is fitted to those families and no 'prediction' is declared from a fitted subset. The only self-citation, [12], is used as an acknowledgment of an idea and the accompanying Lemma 11 is proved in full, so it is not load-bearing. To be explicit, two manuscript passages are flagged as completeness risks rather than circularity: the identities (23) and (24) are asserted with 'Here it is easy to check' rather than derived, and the tables for r_A/r_C appear to have missing or overlapping cases (e.g., i≡1,j≡0 in (10)); these affect correctness checking but do not make the argument circular.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted constants or invented entities. The three families f_A, f_B, f_C are deliberately shaped so that the rational function g_• satisfies equations (23) and (24), so the family design is reverse-engineered rather than derived from first principles; this is a construction assumption, not a hidden fit.

assumptions (4)

standard math Zieve's criterion for f(x)=x^t H(x^{q-1}) over F_{q^2}: permutation iff gcd(t, q-1)=1, H has no roots in µ_{q+1}, and the induced rational function permutes µ_{q+1}.
Invoked in Section 4 (proofs of Theorems 2-4) and attributed to [32]; it is the standard entry point for this polynomial shape.
standard math [9, Lemma 5.1]: a rational function over F_q with exactly two totally ramified points is Möbius-conjugate over the algebraic closure to a power map x^d.
Lemma 10 is claimed as 'a direct consequence of [9, Lemma 5.1] (with a very small variation)' and the proof given in the paper only covers the tame case, which applies because the relevant degree is odd.
ad hoc to paper Equations (23) and (24): N'_•H_•+N_•H'_•=Q(x)^{Q1+Q2} and Q(g_•)H_•^2=Q(x)^{Q1+Q2+1} hold for the three families.
Stated in Section 3 with 'Here it is easy to check' and no derivation; these identities drive the ramification computation in Lemma 10.
standard math Lemmas 6 and 7 from [33] classify degree-one rational functions that permute µ_{q+1} or map µ_{q+1} onto P^1(F_q).
Used in the proofs of Lemmas 10 and 11 to justify the Möbius maps ρ, σ, η.

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Pith. "Pith review of Some permutation pentanomials over finite fields of even characteristic." pith.science (2026). https://pith.science/paper/EU6G2WLA

@misc{pith2026241214641,
  author       = {Pith},
  title        = {Pith review of: Some permutation pentanomials over finite fields of even characteristic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EU6G2WLA}},
  note         = {Machine review of arXiv:2412.14641}
}

abstract

In a recent paper Zhang et al. constructed 17 families of permutation pentanomials of the form $x^t+x^{r_1(q-1)+t}+x^{r_2(q-1)+t}+x^{r_3(q-1)+t}+x^{r_4(q-1)+t}$ over $\mathbb{F}_{q^2}$ where $q=2^m$. In this paper for 14 of these 17 families we provide a simple explanation as to why they are permutations. We also extend these 14 families into three general classes of permutation pentanomials over $\mathbb{F}_{q^2}$.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Some classes of permutation pentanomials
math.NT 2025-01 conditional novelty 8.0 of 10

For every prime p other than 3, this paper builds two large families of five-term permutation polynomials over F_{q^2}, unifying 76 earlier special cases and solving an open problem.

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Works this paper leans on

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [9]

, Determination of a class of permutation quadrinomials , Proc. Lond. Math. Soc. 127 (2023), 221–260. 21

work page 2023

[2] [1]

M. Ayad, K. Belghaba, and O. Kihel, On permutation binomials over ﬁnite ﬁelds , Bull. Aust. Math. Soc. 89 (2014), 112–124

work page 2014

[3] [2]

Bartoli and M

D. Bartoli and M. Timpanella, A family of permutation trinomials over Fq2, Finite Fields Appl. 70 (2021), 101781

work page 2021

[4] [3]

Budaghyan, C

L. Budaghyan, C. Carlet, and A. Pott, New classes of almost bent and almost perfect nonlinear polynomials, IEEE Trans. Inf. Theory 52 (2006), 1141–1152

work page 2006

[5] [4]

Carlet, P

C. Carlet, P. Charpin, and V. Zinoviev, Codes, bent functions and permutations suitable for des-like cryptosystems , Des. Codes Cryptogr. 15 (1998), 125–156

work page 1998

[6] [5]

Deng and D

H. Deng and D. Zheng, More classes of permutation trinomials with niho exponents , Cryptogr. Commun. 11 (2019), 227–236

work page 2019

[7] [6]

Ding and J

C. Ding and J. Yuan, A family of skew hadamard diﬀerence sets , J. Comb. Theory, Ser. A 113 (2006), 1526–1535

work page 2006

[8] [7]

Ding and M

Z. Ding and M. E. Zieve, On a class of permutation quadrinomials , J. Algebra Appl. 0 (0), 2550075

work page